Research Article Model Building and Optimization Analysis of MDF Continuous...
Transcript of Research Article Model Building and Optimization Analysis of MDF Continuous...
Research ArticleModel Building and Optimization Analysis of MDF ContinuousHot-Pressing Process by Neural Network
Qingfa Li1 Yaqiu Liu1 and Liangkuan Zhu2
1School of Information and Computer Engineering Northeast Forestry University Harbin 150040 China2College of Electromechanical Engineering Northeast Forestry University Harbin China
Correspondence should be addressed to Yaqiu Liu yaqiuliugmailcom
Received 8 March 2016 Accepted 9 August 2016
Academic Editor Yakov Strelniker
Copyright copy 2016 Qingfa Li et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We propose a one-layer neural network for solving a class of constrained optimization problems which is brought forward fromthe MDF continuous hot-pressing process The objective function of the optimization problem is the sum of a nonsmooth convexfunction and a smooth nonconvex pseudoconvex function and the feasible set consists of two parts one is a closed convex subsetof 119877119899 and the other is defined by a class of smooth convex functions By the theories of smoothing techniques projection penaltyfunction and regularization term the proposed network is modeled by a differential equation which can be implemented easilyWithout any other condition we prove the global existence of the solutions of the proposed neural network with any initial point inthe closed convex subsetWe show that any accumulation point of the solutions of the proposed neural network is not only a feasiblepoint but also an optimal solution of the considered optimization problem though the objective function is not convex Numericalexperiments on the MDF hot-pressing process including the model building and parameter optimization are tested based on thereal data set which indicate the good performance of the proposed neural network in applications
1 Introduction
Medium density fibreboard (MDF) finds many applicationsin wood industries because of its favorable properties suchas surface characteristics dimensional stability and excel-lent machinability [1 2] In the MDF hot-pressing processmany physical processes are involved and the complexity ofthis operation arises from the fact that they are coupledHot-pressing process is one of the key procedures in theproduction of MDF which influences the utilization ratioof energy and resource With the decreased resource oftimber and the increased demand of MDF it is of greatimportant to analyze the experimental data effectively andreasonably find the main factors among the many indexesof MDF and establish the relation models on the propertiesof slab the parameters in the hot-pressing process and themain indexes of MDF These relation models not only canhelp the staff give reasonable prediction and a reliabilityassessment to the hot-pressing process according to the actualprocess parameters but also provide a theoretical basis forthe setting and adjusting of the main factors in hot-pressing
process according to the actual demand of MDF propertiesSo optimization models and methods have been importanttools for the optimization control and scheduling of the hot-pressing of plates
Real-time online solutions of optimization problems aredesired in many engineering and scientific applications Onepossible and very promising approach to solve the real-timeoptimization problems is to apply artificial neural networks[3ndash5] With the resemblance brains neural networks canbe implemented online by hardware and have become animportant technical toll for solving optimization problemsfor example [3 4 6ndash9] Based on the gradient method theHopfield neural networks proposed in [4 5] are the twoclassical recurrent neural networks for linear and nonlin-ear programming whereafter in addition to the gradientmethod many types of neural networks are designed suchas the Lagrangian neural networks [10] the projection-type neural networks [11 12] the dual network [13] andthe stochastic neural network [14] Projection method isan effective and simple method for solving the constraintsHowever it is impossible to solve the general constraints by
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1327235 16 pageshttpdxdoiorg10115520161327235
2 Mathematical Problems in Engineering
projection method Then Lagrangian and penalty methodsare introduced into networks Based on the Lagrangian func-tionmethod Lagrangian networks were proposed for solvingthe optimization problems [8 10] with general constraintsBut the Lagrangian network increases the dimension of thenetworks along with the number of the constraints In recentyears recurrent neural networks based on penalty methodwere widely investigated for solving optimization problemsThe neural networks for smooth optimization problems cannot solve nonsmooth optimization problems because thegradients of the objective and constrained functions arerequired in such neural networks The generalized nonlinearprogramming circuit (G-NPC) in [15] can be considered asa natural extension of nonlinear programming circuit (NPC)for solving nonsmooth convex optimization problems withinequality constraints But the nonempty interior of feasibleregion and large enough penalty parameters are needed forthe network in [15] In order to overcome the nonemptyassumption of the interior of feasible region Bian and Xue[6] proposed a recurrent neural network for nonsmoothconvex optimization based on penalty function methodThe efficiency of the neural networks for solving convexoptimization problems relies on the convexity of functionsA neural network for nonconvex quadratic optimization ispresented in [16] Some neural networks modeled by differ-ential inclusion were also proposed for some nonsmooth andnonconvex optimization problems [6 17] To overcome thedifferential inclusion smoothing techniques are introducedinto the neural networks The main feature of smoothingmethod is to approximate the nonsmooth functions by a classof smooth functions Thus the neural network constructedby the smoothing techniques is modeled by a differentialequation which can be implemented easily in circuits andmathematical software [18]
In this paper we propose a neural network model forsolving the optimization problem brought forward from theMDF continuous hot-pressing automatic control system InSection 2 some notations and necessary preliminary resultsare listed In Section 3 based on the SVM theory withthe existing linear and nonlinear kernel functions we givean optimization problem which includes the problems forbuilding the models of MDF continuous hot-pressing systemand optimizing the MDF performance indexes as specialcases In order to build up the relation models on theproperties of the slab technical parameters in hot-pressingprocess and the performance indexes of MDF when the ker-nel function is positive definite or semipositive definite thecorresponding optimization problem is a constrained convexproblem otherwise it is a nonconvex problemThe optimiza-tion problem for optimizing the performance parametersis a nonconvex constrained optimization problem but itsobjective function is pseudoconvex due to the appropriatechoice of kernel functions In Section 4 we propose a neuralnetwork based on the penalty function method projectionmethod and smoothing techniques The proposed networkis modeled by a nonautomatic differential equation ByLyapunovmethod we prove that the solution of the proposednetwork is global existent and convergent to the feasible setof the considered optimization problems Moreover due to
the pseudoconvexity of the objective function and the con-vexity of the constraint the proposed network also convergesto the optimal solution set of the optimization problem InSection 5 based on the existing data set we use the proposednetwork into the model building and parameter optimizingproblems of hot-pressing system which validates the goodperformance of the obtained results in this paper
Notations 119877+= [0 +infin) Given column vectors 119909 = (119909
1
1199092 119909
119899)119879 and 119910 = (119910
1 119910
2 119910
119899)119879 ⟨119909 119910⟩ = 119909
119879
119910 =
sum119899
119894=1119909119894119910119894is the scalar product of 119909 and 119910 119909
119894denotes the
119894th element of 119909 119909 denotes the Euclidean 2-norm definedby 119909 = (sum
119899
119894=11199092
119894)12 For a closed convex subset Γ sube 119877
119899dist(119909 Γ) is the distance from 119909 to Γ defined by dist(119909 Γ) =min
119910isinΓ119909 minus 119910
2 Preliminaries
In this section we state some definitions and propertiesneeded in this paper We refer the readers to [19ndash21]
21 Support Vector Regression Kernels were regarded as afunction with the formulation of inner product and havebeen a powerful tool in machine learning for their superiorperformance over a wide range of learning problems suchas isolated handwritten digit recognition text categorizationand face detection [21 22]
LetX be a nonempty set and 119870 X timesX rarr 119877 be a real-valued and symmetric function With the kernel matrix K =
(119896(119909119894 119909
119895))119899
119894119895=1 119870 is said to be a positive semidefinite kernel
ifK is positive semidefinite for any 119899 isin N and 1199091 119909
2 119909
119899isin
XWe call119870 an indefinite kernel if there exist1199091 119909
2 119909
119899isin
X and V 119908 isin 119877119899 such that V119879KV lt 0 and 119908119879K119908 gt 0In what follows we list some widely used kernels
(i) Gaussian radial basis function kernel 119870(119909 119910) =
exp(minus119909 minus 119910221205902)
(ii) Polynomial kernel 119870(119909 119910) = (⟨119909 119910⟩ + 119888)119901 119888 ge 0
119901 ge 1
(iii) Sigmoid kernel 119870(119909 119910) = tanh(120581⟨119909 119910⟩ + V) 120581 gt 0V lt 0
22 Smoothing Approximation Smoothing approximation isan effective method for solving nonsmooth optimizationproblems and has been widely used in the past decades Themain feature of smoothing method is to approximate thenonsmooth functions by a class of parameterized smoothfunctions In this paper we adopt the smoothing functiondefined as follows
Definition 1 (see [23]) Let 119892 119877119899
rarr 119877 be a continuousfunction One calls 119877119899 times119877
+rarr 119877 a smoothing function of
119892 if (sdot 120583) is continuously differentiable for any fixed 120583 gt 0and lim
119911rarr119909120583darr0(119911 120583) = 119892(119909) holds for any 119909 isin 119877119899
Mathematical Problems in Engineering 3
Chen and Mangasarian constructed a class of smoothapproximations of the function (119904)
+fl max0 119904 by convo-
lution [20 24] as follows Let 120588 119877 rarr 119877+be a piecewise
continuous density function satisfying
120588 (119905) = 120588 (minus119905)
120581 fl int
infin
minusinfin
|119905| 120588 (119905) 119889119905 lt infin
(1)
Then
120601 (119905 120583) fl int
infin
minusinfin
(119904 minus 120583119905)+120588 (119905) 119889119905 (2)
from 119877 times 119877+to 119877
+is well defined
By different density functions many popular smoothingfunctions of (119904)
+can be derived such as
1206011(119904 120583) = 119904 + 120583 ln (1 + 119890minus119904120583)
1206012(119904 120583) =
1
2(119904 + radic1199042 + 41205832)
1206013(119904 120583) =
max 0 119904 if |119904| gt 120583
(119904 + 120583)2
4120583if |119904| le 120583
1206014(119904 120583) =
119904 +120583
2119890minus119904120583 if 119904 gt 0
120583
2119890minus119904120583 if 119904 le 0
(3)
where 1206011is the neural networks smoothing function 120601
2is
called the CHKS (Chen-Harker-Kanzow-Smale) smoothingfunction120601
3is called the uniform smoothing function and120601
4
is called the Picard smoothing function The four functionsbelong to the class of the Chen-Mangasarian smoothingfunctions
Many nonsmooth functions can be reformulated by usingthe plus function We list some of them as follows
|119904| = (119904)++ (minus119904)
+
max (119904 119905) = 119904 + (119905 minus 119904)+
min (119904 119905) = 119904 minus (119904 minus 119905)+
mid (119904 ℓ 119906) = min (max (ℓ 119904) 119906) for given ℓ 119906
(4)
So we can define a smoothing function for the abovenonsmooth functions by a smoothing function of (119904)
+
From Theorem 961 and Corollary 847(b) in [20] when119892 119877
119899
rarr 119877 is locally Lipschitz continuous at 119909 thesubdifferential associated with a smoothing function
119866(119909) = con V | nabla
119909 (119909
119896
120583119896) 997888rarr V for 119909119896 997888rarr 119909 120583
119896
darr 0
(5)
is nonempty and bounded and 120597119892(119909) sube 119866(119909) where ldquoconrdquo
denotes the convex hull In [20 23] it is shown that manysmoothing functions satisfy the gradient consistency
120597119892 (119909) = 119866(119909) (6)
which is an important property of the smoothing methodsand guarantees the convergence of smoothing methods withadaptive updating schemes of smoothing parameters to astationary point of the original problem
23 Pseudoconvex Function Pseudoconvex function is a classof functions which may be nonsmooth or nonconvex butbrings us the opportunity to find the optimal solutions
Definition 2 (see [25]) Let X be a nonempty convex subsetof 119877119899 A function 120601 is said to be pseudoconvex on X if forany 119909 119910 isin X one has
exist120585 (119909) isin 120597120601 (119909)
st ⟨120585 (119909) 119910 minus 119909⟩ ge 0 997904rArr
120601 (119910) ge 120601 (119909)
(7)
Many nonconvex functions in application are pseudocon-vex such as the Butterworth filter function fraction functionand density functionOf particular interest in this paper is thefact that the Gaussian function
ℎ (119909) = minus exp(minus119899
sum
119894=1
1199092
119894
1205902
119894
) (8)
with 120590119894gt 0 119894 = 1 2 119899 is pseudoconvex on 119877119899
24 Project Operator LetX be a closed convex subset of 119877119899Then the projection operator toX at 119909 is defined by
119875X (119909) = argmin119906isinX
119906 minus 1199092 (9)
and satisfies the following inequalities
⟨V minus 119875X (V) 119875X (V) minus 119906⟩ ge 0 forallV isin 119877119899 119906 isin X
1003817100381710038171003817119875X (119906) minus 119875X (V)1003817100381710038171003817 le 119906 minus V forall119906 V isin 119877119899
(10)
(i) Suppose Ω = 119909 isin 119877119899 119906 le 119909 le V with 119906 V isin 119877119899 cup plusmninfin
then 119875Ω(119909) = (119901
1 119901
2 119901
119899)119879 can be expressed by
119901119894=
119906119894119909119894lt 119906
119894
119909119894119906119894le 119909
119894le V
119894
V119894
V119894lt 119909
119894
(11)
(ii) Suppose Ω = 119909 119860119909 = 119887 with 119860 isin 119877119903times119899 of full row rank
and 119887 isin 119877119903 then 119875Ω(119909) = 119909 minus 119860
119879
(119860119860119879
)minus1
(119860119909 minus 119887)
3 Optimization Problems in MDF ContinuousHot-Pressing Process
In this section we will give the optimization model consid-ered in this paper First by the optimization and supportvector machine theories we show the optimization modelsfor building up the relationships in MDF continuous hot-pressing process Then another optimization model for
4 Mathematical Problems in Engineering
optimizing the parameters in the MDF continuous hot-pressing process is obtained Thus we express these twokinds of problems into a uniform formulation which is theoptimization problem considered in Section 3
31 Optimization Problem for Building up the Models of MDFContinuousHot-Pressing Process DenoteX sub 119877
119901 andY sub 119877
as two sets where 119909119894 = (1199091198941 119909
119894
2 119909
119894
119901) isin X is the attribute
vector on behalf of the hot-pressing plate properties 119910119894 isin Yindicates the values of the qualities of hot-pressing plate Let119878 = (119904
1
1199042
119904119899
) be the training data set of hot-pressingprocess where 119904119894 = (119909119894 119910119894) obeys the unknown distributionand is IID (independent and identically distributed) Basedon the support vector machine theory and the training dataset we would like to find a nonlinear function 119910 = 119891(119909) suchthat it approximates the training data set as much as possible
With the kernel function 119870 and the Huber loss functionfrom the theory in [21] the dual optimization problem of theprimal optimization problem of SVM is given as
min120572120573
1
2
119899
sum
119894119895=1
(120573119894minus 120572
119894) (120573
119895minus 120572
119895)119870 (119909
119894
119909119895
)
minus
119899
sum
119894=1
(120573119894minus 120572
119894) 119910
119894
+120576
2119862
119899
sum
119894=1
(120573119894minus 120572
119894)2
st119899
sum
119894=1
(120573119894minus 120572
119894) = 0
0 le 120573119894 120572
119894le 119862 119894 = 1 2 119899
(12)
Denote 120572lowast119894and 120573lowast
119894as the optimal solutions of (12) Then the
regression function 119891 can be expressed by
119891 (119909) =
119899
sum
119894=1
(120573lowast
119894minus 120572
lowast
119894)119870 (119909
119894
119909) + 119887 (13)
where 119887 can be calculated by one of the following twomethods
119887 = 119910119895
minus
119899
sum
119894=1
(120573lowast
119894minus 120572
lowast
119894) ⟨119909
119894
119909119895
⟩ + 120576 120572lowast
119895isin (0 119862)
119887 = 119910119895
minus
119899
sum
119894=1
(120573lowast
119894minus 120572
lowast
119894) ⟨119909
119894
119909119895
⟩ minus 120576 120572lowast
119895isin (0 119862)
(14)
Denote 119909119894= 120573
119894minus 120572
119894in (12) then (12) can be reformulated
as
min 1
2119909119879
119876119909 minus 119910119879
119909
st 119890119879
119909 = 0 minus 119862119890 le 119909 le 119862119890
(15)
where 119909 = (1199091 119909
2 119909
119899)119879 119910 = (1199101 1199102 119910119899)119879 119890 = (1 1
1)119879
isin 119877119899 and 119876 = K+(1205762119862)119868
119899withK = (119870(119909
119894
119909119895
))119899
119894119895=1
For the optimal solution 119909lowast = (119909lowast1 119909
lowast
2 119909
lowast
119899) of (15) we call
119909119894 a support vector if 119909lowast
119894= 0 The optimal solution of (15)
solved by the originalmethods often has almost 100 supportvectors which increases the complexity of the relationmodelslargely Thus we introduce the problem deduced by (15) thatis
min 1
2119909119879
119876119909 minus 119910119879
119909 + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816
st 119890119879
119909 = 0 minus 119862119890 le 119909 le 119862119890
(16)
with 120574 gt 0 In (43) 120574sum119899
119894=1|119909119894| is often called the regulation
term which is used to control the number of its supportvectors
Define the penalty function
119875 (119909) =
1003817100381710038171003817100381710038171003817
1
119899119890119890119879
119909
1003817100381710038171003817100381710038171003817=radic119899
119899
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119894=1
119909119894
1003816100381610038161003816100381610038161003816100381610038161003816
(17)
and then 119909 119875(119909) le 0 = 119909 119890119879119909 = 0 From [19 Proposition243] 119909 is an optimal solution of (43) if and only if it is anoptimal solution of the following problem
min 1
2119909119879
119876119909 minus 119910119879
119909 + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816 + 120590
10038161003816100381610038161003816119890119879
11990910038161003816100381610038161003816
st minus 119862119890 le 119909 le 119862119890
(18)
where 120590 = 119862119876 + 119910radic119899 + 120574 + 1 Thus we can build up therelation models of hot-pressing process by solving problem(18)
32 Optimization Problem for Optimizing the Parameters inMDF Hot-Pressing Process Based on the relationships builtup in Section 31 we focus on themodulus of rupture (MOR)modulus of elasticity (MOE) and internal bonding strength(IBS) of hot-pressing plate by optimizing the process param-eters and slab attributes Suppose the regression functions ofMORMOS and IBSwith respect to some relative parametersbased on the Gaussian radial basis function kernel are
1198911(119909) fl
119899
sum
119894=1
120588lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) + 119887
1(MOR)
1198912(119909) fl
119899
sum
119894=1
120579lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) + 119887
2(MOS)
1198913(119909) fl
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) + 119887
3(IBS)
(19)
and based on the linear polynomial kernel are
1198911(119909) fl
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1(MOR)
1198912(119909) fl
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2(MOS)
1198913(119909) fl
119899
sum
119894=1
120581lowast
119894⟨119911
119894 119909⟩ + 119887
3(IBS)
(20)
Mathematical Problems in Engineering 5
where 1199111 119911
2 119911
119899are the variables in the data set for
regression and 119909 = (1199091 119909
2 119909
3 119909
4)119879 indicates the indepen-
dent variable with hot-pressing temperature 1199091 hot-pressing
pressure 1199092 hot-pressing time 119909
3 and moisture content 119909
4
The regression functions in (19) satisfy the following twoproperties
(i) 119891119894is continuously differentiable on 119877119899 119894 = 1 2 3
(ii) 119891119894and minus119891
119894are not convex but minus119891
119894is pseudoconvex on
119877119899 119894 = 1 2 3
And the regression functions in (20) satisfy the following twoproperties
(i) 119891119894is continuously differentiable on 119877119899 119894 = 1 2 3
(ii) 119891119894and minus119891
119894are convex on 119877119899 119894 = 1 2 3
From the physical significance of MOR MOS and IBSwe suppose that the larger the numbers of MOR MOS andIBS the better the quality of hot-pressing plate In order toadopt the different demand on the indexes of the hot-pressingplate in different applications we consider the following twocases in this part
First we focus onmaximizing a single performance indexof the hot-pressing plate when the other two performanceindexes are within the certain areas If we want to optimizethe IBS the corresponding optimization model for this casecan be expressed by
min minus 1198913(119909)
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
minus 1205801le 119891
1(119909) minus 119910
lowast
1le 120576
1
minus 1205802le 119891
2(119909) minus 119910
lowast
2le 120576
2
(21)
where ]1 ]2 ]3 ]4gt 0 indicate the upper bounds of hot-
pressing temperature hot-pressing pressure hot-pressingtime andmoisture content and [119910lowast
1minus120580
1 119910
lowast
1+120576
1] [119910lowast
2minus120580
2 119910
lowast
2+
1205762] are the feasible regions of MOR and MOS respectivelyIn order to let problem (21) be solved effectively we
let the objective function 1198913be with the Gaussian radial
basis function and the regression functions 1198911and 119891
2in the
constraints are with the linear polynomial kernel Then (21)is
min minus
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) minus 119887
3
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
minus 1205801le
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1le 120576
1
minus 1205802le
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2le 120576
2
(22)
which is a pseudoconvex optimization problem with convexconstraints
Second we would like to optimize the MOR MOS andIBS synthetically For this demand we consider the followingoptimization model
min minus1205821
119891lowast
1
1198911(119909) minus
1205822
119891lowast
2
1198912(119909) minus
1205823
119891lowast
3
1198913(119909)
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
(23)
where 1205821 120582
2 120582
3gt 0 indicate the importance of MOR MOS
and IBS ]1 ]2 ]3 ]4are with the same meaning as in (21)
and 119891lowast1 119891lowast
2 and 119891lowast
3are the expected values of MOR MOS
and IBS In particular if MOR MOS and IBS are with thesame importance in the quality of the hot-pressing plate wecan let 120582
1= 120582
2= 120582
3= 13 and we can let 120582
1= 12
1205822= 13 and 120582
3= 16 if the importance of MOR MOS
and IBS is strictly monotone decreasing Similar to the kernelfunctions in problem (21) we let119891
1119891
2 and119891
3in problem (23)
be with the Gaussian radial basis function which means thatproblem (23) is also a pseudoconvex optimization problemwith convex constraints
33 General Model Based on analysis in Sections 31 and32 we consider the following minimization problem in thispaper
minimize 119891 (119909) fl Θ (119909) + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816 + 120590
10038161003816100381610038161003816119890119879
11990910038161003816100381610038161003816
subject to 119909 isin Ω fl 119909 119886 le 119909 le 119887
119892119894(119909) le 0 119894 = 1 119898
(24)
6 Mathematical Problems in Engineering
where 120574 120590 ge 0 119886 119887 isin 119877119899 with 119886 lt 119887 Θ 119877
119899
rarr 119877
is continuously differentiable and pseudoconvex on 119877119899 and119892119894 119877
119899
rarr 119877 (119894 = 1 2 119898) is continuously differentiableand convex on 119877119899
On the one hand when Θ(119909) = (12)119909119879119876119909 minus 119910119879119909 119886 flminus119862119890 119887 fl 119862119890 and 120574 and 120590 are defined as in (18) then problem(24) without 119892
119894reduces to problem (18) On the other hand
if we let
Θ (119909) fl minus
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) minus 119887
3
119886 fl 0
119887 fl (]1 ]2 ]3 ]4)119879
(25)
1198921(119909) fl minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) fl
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
(26)
1198923(119909) fl minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) fl
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(27)
then problem (24) reduces to problem (22) Similar reformu-lation can be done for problem (23) by (24)
Therefore problem (24) considered in this paper includesthe optimizationmodels for building up the relationships andoptimizing the relative parameters in MDF continuous hot-pressing process
Inwhat follows we denoteF as the feasible region of (24)that is
F fl 119909 isin Ω 119892119894(119909) le 0 119894 = 1 2 119898 (28)
andM is the optimal solution set of (24)
4 Main Results
41 Proposed Neural Network In this subsection we proposea one-layer recurrent neural network for solving problem(24) where we combine the penalty function and projectionmethods to solve the constraints and use the smoothingtechniques to overcome the nonsmoothness of the objectivefunction and penalty function
Define the penalty function
119901 (119909) =
119898
sum
119894=1
max 0 119892119894(119909) (29)
Then 119909 isin Ω 119901(119909) le 0 = FFrom the smoothing functions in (25) for the plus
function we define the smoothing function of 119901 as
(119909 120583) =
119898
sum
119894=1
120601 (119892119894(119909) 120583) (30)
where
120601 (119904 120583) =
max 0 119904 if |119904| gt 120583
(119904 + 120583)2
4120583if |119904| le 120583
(31)
Form the results in [23] 120601(119904 120583) owns the followingproperties
Lemma 3 (see [23]) (i) For any 119904 isin 119877 120601(119904 sdot) is continuouslydifferentiable and 120601(sdot 120583) is also continuously differentiable forany fixed 120583 gt 0
(ii) 0 le nabla120583120601(119904 120583) le 1 forall119904 isin 119877 forall120583 isin (0 +infin)
(iii) 0 le 120601(119904 120583) minus |119904| le 1205834 forall119904 isin 119877 forall120583 isin (0 +infin)(iv) 120601(sdot 120583) is convex for any fixed 120583 gt 0 and
lim119904rarr119905120583darr0
nabla119904120601(119904 120583) sube 120597max0 119905
Then has the following properties
Lemma 4 is a smoothing function 119901 and satisfies thefollowing
(i) (sdot 120583) is convex for any fixed 120583 gt 0(ii) lim
119911rarr119909120583darr0nabla119911(119911 120583) sube 120597119901(119909)
(iii) 0 le nabla120583(119909 120583) le 119898 forall119909 isin 119877119899 120583 isin (0 +infin)
Next by the smoothing function for the absolute valuefunction | sdot |
120579 (119904 120583) =
|119904| if |119904| ge 120583
1199042
2120583+120583
2if |119904| lt 120583
(32)
we define
(119909 120583) fl Θ (119909) + 120574
119899
sum
119894=1
120579 (119909119894 120583) + 120590120579 (119890
119879
119909 120583) (33)
Since 120579(119904 120583) = 120601(119904 120583) + 120601(minus119904 120583) 120579(119904 120583) owns allproperties in Lemma 8 and the following results hold
Lemma 5 is a smoothing function 119891 in (24) with thefollowing properties
(i) (sdot 120583) is pseudoconvex for any fixed 120583 gt 0
(ii) lim119911rarr119909120583darr0
nabla119909(119909 120583) sube 120597119891(119909)
(iii) 0 le nabla120583(119909 120583) le 120574119899 + 120590 forall119909 isin 119877119899 120583 isin (0 +infin)
From the projected gradient method and the viscosityregularization method we introduce the following neuralnetwork to solve (24)
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(34)
where 120583(119905) = 119890minus119905 and 120576(119905) = 1(119905 + 1) with 1205760gt 0
Mathematical Problems in Engineering 7
120583intminus1
minus1
xintminus
sumsum PΩminus
minus
++
120576intab
bba
aab
pnablax
xqnabla
Figure 1 Simple block structure of proposed network (37)
By the definitions for (34) can be expressed as
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)
minus
119903
sum
119894=1
nabla119904120601 (119904 120583 (119905))
119904=119892119894(119909(119905))nabla119892
119894(119909 (119905)) minus 120576 (119905)
sdot (nablaΘ (119909 (119905)) + 120574
119899
sum
119894=1
nabla119904120579 (119904 120583)
119904=119909119894(119905)119890119894
+ 120590nabla119904120579 (119904 120583)
119904=119890119879119909(119905)
119890)]
(35)
where
nabla119904120601 (119904 120583) =
1 if 119904 gt 120583119904 + 120583
2120583if |119904| le 120583
0 if 119904 lt minus120583
nabla119904120579 (119904 120583) =
1 if |119904| ge 120583119904
120583if |119904| lt 120583
(36)
To implement (34) by circuits we can use the reformu-lated form of (34) as follows
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
120576 (119905) = minus1205762
(119905)
(119905) = minus120583 (119905)
(37)
Equation (34) can be seen as a network with three input andthree output variables that are 119909(119905) 120583(119905) and 120576(119905) A simpleblock structure of the proposed network (37) implementedby circuits is presented in Figure 1
42 Theoretical Analysis In this subsection we study somenecessary dynamical and optimality properties of proposednetwork (34) for solving (24)
The global existence of the solutions of (34) is a necessarycondition for its usability in optimization With an initial
point 1199090isin Ω the solution of (34) is global existentMoreover
the uniqueness of the solution of (34) with 1199090isin Ω is proved
under some conditions The proposed network (34) can beimplemented in circuits and mathematical software Thenthe feasibility and optimality of network (34) for optimizationproblem (24) are proved theoretically
We call 119909 [0 119879) with T gt 0 a solution of (34) if 119909 isabsolutely continuous on [0 119879) and satisfies (34) everywhere
Proposition 6 For any initial point 1199090isin Ω there is a global
solution of (34) defined on [0 +infin) and it satisfies
119909 (119905) isin Ω forall119905 isin [0 +infin) (38)
Proof Since the right hand function in network (34) iscontinuous about 119909 and 119905 there are 119879 gt 0 and an absolutecontinuous function 119909 [0 119879) rarr 119877
119899 such that 119909(119905) satisfies(34) for all 119905 isin [0 119879)
Denote 120579(119905) = 119875Ω[119909(119905) minus nabla
119909(119909(119905) 120583(119905)) minus 120576(119905)nabla
119909(119909(119905)
120583(119905))] Then (34) can be rewritten as
(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)
A simple integration procedure of the above equation gives
119909 (119905) = 119890minus119905
1199090+ 119890
minus119905
int
119905
0
120579 (119904) 119890119904
119889119904 (40)
which can be rewritten as
119909 (119905) = 119890minus119905
1199090+ (1 minus 119890
minus119905
)int
119905
0
120579 (119904)119890119904
119890119905 minus 1119889119904 (41)
By int1199050
119890119904
(119890119905
minus 1)119889119904 = 1 120579(119904) isin Ω forall0 le 119904 le 119905 and 1199090isin Ω we
obtain that 119909(119905) isin Ω forall119905 isin [0 119879)By the boundedness of Ω and the extension theory this
solution of (34) can be extended Thus the solution of (34)with initial point 119909
0isin Ω is globally existent Similarly we can
obtain that
119909 (119905) isin Ω forall119905 isin [0 +infin) (42)
Some Lipschitz condition is often used to guaranteethe uniqueness of the solution of a neural network Inwhat follows we give a sufficient condition to ensure theuniqueness of the solution of (34) with initial point 119909
0isin Ω
Proposition 7 For any initial point 1199090isin Ω if nabla
119909(sdot 120583) and
nabla119909(sdot 120583) are locally Lipschitz continuous for any fixed 120583 isin
(0 1] then the solution of neural network (34) is unique
Proof By Proposition 6 the solutions of (34) with initialpoint119909
0isin Ω exist globally and satisfy119909(119905) isin Ωforall119905 isin [0 +infin)
Suppose that there exist two solutions 119909 [0 +infin) rarr 119877119899
and 119910 [0 +infin) rarr 119877119899 of (34) with initial point 119909
0isin Ω
and suppose there exists such that = inf119905ge0119909(119905) =119910(119905)
119905 By theboundedness of Ω there is 119877 gt 0 such that 119909(119905) le 119877 and119910(119905) le 119877 forall119905 isin [0 +infin)
8 Mathematical Problems in Engineering
Then there is 119871 gt 0 such that10038161003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla
119909 (119910 (119905) 120583 (119905))
10038161003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
1003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla119909 (119910 (119905) 120583 (119905))1003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
(43)
Differentiating (12)119909(119905)minus119910(119905)2 along the two solutionsof (34) by the Lipschitz continuity of 119875
Ωand (43) we have
119889
119889119905
1
2
1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817
2
= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩
le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817
2
(44)
Applying Gronwallrsquos inequality into the integration of theabove inequality it gives 119909(119905) = 119910(119905) forall119905 isin [119905 + 1] whichleads to a contradiction Therefore the solution of (34) withinitial point 119909
0isin Ω is unique
Lyapunov method is employed to analyze the perfor-mance of (34) Here we introduce the following two Lya-punov energy functions
119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
119866 (119909 119905) =1
2dist2 (119909M) + (119909 120583 (119905))
+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
(45)
The above two Lyapunov functions satisfy the followingestimations along the solutions of (34)
Lemma 8 (i) The derivative of 119864(119909 119905) along the solution of(34) can be calculated by
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
(46)
(ii) The derivative of 119866(119909 119905) along the solution of (34) can becalculated by
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(47)
Proof (i) Differentiating119864(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119864 (119909 (119905) 119905) = ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(48)
Equation (34) can be rewritten as
(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(49)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119909(119905) using (49) and (10) we have
⟨119909 (119905) minus nabla119909 (119909 (119905) 120583 (119905)) minus 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
minus (119905) minus 119909 (119905) (119905) + 119909 (119905) minus 119909 (119905)⟩ ge 0
(50)
which follows the fact that
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
le minus (119905)2
forall119905 ge 0
(51)
Combining (48) and (51) we get that
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
(52)
From Lemmas 4 and 5 (119905) = minus119890minus119905 and 120576(119905) lt 0 we obtainthe estimation in (i)
(ii) Differentiating 119866(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩
+ ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(53)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119875M(119909(119905)) using (10) and (49) we have
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0
(54)
which can be rewritten as
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
minus (119905)2
(55)
Mathematical Problems in Engineering 9
Combining (53) and (55) we get
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905) (119891 (119909 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(56)
Similar to the analysis in (i) we obtain the estimation in(ii)
Next we prove the efficiency of proposed network (34) forsolving optimization problem (24) where the convergencefeasibility of the proposed network is a basic property
Theorem 9 For initial point 1199090isin Ω any solution 119909 [0 +infin)
rarr 119877119899 of (34) satisfies
lim119905rarr+infin
dist (119909 (119905) F) = 0 (57)
Proof Denote 119909 [0 +infin) rarr 119877119899 as a global solution of (34)
with initial point 1199090isin Ω
From 119909(119905) isin Ω forall119905 isin [0 +infin) and Lemma 8 119864(119909(119905) 119905) isnonincreasing along the solution of (34) Using119864(119909(119905) 119905) ge 0forall119905 isin [0 +infin) we confirm that
lim119905rarr+infin
119864 (119909 (119905) 119905) exists (58)
From 119909(119905) isin Ω forall119905 isin [0 +infin) and lim119905rarr+infin
120576(119905) = 0 wehave
lim119905rarr+infin
120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ
119891) = 0 (59)
which implies
lim119905rarr+infin
119864 (119909 (119905) 119905) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) (60)
Inwhat follows wewill prove that lim119905rarr+infin
(119909(119905) 120583(119905)) =
0 Arguing by contradiction we assume that
lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 119888 = 0 (61)
By (119909(119905) 120583(119905)) ge 119901(119909(119905)) ge 0 forall119905 isin [0 +infin) we have 119888 gt 0which follows the fact that there is 119879
1ge 0 such that
(119909 (119905) 120583 (119905)) ge119888
2 forall119905 isin [119879
1 +infin) (62)
From Lemma 4 we obtain
1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)
which implies that there exists 1198792gt 119879
1such that
(119875M (119909 (119905)) 120583 (119905)) le119888
8 (64)
Since Ω is bounded and 119909(119905) isin Ω forall119905 isin [0 +infin) byLemma 5 there is 119877 gt 0 such that10038161003816100381610038161003816⟨nabla
119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
10038161003816100381610038161003816le 119877
forall119905 isin [0 +infin)
(65)
Then
lim119905rarr+infin
120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)
which implies that there is 1198793ge 119879
2such that
10038161003816100381610038161003816120576 (119905) ⟨nabla
119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩
10038161003816100381610038161003816le119888
8
forall119905 isin [1198792 +infin)
(67)
From Lemma 8 and the convexity of (sdot 120583) for any fixed120583 gt 0 we obtain that
119889
119889119905119866 (119909 (119905) 119905) le minus ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)
sdot ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(68)
By (62) (67) and (68) we obtain
119889
119889119905119866 (119909 (119905) 119905) le minus
119888
4 forall119905 isin [119879
3 +infin) (69)
Integrating the above inequality from 1198793to 119905 (gt119879
3) we
have
119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879
3) minus
119888
4(119905 minus 119879
3)
forall119905 isin [1198793 +infin)
(70)
Thus
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (71)
which leads to a contradiction with 119866(119909 119905) ge 0 for all 119909 isin 119877119899and 119905 isin [0 +infin) Therefore
lim119905rarr+infin
119901 (119909 (119905)) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 0 (72)
which guarantees that
lim119905rarr+infin
dist (119909 (119905) F) = 0 (73)
The following theorem indicates that any accumulationpoint of the solutions of (34) is just an optimal solution of(34)
10 Mathematical Problems in Engineering
Theorem 10 For initial point 1199090isin Ω any solution 119909 of (34)
is convergent to the optimal solution setM that is
lim119905rarr+infin
dist (119909 (119905) M) = 0 (74)
Proof From (63) (68) and (119909(119905) 120583(119905)) ge 0 forall119905 isin [0 +infin)we have
119889
119889119905119866 (119909 (119905) 119905)
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
+ 119898120583 (119905)
(75)
by (119905) = minus120583(119905) which can be rewritten as
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(76)
Denote
119868 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le 0
119869 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
gt 0
(77)
Owning to the continuity of ⟨nabla119909(119909(119905) 120583(119905)) 119909(119905)minus119875M(119909(119905))⟩
on [0 +infin) 119868 and 119869 are closed and open in [0 +infin) respec-tively
Case 1 In this case we assume that there exists 119879 ge 0 suchthat 119905 isin 119868 forall119905 isin [119879 +infin)
From the definition on 119868 and the pseudoconvexity of(sdot 120583) on 119877119899 we have
(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))
forall119905 isin [119879 +infin)
(78)
which implies
lim sup119905rarr+infin
119891 (119909 (119905)) le min119909isinF
119891 (119909) (79)
ByTheorem 9 we confirm that
lim119905rarr+infin
dist (119909 (119905) M) = 0 (80)
Case 2 In this case we assume that there exists 119879 ge 0 such
that 119905 isin 119869 forall119905 isin [119879 +infin) which means that
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] lt 0 forall119905 isin [119879 +infin) (81)
Then
lim119905rarr+infin
[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)
Since lim119905rarr+infin
119864(119909(119905) 119905) exists and lim119905rarr+infin
120583(119905) = 0 weobtain that
lim119905rarr+infin
dist (119909 (119905) M) exists (83)
Suppose
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889
gt 0
(84)
Then there is 1198791ge such that
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge
119889
2
forall119905 isin [1198791 +infin)
(85)
Then from (76) we have
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus
119889
2120576 (119905)
forall119905 isin [1198791 +infin)
(86)
Integrating the above inequality from 1198791to 119905 (gt119879
1) we
have
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879
1) + 119898120583 (119879
1)
+ int
119905
1198791
minus119889
2120576 (119904) 119889119904
(87)
Let 119905 rarr +infin in the above inequality then we have
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (88)
which leads to a contraction with the boundedness frombelow 119866(119909(119905) 119905) on [0 +infin) Thus
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)
which follows the fact that there is an increasing sequence 119905119899
such that 119905119899rarr +infin and
lim119899rarr+infin
⟨nabla119909 (119909 (119905
119899) 120583 (119905
119899)) 119909 (119905
119899) minus 119875M (119909 (119905
119899))⟩
= 0
(90)
By 119909(119905119899) sube Ω and Ω which is bounded there are 119909lowastlowast isin
Ω and a subsequence of 119909(119905119899) (denoted as 119909(119905
119899119896)) such that
lim119896rarr+infin
119905119899119896= +infin and lim
119896rarr+infin119909(119905
119899119896) = 119909
lowastlowast By Lemma 5there is 120585(119909lowastlowast) isin 120597119891(119909lowastlowast) such that
⟨120585 (119909lowastlowast
) 119909lowastlowast
minus 119875M (119909lowastlowast
)⟩ = 0 (91)
From the pseudoconvexity of 119891 and the above inequalitywe have
119891 (119909lowastlowast
) le 119891 (119875M (119909lowastlowast
)) (92)
Mathematical Problems in Engineering 11
Table 1 Experimental data set in MDF hot-pressing
Regression parameter 120574 120590 119862 MSE DF SEMOR 0001 5 30 00307 09543 minus08299
MOE 0001 6 40 00642 09063 minus17060
IBS 0001 5 40 00451 09256 minus06388
By Theorem 9 we have 119909lowastlowast isin F Therefore 119909lowastlowast isin Mwhich implies that
lim119896rarr+infin
dist (119909 (119905119899119896) M) = 0 (93)
Combining the above results with (83) we conclude thatlim
119905rarr+infindist(119909(119905)M) = 0
Case 3 In this case we assume that both 119868 and 119869 areunbounded
For 119905 isin 119868 similar to the analysis in Case 1 we have
lim119905rarr+infin119905isin119868
dist (119909 (119905) M) = 0 (94)
For 119905 isin 119869 define 120591(119905) = sup119904le119905119904isin119868
119904 Then 120591(119905) isin 119868 and(120591(119905) 119905] sube 119869 By the unboundedness of 119868 and 119869 we have
lim119905isin119869119905rarr+infin
120591 (119905) = +infin (95)
From (81) and the continuity of 119866(119909(119905) 119905) on [0 +infin) wehave
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))
+ 119898120583 (120591 (119905)) forall119905 isin 119869
(96)
By 120591(119905) isin 119868 (94) and (95) we have
lim sup119905rarr+infin119905isin119869
[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le lim119905rarr+infin119905isin119869
[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0
(97)
which gives
lim119905rarr+infin119905isin119869
119866 (119909 (119905) 119905) = 0 (98)
From the definition of 119866(119909 119905) and the above result wehave
lim119905rarr+infin119905isin119869
dist (119909 (119905) M) = 0 (99)
Therefore from (94) and (99) we obtain
lim119905rarr+infin
dist (119909 (119905) M) = 0 (100)
5 Numerical Experiments
In this section we test the proposed neural network (34)for solving problem (24) which is brought forward from
the MDF continuous hot-pressing process Based on theexisting data set we use the established theories and proposedneural network (34) to build the relationships between themain qualities of the hot-pressing plate and some relativetechnology parameters from optimization problem (18)Then based on optimization problem (22) we will useproposed network (34) to solve the optimal values of thetechnology parameters in hot-pressing system for optimizingthe qualities of the hot-pressing plate All these numericalexperiments validate the good performance of the proposednetwork in this paper
The numerical testing was carried out on a Lenovo PC(300GHz 200GB of RAM) with the use of Matlab 74 Andwe use ode23 to realize the neural network (34) in Matlab
51 Construction Relation Models in MDF Continuous Hot-Pressing Process In this part by considered optimizationproblem (18) and network (34) we build the relation modelwhich takes the hot-pressing temperature (TE) hot-pressingpressure (PR) hot-pressing time (TI) and moisture content(MC) of slab as the argument variables and the MOR MOEand IBS indexes of MDF as the dependent variables Thenumerical results show the good fitting of the built modelsfor the data set where the data set is given in Table 4
In order to use the data in Table 4 we first normalize theminto [0 1] And we use the mean square error (MSE) degreeof fitting (DF) and sufficient evaluation (SE) to evaluate thenumerical results where
MSE MSE = radicsum119899
119894=1(119909
119894minus 119909
lowast
119894)2
119899
DF DF = 1 minus radic(sum119899
119894=1(119909
119894minus 119909
lowast
119894)2
sum119899
119894=11199092
119894
)
SE SE = 12119909119879
119876119909 minus 119910119879
119909
(101)
where 119909 and 119909lowast indicate the actual value and predicted valueand 119876 and 119910 are defined as in (18) The smaller the MSE andthe SE and the closer the DF to 1 the better the regressionresult Moreover the SE function is the objective function in(15)
Based on the Gaussian radial basis function kernel thevalues of the initial parameters in problem (18) are given inTable 1 With a random initial point 119909
0isin Ω the numerical
results with respect to the obtained solution are also listed inTable 1 Figures 2ndash4 illustrate the fitting effect of the MORMOE and IBS values by using the proposed network (34) forsolving problem (18) And the parameters of the regressionfunctions in (19) are shown in Table 5
12 Mathematical Problems in Engineering
Table 2 Criterion of physical and mechanical performance indexes
Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45
IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045
MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700
2520151050
OriginalPredicted
0
05
1
15
2
25
3
35
4
5
45
Figure 2 Normalized sample fitting results of network (34) for theregression of MOR
Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5
52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2
521 Case 1
Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 3 Normalized sample fitting results of network (34) for theregression of MOE
By the information in Table 2 we let ]1= 200 ]
2= 4
]3= 7 and ]
4= 12 and choose 119910lowast
1= 22 119910lowast
2= 2500 120580
1= 8
1205761= 8 120580
2= 500 and 120576
2= 500 in optimization problem (21)
Then Ω = 119909 isin 1198774
0 le 1199091le ]
1 0 le 119909
2le ]
2 0 le 119909
3le
]3 0 le 119909
4le ]
4 and we can let
1198921(119909) = minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) =
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
1198923(119909) = minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) =
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(102)
By the proposed network (34) for solving (21) we obtainthe optimal solution
119909lowast
= (1594725 31022 64866 60061)119879
(103)
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
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2 Mathematical Problems in Engineering
projection method Then Lagrangian and penalty methodsare introduced into networks Based on the Lagrangian func-tionmethod Lagrangian networks were proposed for solvingthe optimization problems [8 10] with general constraintsBut the Lagrangian network increases the dimension of thenetworks along with the number of the constraints In recentyears recurrent neural networks based on penalty methodwere widely investigated for solving optimization problemsThe neural networks for smooth optimization problems cannot solve nonsmooth optimization problems because thegradients of the objective and constrained functions arerequired in such neural networks The generalized nonlinearprogramming circuit (G-NPC) in [15] can be considered asa natural extension of nonlinear programming circuit (NPC)for solving nonsmooth convex optimization problems withinequality constraints But the nonempty interior of feasibleregion and large enough penalty parameters are needed forthe network in [15] In order to overcome the nonemptyassumption of the interior of feasible region Bian and Xue[6] proposed a recurrent neural network for nonsmoothconvex optimization based on penalty function methodThe efficiency of the neural networks for solving convexoptimization problems relies on the convexity of functionsA neural network for nonconvex quadratic optimization ispresented in [16] Some neural networks modeled by differ-ential inclusion were also proposed for some nonsmooth andnonconvex optimization problems [6 17] To overcome thedifferential inclusion smoothing techniques are introducedinto the neural networks The main feature of smoothingmethod is to approximate the nonsmooth functions by a classof smooth functions Thus the neural network constructedby the smoothing techniques is modeled by a differentialequation which can be implemented easily in circuits andmathematical software [18]
In this paper we propose a neural network model forsolving the optimization problem brought forward from theMDF continuous hot-pressing automatic control system InSection 2 some notations and necessary preliminary resultsare listed In Section 3 based on the SVM theory withthe existing linear and nonlinear kernel functions we givean optimization problem which includes the problems forbuilding the models of MDF continuous hot-pressing systemand optimizing the MDF performance indexes as specialcases In order to build up the relation models on theproperties of the slab technical parameters in hot-pressingprocess and the performance indexes of MDF when the ker-nel function is positive definite or semipositive definite thecorresponding optimization problem is a constrained convexproblem otherwise it is a nonconvex problemThe optimiza-tion problem for optimizing the performance parametersis a nonconvex constrained optimization problem but itsobjective function is pseudoconvex due to the appropriatechoice of kernel functions In Section 4 we propose a neuralnetwork based on the penalty function method projectionmethod and smoothing techniques The proposed networkis modeled by a nonautomatic differential equation ByLyapunovmethod we prove that the solution of the proposednetwork is global existent and convergent to the feasible setof the considered optimization problems Moreover due to
the pseudoconvexity of the objective function and the con-vexity of the constraint the proposed network also convergesto the optimal solution set of the optimization problem InSection 5 based on the existing data set we use the proposednetwork into the model building and parameter optimizingproblems of hot-pressing system which validates the goodperformance of the obtained results in this paper
Notations 119877+= [0 +infin) Given column vectors 119909 = (119909
1
1199092 119909
119899)119879 and 119910 = (119910
1 119910
2 119910
119899)119879 ⟨119909 119910⟩ = 119909
119879
119910 =
sum119899
119894=1119909119894119910119894is the scalar product of 119909 and 119910 119909
119894denotes the
119894th element of 119909 119909 denotes the Euclidean 2-norm definedby 119909 = (sum
119899
119894=11199092
119894)12 For a closed convex subset Γ sube 119877
119899dist(119909 Γ) is the distance from 119909 to Γ defined by dist(119909 Γ) =min
119910isinΓ119909 minus 119910
2 Preliminaries
In this section we state some definitions and propertiesneeded in this paper We refer the readers to [19ndash21]
21 Support Vector Regression Kernels were regarded as afunction with the formulation of inner product and havebeen a powerful tool in machine learning for their superiorperformance over a wide range of learning problems suchas isolated handwritten digit recognition text categorizationand face detection [21 22]
LetX be a nonempty set and 119870 X timesX rarr 119877 be a real-valued and symmetric function With the kernel matrix K =
(119896(119909119894 119909
119895))119899
119894119895=1 119870 is said to be a positive semidefinite kernel
ifK is positive semidefinite for any 119899 isin N and 1199091 119909
2 119909
119899isin
XWe call119870 an indefinite kernel if there exist1199091 119909
2 119909
119899isin
X and V 119908 isin 119877119899 such that V119879KV lt 0 and 119908119879K119908 gt 0In what follows we list some widely used kernels
(i) Gaussian radial basis function kernel 119870(119909 119910) =
exp(minus119909 minus 119910221205902)
(ii) Polynomial kernel 119870(119909 119910) = (⟨119909 119910⟩ + 119888)119901 119888 ge 0
119901 ge 1
(iii) Sigmoid kernel 119870(119909 119910) = tanh(120581⟨119909 119910⟩ + V) 120581 gt 0V lt 0
22 Smoothing Approximation Smoothing approximation isan effective method for solving nonsmooth optimizationproblems and has been widely used in the past decades Themain feature of smoothing method is to approximate thenonsmooth functions by a class of parameterized smoothfunctions In this paper we adopt the smoothing functiondefined as follows
Definition 1 (see [23]) Let 119892 119877119899
rarr 119877 be a continuousfunction One calls 119877119899 times119877
+rarr 119877 a smoothing function of
119892 if (sdot 120583) is continuously differentiable for any fixed 120583 gt 0and lim
119911rarr119909120583darr0(119911 120583) = 119892(119909) holds for any 119909 isin 119877119899
Mathematical Problems in Engineering 3
Chen and Mangasarian constructed a class of smoothapproximations of the function (119904)
+fl max0 119904 by convo-
lution [20 24] as follows Let 120588 119877 rarr 119877+be a piecewise
continuous density function satisfying
120588 (119905) = 120588 (minus119905)
120581 fl int
infin
minusinfin
|119905| 120588 (119905) 119889119905 lt infin
(1)
Then
120601 (119905 120583) fl int
infin
minusinfin
(119904 minus 120583119905)+120588 (119905) 119889119905 (2)
from 119877 times 119877+to 119877
+is well defined
By different density functions many popular smoothingfunctions of (119904)
+can be derived such as
1206011(119904 120583) = 119904 + 120583 ln (1 + 119890minus119904120583)
1206012(119904 120583) =
1
2(119904 + radic1199042 + 41205832)
1206013(119904 120583) =
max 0 119904 if |119904| gt 120583
(119904 + 120583)2
4120583if |119904| le 120583
1206014(119904 120583) =
119904 +120583
2119890minus119904120583 if 119904 gt 0
120583
2119890minus119904120583 if 119904 le 0
(3)
where 1206011is the neural networks smoothing function 120601
2is
called the CHKS (Chen-Harker-Kanzow-Smale) smoothingfunction120601
3is called the uniform smoothing function and120601
4
is called the Picard smoothing function The four functionsbelong to the class of the Chen-Mangasarian smoothingfunctions
Many nonsmooth functions can be reformulated by usingthe plus function We list some of them as follows
|119904| = (119904)++ (minus119904)
+
max (119904 119905) = 119904 + (119905 minus 119904)+
min (119904 119905) = 119904 minus (119904 minus 119905)+
mid (119904 ℓ 119906) = min (max (ℓ 119904) 119906) for given ℓ 119906
(4)
So we can define a smoothing function for the abovenonsmooth functions by a smoothing function of (119904)
+
From Theorem 961 and Corollary 847(b) in [20] when119892 119877
119899
rarr 119877 is locally Lipschitz continuous at 119909 thesubdifferential associated with a smoothing function
119866(119909) = con V | nabla
119909 (119909
119896
120583119896) 997888rarr V for 119909119896 997888rarr 119909 120583
119896
darr 0
(5)
is nonempty and bounded and 120597119892(119909) sube 119866(119909) where ldquoconrdquo
denotes the convex hull In [20 23] it is shown that manysmoothing functions satisfy the gradient consistency
120597119892 (119909) = 119866(119909) (6)
which is an important property of the smoothing methodsand guarantees the convergence of smoothing methods withadaptive updating schemes of smoothing parameters to astationary point of the original problem
23 Pseudoconvex Function Pseudoconvex function is a classof functions which may be nonsmooth or nonconvex butbrings us the opportunity to find the optimal solutions
Definition 2 (see [25]) Let X be a nonempty convex subsetof 119877119899 A function 120601 is said to be pseudoconvex on X if forany 119909 119910 isin X one has
exist120585 (119909) isin 120597120601 (119909)
st ⟨120585 (119909) 119910 minus 119909⟩ ge 0 997904rArr
120601 (119910) ge 120601 (119909)
(7)
Many nonconvex functions in application are pseudocon-vex such as the Butterworth filter function fraction functionand density functionOf particular interest in this paper is thefact that the Gaussian function
ℎ (119909) = minus exp(minus119899
sum
119894=1
1199092
119894
1205902
119894
) (8)
with 120590119894gt 0 119894 = 1 2 119899 is pseudoconvex on 119877119899
24 Project Operator LetX be a closed convex subset of 119877119899Then the projection operator toX at 119909 is defined by
119875X (119909) = argmin119906isinX
119906 minus 1199092 (9)
and satisfies the following inequalities
⟨V minus 119875X (V) 119875X (V) minus 119906⟩ ge 0 forallV isin 119877119899 119906 isin X
1003817100381710038171003817119875X (119906) minus 119875X (V)1003817100381710038171003817 le 119906 minus V forall119906 V isin 119877119899
(10)
(i) Suppose Ω = 119909 isin 119877119899 119906 le 119909 le V with 119906 V isin 119877119899 cup plusmninfin
then 119875Ω(119909) = (119901
1 119901
2 119901
119899)119879 can be expressed by
119901119894=
119906119894119909119894lt 119906
119894
119909119894119906119894le 119909
119894le V
119894
V119894
V119894lt 119909
119894
(11)
(ii) Suppose Ω = 119909 119860119909 = 119887 with 119860 isin 119877119903times119899 of full row rank
and 119887 isin 119877119903 then 119875Ω(119909) = 119909 minus 119860
119879
(119860119860119879
)minus1
(119860119909 minus 119887)
3 Optimization Problems in MDF ContinuousHot-Pressing Process
In this section we will give the optimization model consid-ered in this paper First by the optimization and supportvector machine theories we show the optimization modelsfor building up the relationships in MDF continuous hot-pressing process Then another optimization model for
4 Mathematical Problems in Engineering
optimizing the parameters in the MDF continuous hot-pressing process is obtained Thus we express these twokinds of problems into a uniform formulation which is theoptimization problem considered in Section 3
31 Optimization Problem for Building up the Models of MDFContinuousHot-Pressing Process DenoteX sub 119877
119901 andY sub 119877
as two sets where 119909119894 = (1199091198941 119909
119894
2 119909
119894
119901) isin X is the attribute
vector on behalf of the hot-pressing plate properties 119910119894 isin Yindicates the values of the qualities of hot-pressing plate Let119878 = (119904
1
1199042
119904119899
) be the training data set of hot-pressingprocess where 119904119894 = (119909119894 119910119894) obeys the unknown distributionand is IID (independent and identically distributed) Basedon the support vector machine theory and the training dataset we would like to find a nonlinear function 119910 = 119891(119909) suchthat it approximates the training data set as much as possible
With the kernel function 119870 and the Huber loss functionfrom the theory in [21] the dual optimization problem of theprimal optimization problem of SVM is given as
min120572120573
1
2
119899
sum
119894119895=1
(120573119894minus 120572
119894) (120573
119895minus 120572
119895)119870 (119909
119894
119909119895
)
minus
119899
sum
119894=1
(120573119894minus 120572
119894) 119910
119894
+120576
2119862
119899
sum
119894=1
(120573119894minus 120572
119894)2
st119899
sum
119894=1
(120573119894minus 120572
119894) = 0
0 le 120573119894 120572
119894le 119862 119894 = 1 2 119899
(12)
Denote 120572lowast119894and 120573lowast
119894as the optimal solutions of (12) Then the
regression function 119891 can be expressed by
119891 (119909) =
119899
sum
119894=1
(120573lowast
119894minus 120572
lowast
119894)119870 (119909
119894
119909) + 119887 (13)
where 119887 can be calculated by one of the following twomethods
119887 = 119910119895
minus
119899
sum
119894=1
(120573lowast
119894minus 120572
lowast
119894) ⟨119909
119894
119909119895
⟩ + 120576 120572lowast
119895isin (0 119862)
119887 = 119910119895
minus
119899
sum
119894=1
(120573lowast
119894minus 120572
lowast
119894) ⟨119909
119894
119909119895
⟩ minus 120576 120572lowast
119895isin (0 119862)
(14)
Denote 119909119894= 120573
119894minus 120572
119894in (12) then (12) can be reformulated
as
min 1
2119909119879
119876119909 minus 119910119879
119909
st 119890119879
119909 = 0 minus 119862119890 le 119909 le 119862119890
(15)
where 119909 = (1199091 119909
2 119909
119899)119879 119910 = (1199101 1199102 119910119899)119879 119890 = (1 1
1)119879
isin 119877119899 and 119876 = K+(1205762119862)119868
119899withK = (119870(119909
119894
119909119895
))119899
119894119895=1
For the optimal solution 119909lowast = (119909lowast1 119909
lowast
2 119909
lowast
119899) of (15) we call
119909119894 a support vector if 119909lowast
119894= 0 The optimal solution of (15)
solved by the originalmethods often has almost 100 supportvectors which increases the complexity of the relationmodelslargely Thus we introduce the problem deduced by (15) thatis
min 1
2119909119879
119876119909 minus 119910119879
119909 + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816
st 119890119879
119909 = 0 minus 119862119890 le 119909 le 119862119890
(16)
with 120574 gt 0 In (43) 120574sum119899
119894=1|119909119894| is often called the regulation
term which is used to control the number of its supportvectors
Define the penalty function
119875 (119909) =
1003817100381710038171003817100381710038171003817
1
119899119890119890119879
119909
1003817100381710038171003817100381710038171003817=radic119899
119899
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119894=1
119909119894
1003816100381610038161003816100381610038161003816100381610038161003816
(17)
and then 119909 119875(119909) le 0 = 119909 119890119879119909 = 0 From [19 Proposition243] 119909 is an optimal solution of (43) if and only if it is anoptimal solution of the following problem
min 1
2119909119879
119876119909 minus 119910119879
119909 + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816 + 120590
10038161003816100381610038161003816119890119879
11990910038161003816100381610038161003816
st minus 119862119890 le 119909 le 119862119890
(18)
where 120590 = 119862119876 + 119910radic119899 + 120574 + 1 Thus we can build up therelation models of hot-pressing process by solving problem(18)
32 Optimization Problem for Optimizing the Parameters inMDF Hot-Pressing Process Based on the relationships builtup in Section 31 we focus on themodulus of rupture (MOR)modulus of elasticity (MOE) and internal bonding strength(IBS) of hot-pressing plate by optimizing the process param-eters and slab attributes Suppose the regression functions ofMORMOS and IBSwith respect to some relative parametersbased on the Gaussian radial basis function kernel are
1198911(119909) fl
119899
sum
119894=1
120588lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) + 119887
1(MOR)
1198912(119909) fl
119899
sum
119894=1
120579lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) + 119887
2(MOS)
1198913(119909) fl
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) + 119887
3(IBS)
(19)
and based on the linear polynomial kernel are
1198911(119909) fl
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1(MOR)
1198912(119909) fl
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2(MOS)
1198913(119909) fl
119899
sum
119894=1
120581lowast
119894⟨119911
119894 119909⟩ + 119887
3(IBS)
(20)
Mathematical Problems in Engineering 5
where 1199111 119911
2 119911
119899are the variables in the data set for
regression and 119909 = (1199091 119909
2 119909
3 119909
4)119879 indicates the indepen-
dent variable with hot-pressing temperature 1199091 hot-pressing
pressure 1199092 hot-pressing time 119909
3 and moisture content 119909
4
The regression functions in (19) satisfy the following twoproperties
(i) 119891119894is continuously differentiable on 119877119899 119894 = 1 2 3
(ii) 119891119894and minus119891
119894are not convex but minus119891
119894is pseudoconvex on
119877119899 119894 = 1 2 3
And the regression functions in (20) satisfy the following twoproperties
(i) 119891119894is continuously differentiable on 119877119899 119894 = 1 2 3
(ii) 119891119894and minus119891
119894are convex on 119877119899 119894 = 1 2 3
From the physical significance of MOR MOS and IBSwe suppose that the larger the numbers of MOR MOS andIBS the better the quality of hot-pressing plate In order toadopt the different demand on the indexes of the hot-pressingplate in different applications we consider the following twocases in this part
First we focus onmaximizing a single performance indexof the hot-pressing plate when the other two performanceindexes are within the certain areas If we want to optimizethe IBS the corresponding optimization model for this casecan be expressed by
min minus 1198913(119909)
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
minus 1205801le 119891
1(119909) minus 119910
lowast
1le 120576
1
minus 1205802le 119891
2(119909) minus 119910
lowast
2le 120576
2
(21)
where ]1 ]2 ]3 ]4gt 0 indicate the upper bounds of hot-
pressing temperature hot-pressing pressure hot-pressingtime andmoisture content and [119910lowast
1minus120580
1 119910
lowast
1+120576
1] [119910lowast
2minus120580
2 119910
lowast
2+
1205762] are the feasible regions of MOR and MOS respectivelyIn order to let problem (21) be solved effectively we
let the objective function 1198913be with the Gaussian radial
basis function and the regression functions 1198911and 119891
2in the
constraints are with the linear polynomial kernel Then (21)is
min minus
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) minus 119887
3
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
minus 1205801le
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1le 120576
1
minus 1205802le
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2le 120576
2
(22)
which is a pseudoconvex optimization problem with convexconstraints
Second we would like to optimize the MOR MOS andIBS synthetically For this demand we consider the followingoptimization model
min minus1205821
119891lowast
1
1198911(119909) minus
1205822
119891lowast
2
1198912(119909) minus
1205823
119891lowast
3
1198913(119909)
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
(23)
where 1205821 120582
2 120582
3gt 0 indicate the importance of MOR MOS
and IBS ]1 ]2 ]3 ]4are with the same meaning as in (21)
and 119891lowast1 119891lowast
2 and 119891lowast
3are the expected values of MOR MOS
and IBS In particular if MOR MOS and IBS are with thesame importance in the quality of the hot-pressing plate wecan let 120582
1= 120582
2= 120582
3= 13 and we can let 120582
1= 12
1205822= 13 and 120582
3= 16 if the importance of MOR MOS
and IBS is strictly monotone decreasing Similar to the kernelfunctions in problem (21) we let119891
1119891
2 and119891
3in problem (23)
be with the Gaussian radial basis function which means thatproblem (23) is also a pseudoconvex optimization problemwith convex constraints
33 General Model Based on analysis in Sections 31 and32 we consider the following minimization problem in thispaper
minimize 119891 (119909) fl Θ (119909) + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816 + 120590
10038161003816100381610038161003816119890119879
11990910038161003816100381610038161003816
subject to 119909 isin Ω fl 119909 119886 le 119909 le 119887
119892119894(119909) le 0 119894 = 1 119898
(24)
6 Mathematical Problems in Engineering
where 120574 120590 ge 0 119886 119887 isin 119877119899 with 119886 lt 119887 Θ 119877
119899
rarr 119877
is continuously differentiable and pseudoconvex on 119877119899 and119892119894 119877
119899
rarr 119877 (119894 = 1 2 119898) is continuously differentiableand convex on 119877119899
On the one hand when Θ(119909) = (12)119909119879119876119909 minus 119910119879119909 119886 flminus119862119890 119887 fl 119862119890 and 120574 and 120590 are defined as in (18) then problem(24) without 119892
119894reduces to problem (18) On the other hand
if we let
Θ (119909) fl minus
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) minus 119887
3
119886 fl 0
119887 fl (]1 ]2 ]3 ]4)119879
(25)
1198921(119909) fl minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) fl
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
(26)
1198923(119909) fl minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) fl
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(27)
then problem (24) reduces to problem (22) Similar reformu-lation can be done for problem (23) by (24)
Therefore problem (24) considered in this paper includesthe optimizationmodels for building up the relationships andoptimizing the relative parameters in MDF continuous hot-pressing process
Inwhat follows we denoteF as the feasible region of (24)that is
F fl 119909 isin Ω 119892119894(119909) le 0 119894 = 1 2 119898 (28)
andM is the optimal solution set of (24)
4 Main Results
41 Proposed Neural Network In this subsection we proposea one-layer recurrent neural network for solving problem(24) where we combine the penalty function and projectionmethods to solve the constraints and use the smoothingtechniques to overcome the nonsmoothness of the objectivefunction and penalty function
Define the penalty function
119901 (119909) =
119898
sum
119894=1
max 0 119892119894(119909) (29)
Then 119909 isin Ω 119901(119909) le 0 = FFrom the smoothing functions in (25) for the plus
function we define the smoothing function of 119901 as
(119909 120583) =
119898
sum
119894=1
120601 (119892119894(119909) 120583) (30)
where
120601 (119904 120583) =
max 0 119904 if |119904| gt 120583
(119904 + 120583)2
4120583if |119904| le 120583
(31)
Form the results in [23] 120601(119904 120583) owns the followingproperties
Lemma 3 (see [23]) (i) For any 119904 isin 119877 120601(119904 sdot) is continuouslydifferentiable and 120601(sdot 120583) is also continuously differentiable forany fixed 120583 gt 0
(ii) 0 le nabla120583120601(119904 120583) le 1 forall119904 isin 119877 forall120583 isin (0 +infin)
(iii) 0 le 120601(119904 120583) minus |119904| le 1205834 forall119904 isin 119877 forall120583 isin (0 +infin)(iv) 120601(sdot 120583) is convex for any fixed 120583 gt 0 and
lim119904rarr119905120583darr0
nabla119904120601(119904 120583) sube 120597max0 119905
Then has the following properties
Lemma 4 is a smoothing function 119901 and satisfies thefollowing
(i) (sdot 120583) is convex for any fixed 120583 gt 0(ii) lim
119911rarr119909120583darr0nabla119911(119911 120583) sube 120597119901(119909)
(iii) 0 le nabla120583(119909 120583) le 119898 forall119909 isin 119877119899 120583 isin (0 +infin)
Next by the smoothing function for the absolute valuefunction | sdot |
120579 (119904 120583) =
|119904| if |119904| ge 120583
1199042
2120583+120583
2if |119904| lt 120583
(32)
we define
(119909 120583) fl Θ (119909) + 120574
119899
sum
119894=1
120579 (119909119894 120583) + 120590120579 (119890
119879
119909 120583) (33)
Since 120579(119904 120583) = 120601(119904 120583) + 120601(minus119904 120583) 120579(119904 120583) owns allproperties in Lemma 8 and the following results hold
Lemma 5 is a smoothing function 119891 in (24) with thefollowing properties
(i) (sdot 120583) is pseudoconvex for any fixed 120583 gt 0
(ii) lim119911rarr119909120583darr0
nabla119909(119909 120583) sube 120597119891(119909)
(iii) 0 le nabla120583(119909 120583) le 120574119899 + 120590 forall119909 isin 119877119899 120583 isin (0 +infin)
From the projected gradient method and the viscosityregularization method we introduce the following neuralnetwork to solve (24)
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(34)
where 120583(119905) = 119890minus119905 and 120576(119905) = 1(119905 + 1) with 1205760gt 0
Mathematical Problems in Engineering 7
120583intminus1
minus1
xintminus
sumsum PΩminus
minus
++
120576intab
bba
aab
pnablax
xqnabla
Figure 1 Simple block structure of proposed network (37)
By the definitions for (34) can be expressed as
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)
minus
119903
sum
119894=1
nabla119904120601 (119904 120583 (119905))
119904=119892119894(119909(119905))nabla119892
119894(119909 (119905)) minus 120576 (119905)
sdot (nablaΘ (119909 (119905)) + 120574
119899
sum
119894=1
nabla119904120579 (119904 120583)
119904=119909119894(119905)119890119894
+ 120590nabla119904120579 (119904 120583)
119904=119890119879119909(119905)
119890)]
(35)
where
nabla119904120601 (119904 120583) =
1 if 119904 gt 120583119904 + 120583
2120583if |119904| le 120583
0 if 119904 lt minus120583
nabla119904120579 (119904 120583) =
1 if |119904| ge 120583119904
120583if |119904| lt 120583
(36)
To implement (34) by circuits we can use the reformu-lated form of (34) as follows
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
120576 (119905) = minus1205762
(119905)
(119905) = minus120583 (119905)
(37)
Equation (34) can be seen as a network with three input andthree output variables that are 119909(119905) 120583(119905) and 120576(119905) A simpleblock structure of the proposed network (37) implementedby circuits is presented in Figure 1
42 Theoretical Analysis In this subsection we study somenecessary dynamical and optimality properties of proposednetwork (34) for solving (24)
The global existence of the solutions of (34) is a necessarycondition for its usability in optimization With an initial
point 1199090isin Ω the solution of (34) is global existentMoreover
the uniqueness of the solution of (34) with 1199090isin Ω is proved
under some conditions The proposed network (34) can beimplemented in circuits and mathematical software Thenthe feasibility and optimality of network (34) for optimizationproblem (24) are proved theoretically
We call 119909 [0 119879) with T gt 0 a solution of (34) if 119909 isabsolutely continuous on [0 119879) and satisfies (34) everywhere
Proposition 6 For any initial point 1199090isin Ω there is a global
solution of (34) defined on [0 +infin) and it satisfies
119909 (119905) isin Ω forall119905 isin [0 +infin) (38)
Proof Since the right hand function in network (34) iscontinuous about 119909 and 119905 there are 119879 gt 0 and an absolutecontinuous function 119909 [0 119879) rarr 119877
119899 such that 119909(119905) satisfies(34) for all 119905 isin [0 119879)
Denote 120579(119905) = 119875Ω[119909(119905) minus nabla
119909(119909(119905) 120583(119905)) minus 120576(119905)nabla
119909(119909(119905)
120583(119905))] Then (34) can be rewritten as
(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)
A simple integration procedure of the above equation gives
119909 (119905) = 119890minus119905
1199090+ 119890
minus119905
int
119905
0
120579 (119904) 119890119904
119889119904 (40)
which can be rewritten as
119909 (119905) = 119890minus119905
1199090+ (1 minus 119890
minus119905
)int
119905
0
120579 (119904)119890119904
119890119905 minus 1119889119904 (41)
By int1199050
119890119904
(119890119905
minus 1)119889119904 = 1 120579(119904) isin Ω forall0 le 119904 le 119905 and 1199090isin Ω we
obtain that 119909(119905) isin Ω forall119905 isin [0 119879)By the boundedness of Ω and the extension theory this
solution of (34) can be extended Thus the solution of (34)with initial point 119909
0isin Ω is globally existent Similarly we can
obtain that
119909 (119905) isin Ω forall119905 isin [0 +infin) (42)
Some Lipschitz condition is often used to guaranteethe uniqueness of the solution of a neural network Inwhat follows we give a sufficient condition to ensure theuniqueness of the solution of (34) with initial point 119909
0isin Ω
Proposition 7 For any initial point 1199090isin Ω if nabla
119909(sdot 120583) and
nabla119909(sdot 120583) are locally Lipschitz continuous for any fixed 120583 isin
(0 1] then the solution of neural network (34) is unique
Proof By Proposition 6 the solutions of (34) with initialpoint119909
0isin Ω exist globally and satisfy119909(119905) isin Ωforall119905 isin [0 +infin)
Suppose that there exist two solutions 119909 [0 +infin) rarr 119877119899
and 119910 [0 +infin) rarr 119877119899 of (34) with initial point 119909
0isin Ω
and suppose there exists such that = inf119905ge0119909(119905) =119910(119905)
119905 By theboundedness of Ω there is 119877 gt 0 such that 119909(119905) le 119877 and119910(119905) le 119877 forall119905 isin [0 +infin)
8 Mathematical Problems in Engineering
Then there is 119871 gt 0 such that10038161003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla
119909 (119910 (119905) 120583 (119905))
10038161003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
1003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla119909 (119910 (119905) 120583 (119905))1003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
(43)
Differentiating (12)119909(119905)minus119910(119905)2 along the two solutionsof (34) by the Lipschitz continuity of 119875
Ωand (43) we have
119889
119889119905
1
2
1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817
2
= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩
le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817
2
(44)
Applying Gronwallrsquos inequality into the integration of theabove inequality it gives 119909(119905) = 119910(119905) forall119905 isin [119905 + 1] whichleads to a contradiction Therefore the solution of (34) withinitial point 119909
0isin Ω is unique
Lyapunov method is employed to analyze the perfor-mance of (34) Here we introduce the following two Lya-punov energy functions
119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
119866 (119909 119905) =1
2dist2 (119909M) + (119909 120583 (119905))
+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
(45)
The above two Lyapunov functions satisfy the followingestimations along the solutions of (34)
Lemma 8 (i) The derivative of 119864(119909 119905) along the solution of(34) can be calculated by
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
(46)
(ii) The derivative of 119866(119909 119905) along the solution of (34) can becalculated by
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(47)
Proof (i) Differentiating119864(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119864 (119909 (119905) 119905) = ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(48)
Equation (34) can be rewritten as
(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(49)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119909(119905) using (49) and (10) we have
⟨119909 (119905) minus nabla119909 (119909 (119905) 120583 (119905)) minus 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
minus (119905) minus 119909 (119905) (119905) + 119909 (119905) minus 119909 (119905)⟩ ge 0
(50)
which follows the fact that
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
le minus (119905)2
forall119905 ge 0
(51)
Combining (48) and (51) we get that
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
(52)
From Lemmas 4 and 5 (119905) = minus119890minus119905 and 120576(119905) lt 0 we obtainthe estimation in (i)
(ii) Differentiating 119866(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩
+ ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(53)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119875M(119909(119905)) using (10) and (49) we have
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0
(54)
which can be rewritten as
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
minus (119905)2
(55)
Mathematical Problems in Engineering 9
Combining (53) and (55) we get
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905) (119891 (119909 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(56)
Similar to the analysis in (i) we obtain the estimation in(ii)
Next we prove the efficiency of proposed network (34) forsolving optimization problem (24) where the convergencefeasibility of the proposed network is a basic property
Theorem 9 For initial point 1199090isin Ω any solution 119909 [0 +infin)
rarr 119877119899 of (34) satisfies
lim119905rarr+infin
dist (119909 (119905) F) = 0 (57)
Proof Denote 119909 [0 +infin) rarr 119877119899 as a global solution of (34)
with initial point 1199090isin Ω
From 119909(119905) isin Ω forall119905 isin [0 +infin) and Lemma 8 119864(119909(119905) 119905) isnonincreasing along the solution of (34) Using119864(119909(119905) 119905) ge 0forall119905 isin [0 +infin) we confirm that
lim119905rarr+infin
119864 (119909 (119905) 119905) exists (58)
From 119909(119905) isin Ω forall119905 isin [0 +infin) and lim119905rarr+infin
120576(119905) = 0 wehave
lim119905rarr+infin
120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ
119891) = 0 (59)
which implies
lim119905rarr+infin
119864 (119909 (119905) 119905) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) (60)
Inwhat follows wewill prove that lim119905rarr+infin
(119909(119905) 120583(119905)) =
0 Arguing by contradiction we assume that
lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 119888 = 0 (61)
By (119909(119905) 120583(119905)) ge 119901(119909(119905)) ge 0 forall119905 isin [0 +infin) we have 119888 gt 0which follows the fact that there is 119879
1ge 0 such that
(119909 (119905) 120583 (119905)) ge119888
2 forall119905 isin [119879
1 +infin) (62)
From Lemma 4 we obtain
1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)
which implies that there exists 1198792gt 119879
1such that
(119875M (119909 (119905)) 120583 (119905)) le119888
8 (64)
Since Ω is bounded and 119909(119905) isin Ω forall119905 isin [0 +infin) byLemma 5 there is 119877 gt 0 such that10038161003816100381610038161003816⟨nabla
119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
10038161003816100381610038161003816le 119877
forall119905 isin [0 +infin)
(65)
Then
lim119905rarr+infin
120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)
which implies that there is 1198793ge 119879
2such that
10038161003816100381610038161003816120576 (119905) ⟨nabla
119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩
10038161003816100381610038161003816le119888
8
forall119905 isin [1198792 +infin)
(67)
From Lemma 8 and the convexity of (sdot 120583) for any fixed120583 gt 0 we obtain that
119889
119889119905119866 (119909 (119905) 119905) le minus ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)
sdot ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(68)
By (62) (67) and (68) we obtain
119889
119889119905119866 (119909 (119905) 119905) le minus
119888
4 forall119905 isin [119879
3 +infin) (69)
Integrating the above inequality from 1198793to 119905 (gt119879
3) we
have
119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879
3) minus
119888
4(119905 minus 119879
3)
forall119905 isin [1198793 +infin)
(70)
Thus
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (71)
which leads to a contradiction with 119866(119909 119905) ge 0 for all 119909 isin 119877119899and 119905 isin [0 +infin) Therefore
lim119905rarr+infin
119901 (119909 (119905)) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 0 (72)
which guarantees that
lim119905rarr+infin
dist (119909 (119905) F) = 0 (73)
The following theorem indicates that any accumulationpoint of the solutions of (34) is just an optimal solution of(34)
10 Mathematical Problems in Engineering
Theorem 10 For initial point 1199090isin Ω any solution 119909 of (34)
is convergent to the optimal solution setM that is
lim119905rarr+infin
dist (119909 (119905) M) = 0 (74)
Proof From (63) (68) and (119909(119905) 120583(119905)) ge 0 forall119905 isin [0 +infin)we have
119889
119889119905119866 (119909 (119905) 119905)
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
+ 119898120583 (119905)
(75)
by (119905) = minus120583(119905) which can be rewritten as
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(76)
Denote
119868 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le 0
119869 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
gt 0
(77)
Owning to the continuity of ⟨nabla119909(119909(119905) 120583(119905)) 119909(119905)minus119875M(119909(119905))⟩
on [0 +infin) 119868 and 119869 are closed and open in [0 +infin) respec-tively
Case 1 In this case we assume that there exists 119879 ge 0 suchthat 119905 isin 119868 forall119905 isin [119879 +infin)
From the definition on 119868 and the pseudoconvexity of(sdot 120583) on 119877119899 we have
(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))
forall119905 isin [119879 +infin)
(78)
which implies
lim sup119905rarr+infin
119891 (119909 (119905)) le min119909isinF
119891 (119909) (79)
ByTheorem 9 we confirm that
lim119905rarr+infin
dist (119909 (119905) M) = 0 (80)
Case 2 In this case we assume that there exists 119879 ge 0 such
that 119905 isin 119869 forall119905 isin [119879 +infin) which means that
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] lt 0 forall119905 isin [119879 +infin) (81)
Then
lim119905rarr+infin
[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)
Since lim119905rarr+infin
119864(119909(119905) 119905) exists and lim119905rarr+infin
120583(119905) = 0 weobtain that
lim119905rarr+infin
dist (119909 (119905) M) exists (83)
Suppose
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889
gt 0
(84)
Then there is 1198791ge such that
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge
119889
2
forall119905 isin [1198791 +infin)
(85)
Then from (76) we have
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus
119889
2120576 (119905)
forall119905 isin [1198791 +infin)
(86)
Integrating the above inequality from 1198791to 119905 (gt119879
1) we
have
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879
1) + 119898120583 (119879
1)
+ int
119905
1198791
minus119889
2120576 (119904) 119889119904
(87)
Let 119905 rarr +infin in the above inequality then we have
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (88)
which leads to a contraction with the boundedness frombelow 119866(119909(119905) 119905) on [0 +infin) Thus
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)
which follows the fact that there is an increasing sequence 119905119899
such that 119905119899rarr +infin and
lim119899rarr+infin
⟨nabla119909 (119909 (119905
119899) 120583 (119905
119899)) 119909 (119905
119899) minus 119875M (119909 (119905
119899))⟩
= 0
(90)
By 119909(119905119899) sube Ω and Ω which is bounded there are 119909lowastlowast isin
Ω and a subsequence of 119909(119905119899) (denoted as 119909(119905
119899119896)) such that
lim119896rarr+infin
119905119899119896= +infin and lim
119896rarr+infin119909(119905
119899119896) = 119909
lowastlowast By Lemma 5there is 120585(119909lowastlowast) isin 120597119891(119909lowastlowast) such that
⟨120585 (119909lowastlowast
) 119909lowastlowast
minus 119875M (119909lowastlowast
)⟩ = 0 (91)
From the pseudoconvexity of 119891 and the above inequalitywe have
119891 (119909lowastlowast
) le 119891 (119875M (119909lowastlowast
)) (92)
Mathematical Problems in Engineering 11
Table 1 Experimental data set in MDF hot-pressing
Regression parameter 120574 120590 119862 MSE DF SEMOR 0001 5 30 00307 09543 minus08299
MOE 0001 6 40 00642 09063 minus17060
IBS 0001 5 40 00451 09256 minus06388
By Theorem 9 we have 119909lowastlowast isin F Therefore 119909lowastlowast isin Mwhich implies that
lim119896rarr+infin
dist (119909 (119905119899119896) M) = 0 (93)
Combining the above results with (83) we conclude thatlim
119905rarr+infindist(119909(119905)M) = 0
Case 3 In this case we assume that both 119868 and 119869 areunbounded
For 119905 isin 119868 similar to the analysis in Case 1 we have
lim119905rarr+infin119905isin119868
dist (119909 (119905) M) = 0 (94)
For 119905 isin 119869 define 120591(119905) = sup119904le119905119904isin119868
119904 Then 120591(119905) isin 119868 and(120591(119905) 119905] sube 119869 By the unboundedness of 119868 and 119869 we have
lim119905isin119869119905rarr+infin
120591 (119905) = +infin (95)
From (81) and the continuity of 119866(119909(119905) 119905) on [0 +infin) wehave
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))
+ 119898120583 (120591 (119905)) forall119905 isin 119869
(96)
By 120591(119905) isin 119868 (94) and (95) we have
lim sup119905rarr+infin119905isin119869
[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le lim119905rarr+infin119905isin119869
[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0
(97)
which gives
lim119905rarr+infin119905isin119869
119866 (119909 (119905) 119905) = 0 (98)
From the definition of 119866(119909 119905) and the above result wehave
lim119905rarr+infin119905isin119869
dist (119909 (119905) M) = 0 (99)
Therefore from (94) and (99) we obtain
lim119905rarr+infin
dist (119909 (119905) M) = 0 (100)
5 Numerical Experiments
In this section we test the proposed neural network (34)for solving problem (24) which is brought forward from
the MDF continuous hot-pressing process Based on theexisting data set we use the established theories and proposedneural network (34) to build the relationships between themain qualities of the hot-pressing plate and some relativetechnology parameters from optimization problem (18)Then based on optimization problem (22) we will useproposed network (34) to solve the optimal values of thetechnology parameters in hot-pressing system for optimizingthe qualities of the hot-pressing plate All these numericalexperiments validate the good performance of the proposednetwork in this paper
The numerical testing was carried out on a Lenovo PC(300GHz 200GB of RAM) with the use of Matlab 74 Andwe use ode23 to realize the neural network (34) in Matlab
51 Construction Relation Models in MDF Continuous Hot-Pressing Process In this part by considered optimizationproblem (18) and network (34) we build the relation modelwhich takes the hot-pressing temperature (TE) hot-pressingpressure (PR) hot-pressing time (TI) and moisture content(MC) of slab as the argument variables and the MOR MOEand IBS indexes of MDF as the dependent variables Thenumerical results show the good fitting of the built modelsfor the data set where the data set is given in Table 4
In order to use the data in Table 4 we first normalize theminto [0 1] And we use the mean square error (MSE) degreeof fitting (DF) and sufficient evaluation (SE) to evaluate thenumerical results where
MSE MSE = radicsum119899
119894=1(119909
119894minus 119909
lowast
119894)2
119899
DF DF = 1 minus radic(sum119899
119894=1(119909
119894minus 119909
lowast
119894)2
sum119899
119894=11199092
119894
)
SE SE = 12119909119879
119876119909 minus 119910119879
119909
(101)
where 119909 and 119909lowast indicate the actual value and predicted valueand 119876 and 119910 are defined as in (18) The smaller the MSE andthe SE and the closer the DF to 1 the better the regressionresult Moreover the SE function is the objective function in(15)
Based on the Gaussian radial basis function kernel thevalues of the initial parameters in problem (18) are given inTable 1 With a random initial point 119909
0isin Ω the numerical
results with respect to the obtained solution are also listed inTable 1 Figures 2ndash4 illustrate the fitting effect of the MORMOE and IBS values by using the proposed network (34) forsolving problem (18) And the parameters of the regressionfunctions in (19) are shown in Table 5
12 Mathematical Problems in Engineering
Table 2 Criterion of physical and mechanical performance indexes
Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45
IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045
MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700
2520151050
OriginalPredicted
0
05
1
15
2
25
3
35
4
5
45
Figure 2 Normalized sample fitting results of network (34) for theregression of MOR
Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5
52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2
521 Case 1
Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 3 Normalized sample fitting results of network (34) for theregression of MOE
By the information in Table 2 we let ]1= 200 ]
2= 4
]3= 7 and ]
4= 12 and choose 119910lowast
1= 22 119910lowast
2= 2500 120580
1= 8
1205761= 8 120580
2= 500 and 120576
2= 500 in optimization problem (21)
Then Ω = 119909 isin 1198774
0 le 1199091le ]
1 0 le 119909
2le ]
2 0 le 119909
3le
]3 0 le 119909
4le ]
4 and we can let
1198921(119909) = minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) =
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
1198923(119909) = minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) =
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(102)
By the proposed network (34) for solving (21) we obtainthe optimal solution
119909lowast
= (1594725 31022 64866 60061)119879
(103)
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
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Mathematical Problems in Engineering
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Chen and Mangasarian constructed a class of smoothapproximations of the function (119904)
+fl max0 119904 by convo-
lution [20 24] as follows Let 120588 119877 rarr 119877+be a piecewise
continuous density function satisfying
120588 (119905) = 120588 (minus119905)
120581 fl int
infin
minusinfin
|119905| 120588 (119905) 119889119905 lt infin
(1)
Then
120601 (119905 120583) fl int
infin
minusinfin
(119904 minus 120583119905)+120588 (119905) 119889119905 (2)
from 119877 times 119877+to 119877
+is well defined
By different density functions many popular smoothingfunctions of (119904)
+can be derived such as
1206011(119904 120583) = 119904 + 120583 ln (1 + 119890minus119904120583)
1206012(119904 120583) =
1
2(119904 + radic1199042 + 41205832)
1206013(119904 120583) =
max 0 119904 if |119904| gt 120583
(119904 + 120583)2
4120583if |119904| le 120583
1206014(119904 120583) =
119904 +120583
2119890minus119904120583 if 119904 gt 0
120583
2119890minus119904120583 if 119904 le 0
(3)
where 1206011is the neural networks smoothing function 120601
2is
called the CHKS (Chen-Harker-Kanzow-Smale) smoothingfunction120601
3is called the uniform smoothing function and120601
4
is called the Picard smoothing function The four functionsbelong to the class of the Chen-Mangasarian smoothingfunctions
Many nonsmooth functions can be reformulated by usingthe plus function We list some of them as follows
|119904| = (119904)++ (minus119904)
+
max (119904 119905) = 119904 + (119905 minus 119904)+
min (119904 119905) = 119904 minus (119904 minus 119905)+
mid (119904 ℓ 119906) = min (max (ℓ 119904) 119906) for given ℓ 119906
(4)
So we can define a smoothing function for the abovenonsmooth functions by a smoothing function of (119904)
+
From Theorem 961 and Corollary 847(b) in [20] when119892 119877
119899
rarr 119877 is locally Lipschitz continuous at 119909 thesubdifferential associated with a smoothing function
119866(119909) = con V | nabla
119909 (119909
119896
120583119896) 997888rarr V for 119909119896 997888rarr 119909 120583
119896
darr 0
(5)
is nonempty and bounded and 120597119892(119909) sube 119866(119909) where ldquoconrdquo
denotes the convex hull In [20 23] it is shown that manysmoothing functions satisfy the gradient consistency
120597119892 (119909) = 119866(119909) (6)
which is an important property of the smoothing methodsand guarantees the convergence of smoothing methods withadaptive updating schemes of smoothing parameters to astationary point of the original problem
23 Pseudoconvex Function Pseudoconvex function is a classof functions which may be nonsmooth or nonconvex butbrings us the opportunity to find the optimal solutions
Definition 2 (see [25]) Let X be a nonempty convex subsetof 119877119899 A function 120601 is said to be pseudoconvex on X if forany 119909 119910 isin X one has
exist120585 (119909) isin 120597120601 (119909)
st ⟨120585 (119909) 119910 minus 119909⟩ ge 0 997904rArr
120601 (119910) ge 120601 (119909)
(7)
Many nonconvex functions in application are pseudocon-vex such as the Butterworth filter function fraction functionand density functionOf particular interest in this paper is thefact that the Gaussian function
ℎ (119909) = minus exp(minus119899
sum
119894=1
1199092
119894
1205902
119894
) (8)
with 120590119894gt 0 119894 = 1 2 119899 is pseudoconvex on 119877119899
24 Project Operator LetX be a closed convex subset of 119877119899Then the projection operator toX at 119909 is defined by
119875X (119909) = argmin119906isinX
119906 minus 1199092 (9)
and satisfies the following inequalities
⟨V minus 119875X (V) 119875X (V) minus 119906⟩ ge 0 forallV isin 119877119899 119906 isin X
1003817100381710038171003817119875X (119906) minus 119875X (V)1003817100381710038171003817 le 119906 minus V forall119906 V isin 119877119899
(10)
(i) Suppose Ω = 119909 isin 119877119899 119906 le 119909 le V with 119906 V isin 119877119899 cup plusmninfin
then 119875Ω(119909) = (119901
1 119901
2 119901
119899)119879 can be expressed by
119901119894=
119906119894119909119894lt 119906
119894
119909119894119906119894le 119909
119894le V
119894
V119894
V119894lt 119909
119894
(11)
(ii) Suppose Ω = 119909 119860119909 = 119887 with 119860 isin 119877119903times119899 of full row rank
and 119887 isin 119877119903 then 119875Ω(119909) = 119909 minus 119860
119879
(119860119860119879
)minus1
(119860119909 minus 119887)
3 Optimization Problems in MDF ContinuousHot-Pressing Process
In this section we will give the optimization model consid-ered in this paper First by the optimization and supportvector machine theories we show the optimization modelsfor building up the relationships in MDF continuous hot-pressing process Then another optimization model for
4 Mathematical Problems in Engineering
optimizing the parameters in the MDF continuous hot-pressing process is obtained Thus we express these twokinds of problems into a uniform formulation which is theoptimization problem considered in Section 3
31 Optimization Problem for Building up the Models of MDFContinuousHot-Pressing Process DenoteX sub 119877
119901 andY sub 119877
as two sets where 119909119894 = (1199091198941 119909
119894
2 119909
119894
119901) isin X is the attribute
vector on behalf of the hot-pressing plate properties 119910119894 isin Yindicates the values of the qualities of hot-pressing plate Let119878 = (119904
1
1199042
119904119899
) be the training data set of hot-pressingprocess where 119904119894 = (119909119894 119910119894) obeys the unknown distributionand is IID (independent and identically distributed) Basedon the support vector machine theory and the training dataset we would like to find a nonlinear function 119910 = 119891(119909) suchthat it approximates the training data set as much as possible
With the kernel function 119870 and the Huber loss functionfrom the theory in [21] the dual optimization problem of theprimal optimization problem of SVM is given as
min120572120573
1
2
119899
sum
119894119895=1
(120573119894minus 120572
119894) (120573
119895minus 120572
119895)119870 (119909
119894
119909119895
)
minus
119899
sum
119894=1
(120573119894minus 120572
119894) 119910
119894
+120576
2119862
119899
sum
119894=1
(120573119894minus 120572
119894)2
st119899
sum
119894=1
(120573119894minus 120572
119894) = 0
0 le 120573119894 120572
119894le 119862 119894 = 1 2 119899
(12)
Denote 120572lowast119894and 120573lowast
119894as the optimal solutions of (12) Then the
regression function 119891 can be expressed by
119891 (119909) =
119899
sum
119894=1
(120573lowast
119894minus 120572
lowast
119894)119870 (119909
119894
119909) + 119887 (13)
where 119887 can be calculated by one of the following twomethods
119887 = 119910119895
minus
119899
sum
119894=1
(120573lowast
119894minus 120572
lowast
119894) ⟨119909
119894
119909119895
⟩ + 120576 120572lowast
119895isin (0 119862)
119887 = 119910119895
minus
119899
sum
119894=1
(120573lowast
119894minus 120572
lowast
119894) ⟨119909
119894
119909119895
⟩ minus 120576 120572lowast
119895isin (0 119862)
(14)
Denote 119909119894= 120573
119894minus 120572
119894in (12) then (12) can be reformulated
as
min 1
2119909119879
119876119909 minus 119910119879
119909
st 119890119879
119909 = 0 minus 119862119890 le 119909 le 119862119890
(15)
where 119909 = (1199091 119909
2 119909
119899)119879 119910 = (1199101 1199102 119910119899)119879 119890 = (1 1
1)119879
isin 119877119899 and 119876 = K+(1205762119862)119868
119899withK = (119870(119909
119894
119909119895
))119899
119894119895=1
For the optimal solution 119909lowast = (119909lowast1 119909
lowast
2 119909
lowast
119899) of (15) we call
119909119894 a support vector if 119909lowast
119894= 0 The optimal solution of (15)
solved by the originalmethods often has almost 100 supportvectors which increases the complexity of the relationmodelslargely Thus we introduce the problem deduced by (15) thatis
min 1
2119909119879
119876119909 minus 119910119879
119909 + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816
st 119890119879
119909 = 0 minus 119862119890 le 119909 le 119862119890
(16)
with 120574 gt 0 In (43) 120574sum119899
119894=1|119909119894| is often called the regulation
term which is used to control the number of its supportvectors
Define the penalty function
119875 (119909) =
1003817100381710038171003817100381710038171003817
1
119899119890119890119879
119909
1003817100381710038171003817100381710038171003817=radic119899
119899
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119894=1
119909119894
1003816100381610038161003816100381610038161003816100381610038161003816
(17)
and then 119909 119875(119909) le 0 = 119909 119890119879119909 = 0 From [19 Proposition243] 119909 is an optimal solution of (43) if and only if it is anoptimal solution of the following problem
min 1
2119909119879
119876119909 minus 119910119879
119909 + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816 + 120590
10038161003816100381610038161003816119890119879
11990910038161003816100381610038161003816
st minus 119862119890 le 119909 le 119862119890
(18)
where 120590 = 119862119876 + 119910radic119899 + 120574 + 1 Thus we can build up therelation models of hot-pressing process by solving problem(18)
32 Optimization Problem for Optimizing the Parameters inMDF Hot-Pressing Process Based on the relationships builtup in Section 31 we focus on themodulus of rupture (MOR)modulus of elasticity (MOE) and internal bonding strength(IBS) of hot-pressing plate by optimizing the process param-eters and slab attributes Suppose the regression functions ofMORMOS and IBSwith respect to some relative parametersbased on the Gaussian radial basis function kernel are
1198911(119909) fl
119899
sum
119894=1
120588lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) + 119887
1(MOR)
1198912(119909) fl
119899
sum
119894=1
120579lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) + 119887
2(MOS)
1198913(119909) fl
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) + 119887
3(IBS)
(19)
and based on the linear polynomial kernel are
1198911(119909) fl
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1(MOR)
1198912(119909) fl
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2(MOS)
1198913(119909) fl
119899
sum
119894=1
120581lowast
119894⟨119911
119894 119909⟩ + 119887
3(IBS)
(20)
Mathematical Problems in Engineering 5
where 1199111 119911
2 119911
119899are the variables in the data set for
regression and 119909 = (1199091 119909
2 119909
3 119909
4)119879 indicates the indepen-
dent variable with hot-pressing temperature 1199091 hot-pressing
pressure 1199092 hot-pressing time 119909
3 and moisture content 119909
4
The regression functions in (19) satisfy the following twoproperties
(i) 119891119894is continuously differentiable on 119877119899 119894 = 1 2 3
(ii) 119891119894and minus119891
119894are not convex but minus119891
119894is pseudoconvex on
119877119899 119894 = 1 2 3
And the regression functions in (20) satisfy the following twoproperties
(i) 119891119894is continuously differentiable on 119877119899 119894 = 1 2 3
(ii) 119891119894and minus119891
119894are convex on 119877119899 119894 = 1 2 3
From the physical significance of MOR MOS and IBSwe suppose that the larger the numbers of MOR MOS andIBS the better the quality of hot-pressing plate In order toadopt the different demand on the indexes of the hot-pressingplate in different applications we consider the following twocases in this part
First we focus onmaximizing a single performance indexof the hot-pressing plate when the other two performanceindexes are within the certain areas If we want to optimizethe IBS the corresponding optimization model for this casecan be expressed by
min minus 1198913(119909)
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
minus 1205801le 119891
1(119909) minus 119910
lowast
1le 120576
1
minus 1205802le 119891
2(119909) minus 119910
lowast
2le 120576
2
(21)
where ]1 ]2 ]3 ]4gt 0 indicate the upper bounds of hot-
pressing temperature hot-pressing pressure hot-pressingtime andmoisture content and [119910lowast
1minus120580
1 119910
lowast
1+120576
1] [119910lowast
2minus120580
2 119910
lowast
2+
1205762] are the feasible regions of MOR and MOS respectivelyIn order to let problem (21) be solved effectively we
let the objective function 1198913be with the Gaussian radial
basis function and the regression functions 1198911and 119891
2in the
constraints are with the linear polynomial kernel Then (21)is
min minus
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) minus 119887
3
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
minus 1205801le
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1le 120576
1
minus 1205802le
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2le 120576
2
(22)
which is a pseudoconvex optimization problem with convexconstraints
Second we would like to optimize the MOR MOS andIBS synthetically For this demand we consider the followingoptimization model
min minus1205821
119891lowast
1
1198911(119909) minus
1205822
119891lowast
2
1198912(119909) minus
1205823
119891lowast
3
1198913(119909)
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
(23)
where 1205821 120582
2 120582
3gt 0 indicate the importance of MOR MOS
and IBS ]1 ]2 ]3 ]4are with the same meaning as in (21)
and 119891lowast1 119891lowast
2 and 119891lowast
3are the expected values of MOR MOS
and IBS In particular if MOR MOS and IBS are with thesame importance in the quality of the hot-pressing plate wecan let 120582
1= 120582
2= 120582
3= 13 and we can let 120582
1= 12
1205822= 13 and 120582
3= 16 if the importance of MOR MOS
and IBS is strictly monotone decreasing Similar to the kernelfunctions in problem (21) we let119891
1119891
2 and119891
3in problem (23)
be with the Gaussian radial basis function which means thatproblem (23) is also a pseudoconvex optimization problemwith convex constraints
33 General Model Based on analysis in Sections 31 and32 we consider the following minimization problem in thispaper
minimize 119891 (119909) fl Θ (119909) + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816 + 120590
10038161003816100381610038161003816119890119879
11990910038161003816100381610038161003816
subject to 119909 isin Ω fl 119909 119886 le 119909 le 119887
119892119894(119909) le 0 119894 = 1 119898
(24)
6 Mathematical Problems in Engineering
where 120574 120590 ge 0 119886 119887 isin 119877119899 with 119886 lt 119887 Θ 119877
119899
rarr 119877
is continuously differentiable and pseudoconvex on 119877119899 and119892119894 119877
119899
rarr 119877 (119894 = 1 2 119898) is continuously differentiableand convex on 119877119899
On the one hand when Θ(119909) = (12)119909119879119876119909 minus 119910119879119909 119886 flminus119862119890 119887 fl 119862119890 and 120574 and 120590 are defined as in (18) then problem(24) without 119892
119894reduces to problem (18) On the other hand
if we let
Θ (119909) fl minus
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) minus 119887
3
119886 fl 0
119887 fl (]1 ]2 ]3 ]4)119879
(25)
1198921(119909) fl minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) fl
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
(26)
1198923(119909) fl minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) fl
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(27)
then problem (24) reduces to problem (22) Similar reformu-lation can be done for problem (23) by (24)
Therefore problem (24) considered in this paper includesthe optimizationmodels for building up the relationships andoptimizing the relative parameters in MDF continuous hot-pressing process
Inwhat follows we denoteF as the feasible region of (24)that is
F fl 119909 isin Ω 119892119894(119909) le 0 119894 = 1 2 119898 (28)
andM is the optimal solution set of (24)
4 Main Results
41 Proposed Neural Network In this subsection we proposea one-layer recurrent neural network for solving problem(24) where we combine the penalty function and projectionmethods to solve the constraints and use the smoothingtechniques to overcome the nonsmoothness of the objectivefunction and penalty function
Define the penalty function
119901 (119909) =
119898
sum
119894=1
max 0 119892119894(119909) (29)
Then 119909 isin Ω 119901(119909) le 0 = FFrom the smoothing functions in (25) for the plus
function we define the smoothing function of 119901 as
(119909 120583) =
119898
sum
119894=1
120601 (119892119894(119909) 120583) (30)
where
120601 (119904 120583) =
max 0 119904 if |119904| gt 120583
(119904 + 120583)2
4120583if |119904| le 120583
(31)
Form the results in [23] 120601(119904 120583) owns the followingproperties
Lemma 3 (see [23]) (i) For any 119904 isin 119877 120601(119904 sdot) is continuouslydifferentiable and 120601(sdot 120583) is also continuously differentiable forany fixed 120583 gt 0
(ii) 0 le nabla120583120601(119904 120583) le 1 forall119904 isin 119877 forall120583 isin (0 +infin)
(iii) 0 le 120601(119904 120583) minus |119904| le 1205834 forall119904 isin 119877 forall120583 isin (0 +infin)(iv) 120601(sdot 120583) is convex for any fixed 120583 gt 0 and
lim119904rarr119905120583darr0
nabla119904120601(119904 120583) sube 120597max0 119905
Then has the following properties
Lemma 4 is a smoothing function 119901 and satisfies thefollowing
(i) (sdot 120583) is convex for any fixed 120583 gt 0(ii) lim
119911rarr119909120583darr0nabla119911(119911 120583) sube 120597119901(119909)
(iii) 0 le nabla120583(119909 120583) le 119898 forall119909 isin 119877119899 120583 isin (0 +infin)
Next by the smoothing function for the absolute valuefunction | sdot |
120579 (119904 120583) =
|119904| if |119904| ge 120583
1199042
2120583+120583
2if |119904| lt 120583
(32)
we define
(119909 120583) fl Θ (119909) + 120574
119899
sum
119894=1
120579 (119909119894 120583) + 120590120579 (119890
119879
119909 120583) (33)
Since 120579(119904 120583) = 120601(119904 120583) + 120601(minus119904 120583) 120579(119904 120583) owns allproperties in Lemma 8 and the following results hold
Lemma 5 is a smoothing function 119891 in (24) with thefollowing properties
(i) (sdot 120583) is pseudoconvex for any fixed 120583 gt 0
(ii) lim119911rarr119909120583darr0
nabla119909(119909 120583) sube 120597119891(119909)
(iii) 0 le nabla120583(119909 120583) le 120574119899 + 120590 forall119909 isin 119877119899 120583 isin (0 +infin)
From the projected gradient method and the viscosityregularization method we introduce the following neuralnetwork to solve (24)
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(34)
where 120583(119905) = 119890minus119905 and 120576(119905) = 1(119905 + 1) with 1205760gt 0
Mathematical Problems in Engineering 7
120583intminus1
minus1
xintminus
sumsum PΩminus
minus
++
120576intab
bba
aab
pnablax
xqnabla
Figure 1 Simple block structure of proposed network (37)
By the definitions for (34) can be expressed as
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)
minus
119903
sum
119894=1
nabla119904120601 (119904 120583 (119905))
119904=119892119894(119909(119905))nabla119892
119894(119909 (119905)) minus 120576 (119905)
sdot (nablaΘ (119909 (119905)) + 120574
119899
sum
119894=1
nabla119904120579 (119904 120583)
119904=119909119894(119905)119890119894
+ 120590nabla119904120579 (119904 120583)
119904=119890119879119909(119905)
119890)]
(35)
where
nabla119904120601 (119904 120583) =
1 if 119904 gt 120583119904 + 120583
2120583if |119904| le 120583
0 if 119904 lt minus120583
nabla119904120579 (119904 120583) =
1 if |119904| ge 120583119904
120583if |119904| lt 120583
(36)
To implement (34) by circuits we can use the reformu-lated form of (34) as follows
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
120576 (119905) = minus1205762
(119905)
(119905) = minus120583 (119905)
(37)
Equation (34) can be seen as a network with three input andthree output variables that are 119909(119905) 120583(119905) and 120576(119905) A simpleblock structure of the proposed network (37) implementedby circuits is presented in Figure 1
42 Theoretical Analysis In this subsection we study somenecessary dynamical and optimality properties of proposednetwork (34) for solving (24)
The global existence of the solutions of (34) is a necessarycondition for its usability in optimization With an initial
point 1199090isin Ω the solution of (34) is global existentMoreover
the uniqueness of the solution of (34) with 1199090isin Ω is proved
under some conditions The proposed network (34) can beimplemented in circuits and mathematical software Thenthe feasibility and optimality of network (34) for optimizationproblem (24) are proved theoretically
We call 119909 [0 119879) with T gt 0 a solution of (34) if 119909 isabsolutely continuous on [0 119879) and satisfies (34) everywhere
Proposition 6 For any initial point 1199090isin Ω there is a global
solution of (34) defined on [0 +infin) and it satisfies
119909 (119905) isin Ω forall119905 isin [0 +infin) (38)
Proof Since the right hand function in network (34) iscontinuous about 119909 and 119905 there are 119879 gt 0 and an absolutecontinuous function 119909 [0 119879) rarr 119877
119899 such that 119909(119905) satisfies(34) for all 119905 isin [0 119879)
Denote 120579(119905) = 119875Ω[119909(119905) minus nabla
119909(119909(119905) 120583(119905)) minus 120576(119905)nabla
119909(119909(119905)
120583(119905))] Then (34) can be rewritten as
(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)
A simple integration procedure of the above equation gives
119909 (119905) = 119890minus119905
1199090+ 119890
minus119905
int
119905
0
120579 (119904) 119890119904
119889119904 (40)
which can be rewritten as
119909 (119905) = 119890minus119905
1199090+ (1 minus 119890
minus119905
)int
119905
0
120579 (119904)119890119904
119890119905 minus 1119889119904 (41)
By int1199050
119890119904
(119890119905
minus 1)119889119904 = 1 120579(119904) isin Ω forall0 le 119904 le 119905 and 1199090isin Ω we
obtain that 119909(119905) isin Ω forall119905 isin [0 119879)By the boundedness of Ω and the extension theory this
solution of (34) can be extended Thus the solution of (34)with initial point 119909
0isin Ω is globally existent Similarly we can
obtain that
119909 (119905) isin Ω forall119905 isin [0 +infin) (42)
Some Lipschitz condition is often used to guaranteethe uniqueness of the solution of a neural network Inwhat follows we give a sufficient condition to ensure theuniqueness of the solution of (34) with initial point 119909
0isin Ω
Proposition 7 For any initial point 1199090isin Ω if nabla
119909(sdot 120583) and
nabla119909(sdot 120583) are locally Lipschitz continuous for any fixed 120583 isin
(0 1] then the solution of neural network (34) is unique
Proof By Proposition 6 the solutions of (34) with initialpoint119909
0isin Ω exist globally and satisfy119909(119905) isin Ωforall119905 isin [0 +infin)
Suppose that there exist two solutions 119909 [0 +infin) rarr 119877119899
and 119910 [0 +infin) rarr 119877119899 of (34) with initial point 119909
0isin Ω
and suppose there exists such that = inf119905ge0119909(119905) =119910(119905)
119905 By theboundedness of Ω there is 119877 gt 0 such that 119909(119905) le 119877 and119910(119905) le 119877 forall119905 isin [0 +infin)
8 Mathematical Problems in Engineering
Then there is 119871 gt 0 such that10038161003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla
119909 (119910 (119905) 120583 (119905))
10038161003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
1003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla119909 (119910 (119905) 120583 (119905))1003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
(43)
Differentiating (12)119909(119905)minus119910(119905)2 along the two solutionsof (34) by the Lipschitz continuity of 119875
Ωand (43) we have
119889
119889119905
1
2
1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817
2
= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩
le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817
2
(44)
Applying Gronwallrsquos inequality into the integration of theabove inequality it gives 119909(119905) = 119910(119905) forall119905 isin [119905 + 1] whichleads to a contradiction Therefore the solution of (34) withinitial point 119909
0isin Ω is unique
Lyapunov method is employed to analyze the perfor-mance of (34) Here we introduce the following two Lya-punov energy functions
119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
119866 (119909 119905) =1
2dist2 (119909M) + (119909 120583 (119905))
+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
(45)
The above two Lyapunov functions satisfy the followingestimations along the solutions of (34)
Lemma 8 (i) The derivative of 119864(119909 119905) along the solution of(34) can be calculated by
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
(46)
(ii) The derivative of 119866(119909 119905) along the solution of (34) can becalculated by
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(47)
Proof (i) Differentiating119864(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119864 (119909 (119905) 119905) = ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(48)
Equation (34) can be rewritten as
(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(49)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119909(119905) using (49) and (10) we have
⟨119909 (119905) minus nabla119909 (119909 (119905) 120583 (119905)) minus 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
minus (119905) minus 119909 (119905) (119905) + 119909 (119905) minus 119909 (119905)⟩ ge 0
(50)
which follows the fact that
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
le minus (119905)2
forall119905 ge 0
(51)
Combining (48) and (51) we get that
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
(52)
From Lemmas 4 and 5 (119905) = minus119890minus119905 and 120576(119905) lt 0 we obtainthe estimation in (i)
(ii) Differentiating 119866(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩
+ ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(53)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119875M(119909(119905)) using (10) and (49) we have
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0
(54)
which can be rewritten as
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
minus (119905)2
(55)
Mathematical Problems in Engineering 9
Combining (53) and (55) we get
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905) (119891 (119909 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(56)
Similar to the analysis in (i) we obtain the estimation in(ii)
Next we prove the efficiency of proposed network (34) forsolving optimization problem (24) where the convergencefeasibility of the proposed network is a basic property
Theorem 9 For initial point 1199090isin Ω any solution 119909 [0 +infin)
rarr 119877119899 of (34) satisfies
lim119905rarr+infin
dist (119909 (119905) F) = 0 (57)
Proof Denote 119909 [0 +infin) rarr 119877119899 as a global solution of (34)
with initial point 1199090isin Ω
From 119909(119905) isin Ω forall119905 isin [0 +infin) and Lemma 8 119864(119909(119905) 119905) isnonincreasing along the solution of (34) Using119864(119909(119905) 119905) ge 0forall119905 isin [0 +infin) we confirm that
lim119905rarr+infin
119864 (119909 (119905) 119905) exists (58)
From 119909(119905) isin Ω forall119905 isin [0 +infin) and lim119905rarr+infin
120576(119905) = 0 wehave
lim119905rarr+infin
120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ
119891) = 0 (59)
which implies
lim119905rarr+infin
119864 (119909 (119905) 119905) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) (60)
Inwhat follows wewill prove that lim119905rarr+infin
(119909(119905) 120583(119905)) =
0 Arguing by contradiction we assume that
lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 119888 = 0 (61)
By (119909(119905) 120583(119905)) ge 119901(119909(119905)) ge 0 forall119905 isin [0 +infin) we have 119888 gt 0which follows the fact that there is 119879
1ge 0 such that
(119909 (119905) 120583 (119905)) ge119888
2 forall119905 isin [119879
1 +infin) (62)
From Lemma 4 we obtain
1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)
which implies that there exists 1198792gt 119879
1such that
(119875M (119909 (119905)) 120583 (119905)) le119888
8 (64)
Since Ω is bounded and 119909(119905) isin Ω forall119905 isin [0 +infin) byLemma 5 there is 119877 gt 0 such that10038161003816100381610038161003816⟨nabla
119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
10038161003816100381610038161003816le 119877
forall119905 isin [0 +infin)
(65)
Then
lim119905rarr+infin
120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)
which implies that there is 1198793ge 119879
2such that
10038161003816100381610038161003816120576 (119905) ⟨nabla
119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩
10038161003816100381610038161003816le119888
8
forall119905 isin [1198792 +infin)
(67)
From Lemma 8 and the convexity of (sdot 120583) for any fixed120583 gt 0 we obtain that
119889
119889119905119866 (119909 (119905) 119905) le minus ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)
sdot ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(68)
By (62) (67) and (68) we obtain
119889
119889119905119866 (119909 (119905) 119905) le minus
119888
4 forall119905 isin [119879
3 +infin) (69)
Integrating the above inequality from 1198793to 119905 (gt119879
3) we
have
119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879
3) minus
119888
4(119905 minus 119879
3)
forall119905 isin [1198793 +infin)
(70)
Thus
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (71)
which leads to a contradiction with 119866(119909 119905) ge 0 for all 119909 isin 119877119899and 119905 isin [0 +infin) Therefore
lim119905rarr+infin
119901 (119909 (119905)) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 0 (72)
which guarantees that
lim119905rarr+infin
dist (119909 (119905) F) = 0 (73)
The following theorem indicates that any accumulationpoint of the solutions of (34) is just an optimal solution of(34)
10 Mathematical Problems in Engineering
Theorem 10 For initial point 1199090isin Ω any solution 119909 of (34)
is convergent to the optimal solution setM that is
lim119905rarr+infin
dist (119909 (119905) M) = 0 (74)
Proof From (63) (68) and (119909(119905) 120583(119905)) ge 0 forall119905 isin [0 +infin)we have
119889
119889119905119866 (119909 (119905) 119905)
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
+ 119898120583 (119905)
(75)
by (119905) = minus120583(119905) which can be rewritten as
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(76)
Denote
119868 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le 0
119869 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
gt 0
(77)
Owning to the continuity of ⟨nabla119909(119909(119905) 120583(119905)) 119909(119905)minus119875M(119909(119905))⟩
on [0 +infin) 119868 and 119869 are closed and open in [0 +infin) respec-tively
Case 1 In this case we assume that there exists 119879 ge 0 suchthat 119905 isin 119868 forall119905 isin [119879 +infin)
From the definition on 119868 and the pseudoconvexity of(sdot 120583) on 119877119899 we have
(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))
forall119905 isin [119879 +infin)
(78)
which implies
lim sup119905rarr+infin
119891 (119909 (119905)) le min119909isinF
119891 (119909) (79)
ByTheorem 9 we confirm that
lim119905rarr+infin
dist (119909 (119905) M) = 0 (80)
Case 2 In this case we assume that there exists 119879 ge 0 such
that 119905 isin 119869 forall119905 isin [119879 +infin) which means that
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] lt 0 forall119905 isin [119879 +infin) (81)
Then
lim119905rarr+infin
[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)
Since lim119905rarr+infin
119864(119909(119905) 119905) exists and lim119905rarr+infin
120583(119905) = 0 weobtain that
lim119905rarr+infin
dist (119909 (119905) M) exists (83)
Suppose
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889
gt 0
(84)
Then there is 1198791ge such that
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge
119889
2
forall119905 isin [1198791 +infin)
(85)
Then from (76) we have
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus
119889
2120576 (119905)
forall119905 isin [1198791 +infin)
(86)
Integrating the above inequality from 1198791to 119905 (gt119879
1) we
have
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879
1) + 119898120583 (119879
1)
+ int
119905
1198791
minus119889
2120576 (119904) 119889119904
(87)
Let 119905 rarr +infin in the above inequality then we have
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (88)
which leads to a contraction with the boundedness frombelow 119866(119909(119905) 119905) on [0 +infin) Thus
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)
which follows the fact that there is an increasing sequence 119905119899
such that 119905119899rarr +infin and
lim119899rarr+infin
⟨nabla119909 (119909 (119905
119899) 120583 (119905
119899)) 119909 (119905
119899) minus 119875M (119909 (119905
119899))⟩
= 0
(90)
By 119909(119905119899) sube Ω and Ω which is bounded there are 119909lowastlowast isin
Ω and a subsequence of 119909(119905119899) (denoted as 119909(119905
119899119896)) such that
lim119896rarr+infin
119905119899119896= +infin and lim
119896rarr+infin119909(119905
119899119896) = 119909
lowastlowast By Lemma 5there is 120585(119909lowastlowast) isin 120597119891(119909lowastlowast) such that
⟨120585 (119909lowastlowast
) 119909lowastlowast
minus 119875M (119909lowastlowast
)⟩ = 0 (91)
From the pseudoconvexity of 119891 and the above inequalitywe have
119891 (119909lowastlowast
) le 119891 (119875M (119909lowastlowast
)) (92)
Mathematical Problems in Engineering 11
Table 1 Experimental data set in MDF hot-pressing
Regression parameter 120574 120590 119862 MSE DF SEMOR 0001 5 30 00307 09543 minus08299
MOE 0001 6 40 00642 09063 minus17060
IBS 0001 5 40 00451 09256 minus06388
By Theorem 9 we have 119909lowastlowast isin F Therefore 119909lowastlowast isin Mwhich implies that
lim119896rarr+infin
dist (119909 (119905119899119896) M) = 0 (93)
Combining the above results with (83) we conclude thatlim
119905rarr+infindist(119909(119905)M) = 0
Case 3 In this case we assume that both 119868 and 119869 areunbounded
For 119905 isin 119868 similar to the analysis in Case 1 we have
lim119905rarr+infin119905isin119868
dist (119909 (119905) M) = 0 (94)
For 119905 isin 119869 define 120591(119905) = sup119904le119905119904isin119868
119904 Then 120591(119905) isin 119868 and(120591(119905) 119905] sube 119869 By the unboundedness of 119868 and 119869 we have
lim119905isin119869119905rarr+infin
120591 (119905) = +infin (95)
From (81) and the continuity of 119866(119909(119905) 119905) on [0 +infin) wehave
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))
+ 119898120583 (120591 (119905)) forall119905 isin 119869
(96)
By 120591(119905) isin 119868 (94) and (95) we have
lim sup119905rarr+infin119905isin119869
[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le lim119905rarr+infin119905isin119869
[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0
(97)
which gives
lim119905rarr+infin119905isin119869
119866 (119909 (119905) 119905) = 0 (98)
From the definition of 119866(119909 119905) and the above result wehave
lim119905rarr+infin119905isin119869
dist (119909 (119905) M) = 0 (99)
Therefore from (94) and (99) we obtain
lim119905rarr+infin
dist (119909 (119905) M) = 0 (100)
5 Numerical Experiments
In this section we test the proposed neural network (34)for solving problem (24) which is brought forward from
the MDF continuous hot-pressing process Based on theexisting data set we use the established theories and proposedneural network (34) to build the relationships between themain qualities of the hot-pressing plate and some relativetechnology parameters from optimization problem (18)Then based on optimization problem (22) we will useproposed network (34) to solve the optimal values of thetechnology parameters in hot-pressing system for optimizingthe qualities of the hot-pressing plate All these numericalexperiments validate the good performance of the proposednetwork in this paper
The numerical testing was carried out on a Lenovo PC(300GHz 200GB of RAM) with the use of Matlab 74 Andwe use ode23 to realize the neural network (34) in Matlab
51 Construction Relation Models in MDF Continuous Hot-Pressing Process In this part by considered optimizationproblem (18) and network (34) we build the relation modelwhich takes the hot-pressing temperature (TE) hot-pressingpressure (PR) hot-pressing time (TI) and moisture content(MC) of slab as the argument variables and the MOR MOEand IBS indexes of MDF as the dependent variables Thenumerical results show the good fitting of the built modelsfor the data set where the data set is given in Table 4
In order to use the data in Table 4 we first normalize theminto [0 1] And we use the mean square error (MSE) degreeof fitting (DF) and sufficient evaluation (SE) to evaluate thenumerical results where
MSE MSE = radicsum119899
119894=1(119909
119894minus 119909
lowast
119894)2
119899
DF DF = 1 minus radic(sum119899
119894=1(119909
119894minus 119909
lowast
119894)2
sum119899
119894=11199092
119894
)
SE SE = 12119909119879
119876119909 minus 119910119879
119909
(101)
where 119909 and 119909lowast indicate the actual value and predicted valueand 119876 and 119910 are defined as in (18) The smaller the MSE andthe SE and the closer the DF to 1 the better the regressionresult Moreover the SE function is the objective function in(15)
Based on the Gaussian radial basis function kernel thevalues of the initial parameters in problem (18) are given inTable 1 With a random initial point 119909
0isin Ω the numerical
results with respect to the obtained solution are also listed inTable 1 Figures 2ndash4 illustrate the fitting effect of the MORMOE and IBS values by using the proposed network (34) forsolving problem (18) And the parameters of the regressionfunctions in (19) are shown in Table 5
12 Mathematical Problems in Engineering
Table 2 Criterion of physical and mechanical performance indexes
Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45
IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045
MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700
2520151050
OriginalPredicted
0
05
1
15
2
25
3
35
4
5
45
Figure 2 Normalized sample fitting results of network (34) for theregression of MOR
Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5
52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2
521 Case 1
Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 3 Normalized sample fitting results of network (34) for theregression of MOE
By the information in Table 2 we let ]1= 200 ]
2= 4
]3= 7 and ]
4= 12 and choose 119910lowast
1= 22 119910lowast
2= 2500 120580
1= 8
1205761= 8 120580
2= 500 and 120576
2= 500 in optimization problem (21)
Then Ω = 119909 isin 1198774
0 le 1199091le ]
1 0 le 119909
2le ]
2 0 le 119909
3le
]3 0 le 119909
4le ]
4 and we can let
1198921(119909) = minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) =
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
1198923(119909) = minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) =
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(102)
By the proposed network (34) for solving (21) we obtainthe optimal solution
119909lowast
= (1594725 31022 64866 60061)119879
(103)
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
optimizing the parameters in the MDF continuous hot-pressing process is obtained Thus we express these twokinds of problems into a uniform formulation which is theoptimization problem considered in Section 3
31 Optimization Problem for Building up the Models of MDFContinuousHot-Pressing Process DenoteX sub 119877
119901 andY sub 119877
as two sets where 119909119894 = (1199091198941 119909
119894
2 119909
119894
119901) isin X is the attribute
vector on behalf of the hot-pressing plate properties 119910119894 isin Yindicates the values of the qualities of hot-pressing plate Let119878 = (119904
1
1199042
119904119899
) be the training data set of hot-pressingprocess where 119904119894 = (119909119894 119910119894) obeys the unknown distributionand is IID (independent and identically distributed) Basedon the support vector machine theory and the training dataset we would like to find a nonlinear function 119910 = 119891(119909) suchthat it approximates the training data set as much as possible
With the kernel function 119870 and the Huber loss functionfrom the theory in [21] the dual optimization problem of theprimal optimization problem of SVM is given as
min120572120573
1
2
119899
sum
119894119895=1
(120573119894minus 120572
119894) (120573
119895minus 120572
119895)119870 (119909
119894
119909119895
)
minus
119899
sum
119894=1
(120573119894minus 120572
119894) 119910
119894
+120576
2119862
119899
sum
119894=1
(120573119894minus 120572
119894)2
st119899
sum
119894=1
(120573119894minus 120572
119894) = 0
0 le 120573119894 120572
119894le 119862 119894 = 1 2 119899
(12)
Denote 120572lowast119894and 120573lowast
119894as the optimal solutions of (12) Then the
regression function 119891 can be expressed by
119891 (119909) =
119899
sum
119894=1
(120573lowast
119894minus 120572
lowast
119894)119870 (119909
119894
119909) + 119887 (13)
where 119887 can be calculated by one of the following twomethods
119887 = 119910119895
minus
119899
sum
119894=1
(120573lowast
119894minus 120572
lowast
119894) ⟨119909
119894
119909119895
⟩ + 120576 120572lowast
119895isin (0 119862)
119887 = 119910119895
minus
119899
sum
119894=1
(120573lowast
119894minus 120572
lowast
119894) ⟨119909
119894
119909119895
⟩ minus 120576 120572lowast
119895isin (0 119862)
(14)
Denote 119909119894= 120573
119894minus 120572
119894in (12) then (12) can be reformulated
as
min 1
2119909119879
119876119909 minus 119910119879
119909
st 119890119879
119909 = 0 minus 119862119890 le 119909 le 119862119890
(15)
where 119909 = (1199091 119909
2 119909
119899)119879 119910 = (1199101 1199102 119910119899)119879 119890 = (1 1
1)119879
isin 119877119899 and 119876 = K+(1205762119862)119868
119899withK = (119870(119909
119894
119909119895
))119899
119894119895=1
For the optimal solution 119909lowast = (119909lowast1 119909
lowast
2 119909
lowast
119899) of (15) we call
119909119894 a support vector if 119909lowast
119894= 0 The optimal solution of (15)
solved by the originalmethods often has almost 100 supportvectors which increases the complexity of the relationmodelslargely Thus we introduce the problem deduced by (15) thatis
min 1
2119909119879
119876119909 minus 119910119879
119909 + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816
st 119890119879
119909 = 0 minus 119862119890 le 119909 le 119862119890
(16)
with 120574 gt 0 In (43) 120574sum119899
119894=1|119909119894| is often called the regulation
term which is used to control the number of its supportvectors
Define the penalty function
119875 (119909) =
1003817100381710038171003817100381710038171003817
1
119899119890119890119879
119909
1003817100381710038171003817100381710038171003817=radic119899
119899
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119894=1
119909119894
1003816100381610038161003816100381610038161003816100381610038161003816
(17)
and then 119909 119875(119909) le 0 = 119909 119890119879119909 = 0 From [19 Proposition243] 119909 is an optimal solution of (43) if and only if it is anoptimal solution of the following problem
min 1
2119909119879
119876119909 minus 119910119879
119909 + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816 + 120590
10038161003816100381610038161003816119890119879
11990910038161003816100381610038161003816
st minus 119862119890 le 119909 le 119862119890
(18)
where 120590 = 119862119876 + 119910radic119899 + 120574 + 1 Thus we can build up therelation models of hot-pressing process by solving problem(18)
32 Optimization Problem for Optimizing the Parameters inMDF Hot-Pressing Process Based on the relationships builtup in Section 31 we focus on themodulus of rupture (MOR)modulus of elasticity (MOE) and internal bonding strength(IBS) of hot-pressing plate by optimizing the process param-eters and slab attributes Suppose the regression functions ofMORMOS and IBSwith respect to some relative parametersbased on the Gaussian radial basis function kernel are
1198911(119909) fl
119899
sum
119894=1
120588lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) + 119887
1(MOR)
1198912(119909) fl
119899
sum
119894=1
120579lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) + 119887
2(MOS)
1198913(119909) fl
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) + 119887
3(IBS)
(19)
and based on the linear polynomial kernel are
1198911(119909) fl
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1(MOR)
1198912(119909) fl
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2(MOS)
1198913(119909) fl
119899
sum
119894=1
120581lowast
119894⟨119911
119894 119909⟩ + 119887
3(IBS)
(20)
Mathematical Problems in Engineering 5
where 1199111 119911
2 119911
119899are the variables in the data set for
regression and 119909 = (1199091 119909
2 119909
3 119909
4)119879 indicates the indepen-
dent variable with hot-pressing temperature 1199091 hot-pressing
pressure 1199092 hot-pressing time 119909
3 and moisture content 119909
4
The regression functions in (19) satisfy the following twoproperties
(i) 119891119894is continuously differentiable on 119877119899 119894 = 1 2 3
(ii) 119891119894and minus119891
119894are not convex but minus119891
119894is pseudoconvex on
119877119899 119894 = 1 2 3
And the regression functions in (20) satisfy the following twoproperties
(i) 119891119894is continuously differentiable on 119877119899 119894 = 1 2 3
(ii) 119891119894and minus119891
119894are convex on 119877119899 119894 = 1 2 3
From the physical significance of MOR MOS and IBSwe suppose that the larger the numbers of MOR MOS andIBS the better the quality of hot-pressing plate In order toadopt the different demand on the indexes of the hot-pressingplate in different applications we consider the following twocases in this part
First we focus onmaximizing a single performance indexof the hot-pressing plate when the other two performanceindexes are within the certain areas If we want to optimizethe IBS the corresponding optimization model for this casecan be expressed by
min minus 1198913(119909)
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
minus 1205801le 119891
1(119909) minus 119910
lowast
1le 120576
1
minus 1205802le 119891
2(119909) minus 119910
lowast
2le 120576
2
(21)
where ]1 ]2 ]3 ]4gt 0 indicate the upper bounds of hot-
pressing temperature hot-pressing pressure hot-pressingtime andmoisture content and [119910lowast
1minus120580
1 119910
lowast
1+120576
1] [119910lowast
2minus120580
2 119910
lowast
2+
1205762] are the feasible regions of MOR and MOS respectivelyIn order to let problem (21) be solved effectively we
let the objective function 1198913be with the Gaussian radial
basis function and the regression functions 1198911and 119891
2in the
constraints are with the linear polynomial kernel Then (21)is
min minus
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) minus 119887
3
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
minus 1205801le
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1le 120576
1
minus 1205802le
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2le 120576
2
(22)
which is a pseudoconvex optimization problem with convexconstraints
Second we would like to optimize the MOR MOS andIBS synthetically For this demand we consider the followingoptimization model
min minus1205821
119891lowast
1
1198911(119909) minus
1205822
119891lowast
2
1198912(119909) minus
1205823
119891lowast
3
1198913(119909)
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
(23)
where 1205821 120582
2 120582
3gt 0 indicate the importance of MOR MOS
and IBS ]1 ]2 ]3 ]4are with the same meaning as in (21)
and 119891lowast1 119891lowast
2 and 119891lowast
3are the expected values of MOR MOS
and IBS In particular if MOR MOS and IBS are with thesame importance in the quality of the hot-pressing plate wecan let 120582
1= 120582
2= 120582
3= 13 and we can let 120582
1= 12
1205822= 13 and 120582
3= 16 if the importance of MOR MOS
and IBS is strictly monotone decreasing Similar to the kernelfunctions in problem (21) we let119891
1119891
2 and119891
3in problem (23)
be with the Gaussian radial basis function which means thatproblem (23) is also a pseudoconvex optimization problemwith convex constraints
33 General Model Based on analysis in Sections 31 and32 we consider the following minimization problem in thispaper
minimize 119891 (119909) fl Θ (119909) + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816 + 120590
10038161003816100381610038161003816119890119879
11990910038161003816100381610038161003816
subject to 119909 isin Ω fl 119909 119886 le 119909 le 119887
119892119894(119909) le 0 119894 = 1 119898
(24)
6 Mathematical Problems in Engineering
where 120574 120590 ge 0 119886 119887 isin 119877119899 with 119886 lt 119887 Θ 119877
119899
rarr 119877
is continuously differentiable and pseudoconvex on 119877119899 and119892119894 119877
119899
rarr 119877 (119894 = 1 2 119898) is continuously differentiableand convex on 119877119899
On the one hand when Θ(119909) = (12)119909119879119876119909 minus 119910119879119909 119886 flminus119862119890 119887 fl 119862119890 and 120574 and 120590 are defined as in (18) then problem(24) without 119892
119894reduces to problem (18) On the other hand
if we let
Θ (119909) fl minus
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) minus 119887
3
119886 fl 0
119887 fl (]1 ]2 ]3 ]4)119879
(25)
1198921(119909) fl minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) fl
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
(26)
1198923(119909) fl minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) fl
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(27)
then problem (24) reduces to problem (22) Similar reformu-lation can be done for problem (23) by (24)
Therefore problem (24) considered in this paper includesthe optimizationmodels for building up the relationships andoptimizing the relative parameters in MDF continuous hot-pressing process
Inwhat follows we denoteF as the feasible region of (24)that is
F fl 119909 isin Ω 119892119894(119909) le 0 119894 = 1 2 119898 (28)
andM is the optimal solution set of (24)
4 Main Results
41 Proposed Neural Network In this subsection we proposea one-layer recurrent neural network for solving problem(24) where we combine the penalty function and projectionmethods to solve the constraints and use the smoothingtechniques to overcome the nonsmoothness of the objectivefunction and penalty function
Define the penalty function
119901 (119909) =
119898
sum
119894=1
max 0 119892119894(119909) (29)
Then 119909 isin Ω 119901(119909) le 0 = FFrom the smoothing functions in (25) for the plus
function we define the smoothing function of 119901 as
(119909 120583) =
119898
sum
119894=1
120601 (119892119894(119909) 120583) (30)
where
120601 (119904 120583) =
max 0 119904 if |119904| gt 120583
(119904 + 120583)2
4120583if |119904| le 120583
(31)
Form the results in [23] 120601(119904 120583) owns the followingproperties
Lemma 3 (see [23]) (i) For any 119904 isin 119877 120601(119904 sdot) is continuouslydifferentiable and 120601(sdot 120583) is also continuously differentiable forany fixed 120583 gt 0
(ii) 0 le nabla120583120601(119904 120583) le 1 forall119904 isin 119877 forall120583 isin (0 +infin)
(iii) 0 le 120601(119904 120583) minus |119904| le 1205834 forall119904 isin 119877 forall120583 isin (0 +infin)(iv) 120601(sdot 120583) is convex for any fixed 120583 gt 0 and
lim119904rarr119905120583darr0
nabla119904120601(119904 120583) sube 120597max0 119905
Then has the following properties
Lemma 4 is a smoothing function 119901 and satisfies thefollowing
(i) (sdot 120583) is convex for any fixed 120583 gt 0(ii) lim
119911rarr119909120583darr0nabla119911(119911 120583) sube 120597119901(119909)
(iii) 0 le nabla120583(119909 120583) le 119898 forall119909 isin 119877119899 120583 isin (0 +infin)
Next by the smoothing function for the absolute valuefunction | sdot |
120579 (119904 120583) =
|119904| if |119904| ge 120583
1199042
2120583+120583
2if |119904| lt 120583
(32)
we define
(119909 120583) fl Θ (119909) + 120574
119899
sum
119894=1
120579 (119909119894 120583) + 120590120579 (119890
119879
119909 120583) (33)
Since 120579(119904 120583) = 120601(119904 120583) + 120601(minus119904 120583) 120579(119904 120583) owns allproperties in Lemma 8 and the following results hold
Lemma 5 is a smoothing function 119891 in (24) with thefollowing properties
(i) (sdot 120583) is pseudoconvex for any fixed 120583 gt 0
(ii) lim119911rarr119909120583darr0
nabla119909(119909 120583) sube 120597119891(119909)
(iii) 0 le nabla120583(119909 120583) le 120574119899 + 120590 forall119909 isin 119877119899 120583 isin (0 +infin)
From the projected gradient method and the viscosityregularization method we introduce the following neuralnetwork to solve (24)
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(34)
where 120583(119905) = 119890minus119905 and 120576(119905) = 1(119905 + 1) with 1205760gt 0
Mathematical Problems in Engineering 7
120583intminus1
minus1
xintminus
sumsum PΩminus
minus
++
120576intab
bba
aab
pnablax
xqnabla
Figure 1 Simple block structure of proposed network (37)
By the definitions for (34) can be expressed as
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)
minus
119903
sum
119894=1
nabla119904120601 (119904 120583 (119905))
119904=119892119894(119909(119905))nabla119892
119894(119909 (119905)) minus 120576 (119905)
sdot (nablaΘ (119909 (119905)) + 120574
119899
sum
119894=1
nabla119904120579 (119904 120583)
119904=119909119894(119905)119890119894
+ 120590nabla119904120579 (119904 120583)
119904=119890119879119909(119905)
119890)]
(35)
where
nabla119904120601 (119904 120583) =
1 if 119904 gt 120583119904 + 120583
2120583if |119904| le 120583
0 if 119904 lt minus120583
nabla119904120579 (119904 120583) =
1 if |119904| ge 120583119904
120583if |119904| lt 120583
(36)
To implement (34) by circuits we can use the reformu-lated form of (34) as follows
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
120576 (119905) = minus1205762
(119905)
(119905) = minus120583 (119905)
(37)
Equation (34) can be seen as a network with three input andthree output variables that are 119909(119905) 120583(119905) and 120576(119905) A simpleblock structure of the proposed network (37) implementedby circuits is presented in Figure 1
42 Theoretical Analysis In this subsection we study somenecessary dynamical and optimality properties of proposednetwork (34) for solving (24)
The global existence of the solutions of (34) is a necessarycondition for its usability in optimization With an initial
point 1199090isin Ω the solution of (34) is global existentMoreover
the uniqueness of the solution of (34) with 1199090isin Ω is proved
under some conditions The proposed network (34) can beimplemented in circuits and mathematical software Thenthe feasibility and optimality of network (34) for optimizationproblem (24) are proved theoretically
We call 119909 [0 119879) with T gt 0 a solution of (34) if 119909 isabsolutely continuous on [0 119879) and satisfies (34) everywhere
Proposition 6 For any initial point 1199090isin Ω there is a global
solution of (34) defined on [0 +infin) and it satisfies
119909 (119905) isin Ω forall119905 isin [0 +infin) (38)
Proof Since the right hand function in network (34) iscontinuous about 119909 and 119905 there are 119879 gt 0 and an absolutecontinuous function 119909 [0 119879) rarr 119877
119899 such that 119909(119905) satisfies(34) for all 119905 isin [0 119879)
Denote 120579(119905) = 119875Ω[119909(119905) minus nabla
119909(119909(119905) 120583(119905)) minus 120576(119905)nabla
119909(119909(119905)
120583(119905))] Then (34) can be rewritten as
(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)
A simple integration procedure of the above equation gives
119909 (119905) = 119890minus119905
1199090+ 119890
minus119905
int
119905
0
120579 (119904) 119890119904
119889119904 (40)
which can be rewritten as
119909 (119905) = 119890minus119905
1199090+ (1 minus 119890
minus119905
)int
119905
0
120579 (119904)119890119904
119890119905 minus 1119889119904 (41)
By int1199050
119890119904
(119890119905
minus 1)119889119904 = 1 120579(119904) isin Ω forall0 le 119904 le 119905 and 1199090isin Ω we
obtain that 119909(119905) isin Ω forall119905 isin [0 119879)By the boundedness of Ω and the extension theory this
solution of (34) can be extended Thus the solution of (34)with initial point 119909
0isin Ω is globally existent Similarly we can
obtain that
119909 (119905) isin Ω forall119905 isin [0 +infin) (42)
Some Lipschitz condition is often used to guaranteethe uniqueness of the solution of a neural network Inwhat follows we give a sufficient condition to ensure theuniqueness of the solution of (34) with initial point 119909
0isin Ω
Proposition 7 For any initial point 1199090isin Ω if nabla
119909(sdot 120583) and
nabla119909(sdot 120583) are locally Lipschitz continuous for any fixed 120583 isin
(0 1] then the solution of neural network (34) is unique
Proof By Proposition 6 the solutions of (34) with initialpoint119909
0isin Ω exist globally and satisfy119909(119905) isin Ωforall119905 isin [0 +infin)
Suppose that there exist two solutions 119909 [0 +infin) rarr 119877119899
and 119910 [0 +infin) rarr 119877119899 of (34) with initial point 119909
0isin Ω
and suppose there exists such that = inf119905ge0119909(119905) =119910(119905)
119905 By theboundedness of Ω there is 119877 gt 0 such that 119909(119905) le 119877 and119910(119905) le 119877 forall119905 isin [0 +infin)
8 Mathematical Problems in Engineering
Then there is 119871 gt 0 such that10038161003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla
119909 (119910 (119905) 120583 (119905))
10038161003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
1003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla119909 (119910 (119905) 120583 (119905))1003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
(43)
Differentiating (12)119909(119905)minus119910(119905)2 along the two solutionsof (34) by the Lipschitz continuity of 119875
Ωand (43) we have
119889
119889119905
1
2
1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817
2
= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩
le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817
2
(44)
Applying Gronwallrsquos inequality into the integration of theabove inequality it gives 119909(119905) = 119910(119905) forall119905 isin [119905 + 1] whichleads to a contradiction Therefore the solution of (34) withinitial point 119909
0isin Ω is unique
Lyapunov method is employed to analyze the perfor-mance of (34) Here we introduce the following two Lya-punov energy functions
119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
119866 (119909 119905) =1
2dist2 (119909M) + (119909 120583 (119905))
+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
(45)
The above two Lyapunov functions satisfy the followingestimations along the solutions of (34)
Lemma 8 (i) The derivative of 119864(119909 119905) along the solution of(34) can be calculated by
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
(46)
(ii) The derivative of 119866(119909 119905) along the solution of (34) can becalculated by
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(47)
Proof (i) Differentiating119864(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119864 (119909 (119905) 119905) = ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(48)
Equation (34) can be rewritten as
(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(49)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119909(119905) using (49) and (10) we have
⟨119909 (119905) minus nabla119909 (119909 (119905) 120583 (119905)) minus 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
minus (119905) minus 119909 (119905) (119905) + 119909 (119905) minus 119909 (119905)⟩ ge 0
(50)
which follows the fact that
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
le minus (119905)2
forall119905 ge 0
(51)
Combining (48) and (51) we get that
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
(52)
From Lemmas 4 and 5 (119905) = minus119890minus119905 and 120576(119905) lt 0 we obtainthe estimation in (i)
(ii) Differentiating 119866(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩
+ ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(53)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119875M(119909(119905)) using (10) and (49) we have
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0
(54)
which can be rewritten as
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
minus (119905)2
(55)
Mathematical Problems in Engineering 9
Combining (53) and (55) we get
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905) (119891 (119909 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(56)
Similar to the analysis in (i) we obtain the estimation in(ii)
Next we prove the efficiency of proposed network (34) forsolving optimization problem (24) where the convergencefeasibility of the proposed network is a basic property
Theorem 9 For initial point 1199090isin Ω any solution 119909 [0 +infin)
rarr 119877119899 of (34) satisfies
lim119905rarr+infin
dist (119909 (119905) F) = 0 (57)
Proof Denote 119909 [0 +infin) rarr 119877119899 as a global solution of (34)
with initial point 1199090isin Ω
From 119909(119905) isin Ω forall119905 isin [0 +infin) and Lemma 8 119864(119909(119905) 119905) isnonincreasing along the solution of (34) Using119864(119909(119905) 119905) ge 0forall119905 isin [0 +infin) we confirm that
lim119905rarr+infin
119864 (119909 (119905) 119905) exists (58)
From 119909(119905) isin Ω forall119905 isin [0 +infin) and lim119905rarr+infin
120576(119905) = 0 wehave
lim119905rarr+infin
120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ
119891) = 0 (59)
which implies
lim119905rarr+infin
119864 (119909 (119905) 119905) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) (60)
Inwhat follows wewill prove that lim119905rarr+infin
(119909(119905) 120583(119905)) =
0 Arguing by contradiction we assume that
lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 119888 = 0 (61)
By (119909(119905) 120583(119905)) ge 119901(119909(119905)) ge 0 forall119905 isin [0 +infin) we have 119888 gt 0which follows the fact that there is 119879
1ge 0 such that
(119909 (119905) 120583 (119905)) ge119888
2 forall119905 isin [119879
1 +infin) (62)
From Lemma 4 we obtain
1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)
which implies that there exists 1198792gt 119879
1such that
(119875M (119909 (119905)) 120583 (119905)) le119888
8 (64)
Since Ω is bounded and 119909(119905) isin Ω forall119905 isin [0 +infin) byLemma 5 there is 119877 gt 0 such that10038161003816100381610038161003816⟨nabla
119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
10038161003816100381610038161003816le 119877
forall119905 isin [0 +infin)
(65)
Then
lim119905rarr+infin
120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)
which implies that there is 1198793ge 119879
2such that
10038161003816100381610038161003816120576 (119905) ⟨nabla
119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩
10038161003816100381610038161003816le119888
8
forall119905 isin [1198792 +infin)
(67)
From Lemma 8 and the convexity of (sdot 120583) for any fixed120583 gt 0 we obtain that
119889
119889119905119866 (119909 (119905) 119905) le minus ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)
sdot ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(68)
By (62) (67) and (68) we obtain
119889
119889119905119866 (119909 (119905) 119905) le minus
119888
4 forall119905 isin [119879
3 +infin) (69)
Integrating the above inequality from 1198793to 119905 (gt119879
3) we
have
119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879
3) minus
119888
4(119905 minus 119879
3)
forall119905 isin [1198793 +infin)
(70)
Thus
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (71)
which leads to a contradiction with 119866(119909 119905) ge 0 for all 119909 isin 119877119899and 119905 isin [0 +infin) Therefore
lim119905rarr+infin
119901 (119909 (119905)) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 0 (72)
which guarantees that
lim119905rarr+infin
dist (119909 (119905) F) = 0 (73)
The following theorem indicates that any accumulationpoint of the solutions of (34) is just an optimal solution of(34)
10 Mathematical Problems in Engineering
Theorem 10 For initial point 1199090isin Ω any solution 119909 of (34)
is convergent to the optimal solution setM that is
lim119905rarr+infin
dist (119909 (119905) M) = 0 (74)
Proof From (63) (68) and (119909(119905) 120583(119905)) ge 0 forall119905 isin [0 +infin)we have
119889
119889119905119866 (119909 (119905) 119905)
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
+ 119898120583 (119905)
(75)
by (119905) = minus120583(119905) which can be rewritten as
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(76)
Denote
119868 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le 0
119869 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
gt 0
(77)
Owning to the continuity of ⟨nabla119909(119909(119905) 120583(119905)) 119909(119905)minus119875M(119909(119905))⟩
on [0 +infin) 119868 and 119869 are closed and open in [0 +infin) respec-tively
Case 1 In this case we assume that there exists 119879 ge 0 suchthat 119905 isin 119868 forall119905 isin [119879 +infin)
From the definition on 119868 and the pseudoconvexity of(sdot 120583) on 119877119899 we have
(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))
forall119905 isin [119879 +infin)
(78)
which implies
lim sup119905rarr+infin
119891 (119909 (119905)) le min119909isinF
119891 (119909) (79)
ByTheorem 9 we confirm that
lim119905rarr+infin
dist (119909 (119905) M) = 0 (80)
Case 2 In this case we assume that there exists 119879 ge 0 such
that 119905 isin 119869 forall119905 isin [119879 +infin) which means that
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] lt 0 forall119905 isin [119879 +infin) (81)
Then
lim119905rarr+infin
[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)
Since lim119905rarr+infin
119864(119909(119905) 119905) exists and lim119905rarr+infin
120583(119905) = 0 weobtain that
lim119905rarr+infin
dist (119909 (119905) M) exists (83)
Suppose
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889
gt 0
(84)
Then there is 1198791ge such that
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge
119889
2
forall119905 isin [1198791 +infin)
(85)
Then from (76) we have
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus
119889
2120576 (119905)
forall119905 isin [1198791 +infin)
(86)
Integrating the above inequality from 1198791to 119905 (gt119879
1) we
have
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879
1) + 119898120583 (119879
1)
+ int
119905
1198791
minus119889
2120576 (119904) 119889119904
(87)
Let 119905 rarr +infin in the above inequality then we have
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (88)
which leads to a contraction with the boundedness frombelow 119866(119909(119905) 119905) on [0 +infin) Thus
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)
which follows the fact that there is an increasing sequence 119905119899
such that 119905119899rarr +infin and
lim119899rarr+infin
⟨nabla119909 (119909 (119905
119899) 120583 (119905
119899)) 119909 (119905
119899) minus 119875M (119909 (119905
119899))⟩
= 0
(90)
By 119909(119905119899) sube Ω and Ω which is bounded there are 119909lowastlowast isin
Ω and a subsequence of 119909(119905119899) (denoted as 119909(119905
119899119896)) such that
lim119896rarr+infin
119905119899119896= +infin and lim
119896rarr+infin119909(119905
119899119896) = 119909
lowastlowast By Lemma 5there is 120585(119909lowastlowast) isin 120597119891(119909lowastlowast) such that
⟨120585 (119909lowastlowast
) 119909lowastlowast
minus 119875M (119909lowastlowast
)⟩ = 0 (91)
From the pseudoconvexity of 119891 and the above inequalitywe have
119891 (119909lowastlowast
) le 119891 (119875M (119909lowastlowast
)) (92)
Mathematical Problems in Engineering 11
Table 1 Experimental data set in MDF hot-pressing
Regression parameter 120574 120590 119862 MSE DF SEMOR 0001 5 30 00307 09543 minus08299
MOE 0001 6 40 00642 09063 minus17060
IBS 0001 5 40 00451 09256 minus06388
By Theorem 9 we have 119909lowastlowast isin F Therefore 119909lowastlowast isin Mwhich implies that
lim119896rarr+infin
dist (119909 (119905119899119896) M) = 0 (93)
Combining the above results with (83) we conclude thatlim
119905rarr+infindist(119909(119905)M) = 0
Case 3 In this case we assume that both 119868 and 119869 areunbounded
For 119905 isin 119868 similar to the analysis in Case 1 we have
lim119905rarr+infin119905isin119868
dist (119909 (119905) M) = 0 (94)
For 119905 isin 119869 define 120591(119905) = sup119904le119905119904isin119868
119904 Then 120591(119905) isin 119868 and(120591(119905) 119905] sube 119869 By the unboundedness of 119868 and 119869 we have
lim119905isin119869119905rarr+infin
120591 (119905) = +infin (95)
From (81) and the continuity of 119866(119909(119905) 119905) on [0 +infin) wehave
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))
+ 119898120583 (120591 (119905)) forall119905 isin 119869
(96)
By 120591(119905) isin 119868 (94) and (95) we have
lim sup119905rarr+infin119905isin119869
[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le lim119905rarr+infin119905isin119869
[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0
(97)
which gives
lim119905rarr+infin119905isin119869
119866 (119909 (119905) 119905) = 0 (98)
From the definition of 119866(119909 119905) and the above result wehave
lim119905rarr+infin119905isin119869
dist (119909 (119905) M) = 0 (99)
Therefore from (94) and (99) we obtain
lim119905rarr+infin
dist (119909 (119905) M) = 0 (100)
5 Numerical Experiments
In this section we test the proposed neural network (34)for solving problem (24) which is brought forward from
the MDF continuous hot-pressing process Based on theexisting data set we use the established theories and proposedneural network (34) to build the relationships between themain qualities of the hot-pressing plate and some relativetechnology parameters from optimization problem (18)Then based on optimization problem (22) we will useproposed network (34) to solve the optimal values of thetechnology parameters in hot-pressing system for optimizingthe qualities of the hot-pressing plate All these numericalexperiments validate the good performance of the proposednetwork in this paper
The numerical testing was carried out on a Lenovo PC(300GHz 200GB of RAM) with the use of Matlab 74 Andwe use ode23 to realize the neural network (34) in Matlab
51 Construction Relation Models in MDF Continuous Hot-Pressing Process In this part by considered optimizationproblem (18) and network (34) we build the relation modelwhich takes the hot-pressing temperature (TE) hot-pressingpressure (PR) hot-pressing time (TI) and moisture content(MC) of slab as the argument variables and the MOR MOEand IBS indexes of MDF as the dependent variables Thenumerical results show the good fitting of the built modelsfor the data set where the data set is given in Table 4
In order to use the data in Table 4 we first normalize theminto [0 1] And we use the mean square error (MSE) degreeof fitting (DF) and sufficient evaluation (SE) to evaluate thenumerical results where
MSE MSE = radicsum119899
119894=1(119909
119894minus 119909
lowast
119894)2
119899
DF DF = 1 minus radic(sum119899
119894=1(119909
119894minus 119909
lowast
119894)2
sum119899
119894=11199092
119894
)
SE SE = 12119909119879
119876119909 minus 119910119879
119909
(101)
where 119909 and 119909lowast indicate the actual value and predicted valueand 119876 and 119910 are defined as in (18) The smaller the MSE andthe SE and the closer the DF to 1 the better the regressionresult Moreover the SE function is the objective function in(15)
Based on the Gaussian radial basis function kernel thevalues of the initial parameters in problem (18) are given inTable 1 With a random initial point 119909
0isin Ω the numerical
results with respect to the obtained solution are also listed inTable 1 Figures 2ndash4 illustrate the fitting effect of the MORMOE and IBS values by using the proposed network (34) forsolving problem (18) And the parameters of the regressionfunctions in (19) are shown in Table 5
12 Mathematical Problems in Engineering
Table 2 Criterion of physical and mechanical performance indexes
Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45
IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045
MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700
2520151050
OriginalPredicted
0
05
1
15
2
25
3
35
4
5
45
Figure 2 Normalized sample fitting results of network (34) for theregression of MOR
Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5
52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2
521 Case 1
Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 3 Normalized sample fitting results of network (34) for theregression of MOE
By the information in Table 2 we let ]1= 200 ]
2= 4
]3= 7 and ]
4= 12 and choose 119910lowast
1= 22 119910lowast
2= 2500 120580
1= 8
1205761= 8 120580
2= 500 and 120576
2= 500 in optimization problem (21)
Then Ω = 119909 isin 1198774
0 le 1199091le ]
1 0 le 119909
2le ]
2 0 le 119909
3le
]3 0 le 119909
4le ]
4 and we can let
1198921(119909) = minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) =
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
1198923(119909) = minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) =
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(102)
By the proposed network (34) for solving (21) we obtainthe optimal solution
119909lowast
= (1594725 31022 64866 60061)119879
(103)
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
where 1199111 119911
2 119911
119899are the variables in the data set for
regression and 119909 = (1199091 119909
2 119909
3 119909
4)119879 indicates the indepen-
dent variable with hot-pressing temperature 1199091 hot-pressing
pressure 1199092 hot-pressing time 119909
3 and moisture content 119909
4
The regression functions in (19) satisfy the following twoproperties
(i) 119891119894is continuously differentiable on 119877119899 119894 = 1 2 3
(ii) 119891119894and minus119891
119894are not convex but minus119891
119894is pseudoconvex on
119877119899 119894 = 1 2 3
And the regression functions in (20) satisfy the following twoproperties
(i) 119891119894is continuously differentiable on 119877119899 119894 = 1 2 3
(ii) 119891119894and minus119891
119894are convex on 119877119899 119894 = 1 2 3
From the physical significance of MOR MOS and IBSwe suppose that the larger the numbers of MOR MOS andIBS the better the quality of hot-pressing plate In order toadopt the different demand on the indexes of the hot-pressingplate in different applications we consider the following twocases in this part
First we focus onmaximizing a single performance indexof the hot-pressing plate when the other two performanceindexes are within the certain areas If we want to optimizethe IBS the corresponding optimization model for this casecan be expressed by
min minus 1198913(119909)
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
minus 1205801le 119891
1(119909) minus 119910
lowast
1le 120576
1
minus 1205802le 119891
2(119909) minus 119910
lowast
2le 120576
2
(21)
where ]1 ]2 ]3 ]4gt 0 indicate the upper bounds of hot-
pressing temperature hot-pressing pressure hot-pressingtime andmoisture content and [119910lowast
1minus120580
1 119910
lowast
1+120576
1] [119910lowast
2minus120580
2 119910
lowast
2+
1205762] are the feasible regions of MOR and MOS respectivelyIn order to let problem (21) be solved effectively we
let the objective function 1198913be with the Gaussian radial
basis function and the regression functions 1198911and 119891
2in the
constraints are with the linear polynomial kernel Then (21)is
min minus
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) minus 119887
3
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
minus 1205801le
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1le 120576
1
minus 1205802le
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2le 120576
2
(22)
which is a pseudoconvex optimization problem with convexconstraints
Second we would like to optimize the MOR MOS andIBS synthetically For this demand we consider the followingoptimization model
min minus1205821
119891lowast
1
1198911(119909) minus
1205822
119891lowast
2
1198912(119909) minus
1205823
119891lowast
3
1198913(119909)
st 0 le 1199091le ]
1
0 le 1199092le ]
2
0 le 1199093le ]
3
0 le 1199094le ]
4
(23)
where 1205821 120582
2 120582
3gt 0 indicate the importance of MOR MOS
and IBS ]1 ]2 ]3 ]4are with the same meaning as in (21)
and 119891lowast1 119891lowast
2 and 119891lowast
3are the expected values of MOR MOS
and IBS In particular if MOR MOS and IBS are with thesame importance in the quality of the hot-pressing plate wecan let 120582
1= 120582
2= 120582
3= 13 and we can let 120582
1= 12
1205822= 13 and 120582
3= 16 if the importance of MOR MOS
and IBS is strictly monotone decreasing Similar to the kernelfunctions in problem (21) we let119891
1119891
2 and119891
3in problem (23)
be with the Gaussian radial basis function which means thatproblem (23) is also a pseudoconvex optimization problemwith convex constraints
33 General Model Based on analysis in Sections 31 and32 we consider the following minimization problem in thispaper
minimize 119891 (119909) fl Θ (119909) + 120574
119899
sum
119894=1
10038161003816100381610038161199091198941003816100381610038161003816 + 120590
10038161003816100381610038161003816119890119879
11990910038161003816100381610038161003816
subject to 119909 isin Ω fl 119909 119886 le 119909 le 119887
119892119894(119909) le 0 119894 = 1 119898
(24)
6 Mathematical Problems in Engineering
where 120574 120590 ge 0 119886 119887 isin 119877119899 with 119886 lt 119887 Θ 119877
119899
rarr 119877
is continuously differentiable and pseudoconvex on 119877119899 and119892119894 119877
119899
rarr 119877 (119894 = 1 2 119898) is continuously differentiableand convex on 119877119899
On the one hand when Θ(119909) = (12)119909119879119876119909 minus 119910119879119909 119886 flminus119862119890 119887 fl 119862119890 and 120574 and 120590 are defined as in (18) then problem(24) without 119892
119894reduces to problem (18) On the other hand
if we let
Θ (119909) fl minus
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) minus 119887
3
119886 fl 0
119887 fl (]1 ]2 ]3 ]4)119879
(25)
1198921(119909) fl minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) fl
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
(26)
1198923(119909) fl minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) fl
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(27)
then problem (24) reduces to problem (22) Similar reformu-lation can be done for problem (23) by (24)
Therefore problem (24) considered in this paper includesthe optimizationmodels for building up the relationships andoptimizing the relative parameters in MDF continuous hot-pressing process
Inwhat follows we denoteF as the feasible region of (24)that is
F fl 119909 isin Ω 119892119894(119909) le 0 119894 = 1 2 119898 (28)
andM is the optimal solution set of (24)
4 Main Results
41 Proposed Neural Network In this subsection we proposea one-layer recurrent neural network for solving problem(24) where we combine the penalty function and projectionmethods to solve the constraints and use the smoothingtechniques to overcome the nonsmoothness of the objectivefunction and penalty function
Define the penalty function
119901 (119909) =
119898
sum
119894=1
max 0 119892119894(119909) (29)
Then 119909 isin Ω 119901(119909) le 0 = FFrom the smoothing functions in (25) for the plus
function we define the smoothing function of 119901 as
(119909 120583) =
119898
sum
119894=1
120601 (119892119894(119909) 120583) (30)
where
120601 (119904 120583) =
max 0 119904 if |119904| gt 120583
(119904 + 120583)2
4120583if |119904| le 120583
(31)
Form the results in [23] 120601(119904 120583) owns the followingproperties
Lemma 3 (see [23]) (i) For any 119904 isin 119877 120601(119904 sdot) is continuouslydifferentiable and 120601(sdot 120583) is also continuously differentiable forany fixed 120583 gt 0
(ii) 0 le nabla120583120601(119904 120583) le 1 forall119904 isin 119877 forall120583 isin (0 +infin)
(iii) 0 le 120601(119904 120583) minus |119904| le 1205834 forall119904 isin 119877 forall120583 isin (0 +infin)(iv) 120601(sdot 120583) is convex for any fixed 120583 gt 0 and
lim119904rarr119905120583darr0
nabla119904120601(119904 120583) sube 120597max0 119905
Then has the following properties
Lemma 4 is a smoothing function 119901 and satisfies thefollowing
(i) (sdot 120583) is convex for any fixed 120583 gt 0(ii) lim
119911rarr119909120583darr0nabla119911(119911 120583) sube 120597119901(119909)
(iii) 0 le nabla120583(119909 120583) le 119898 forall119909 isin 119877119899 120583 isin (0 +infin)
Next by the smoothing function for the absolute valuefunction | sdot |
120579 (119904 120583) =
|119904| if |119904| ge 120583
1199042
2120583+120583
2if |119904| lt 120583
(32)
we define
(119909 120583) fl Θ (119909) + 120574
119899
sum
119894=1
120579 (119909119894 120583) + 120590120579 (119890
119879
119909 120583) (33)
Since 120579(119904 120583) = 120601(119904 120583) + 120601(minus119904 120583) 120579(119904 120583) owns allproperties in Lemma 8 and the following results hold
Lemma 5 is a smoothing function 119891 in (24) with thefollowing properties
(i) (sdot 120583) is pseudoconvex for any fixed 120583 gt 0
(ii) lim119911rarr119909120583darr0
nabla119909(119909 120583) sube 120597119891(119909)
(iii) 0 le nabla120583(119909 120583) le 120574119899 + 120590 forall119909 isin 119877119899 120583 isin (0 +infin)
From the projected gradient method and the viscosityregularization method we introduce the following neuralnetwork to solve (24)
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(34)
where 120583(119905) = 119890minus119905 and 120576(119905) = 1(119905 + 1) with 1205760gt 0
Mathematical Problems in Engineering 7
120583intminus1
minus1
xintminus
sumsum PΩminus
minus
++
120576intab
bba
aab
pnablax
xqnabla
Figure 1 Simple block structure of proposed network (37)
By the definitions for (34) can be expressed as
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)
minus
119903
sum
119894=1
nabla119904120601 (119904 120583 (119905))
119904=119892119894(119909(119905))nabla119892
119894(119909 (119905)) minus 120576 (119905)
sdot (nablaΘ (119909 (119905)) + 120574
119899
sum
119894=1
nabla119904120579 (119904 120583)
119904=119909119894(119905)119890119894
+ 120590nabla119904120579 (119904 120583)
119904=119890119879119909(119905)
119890)]
(35)
where
nabla119904120601 (119904 120583) =
1 if 119904 gt 120583119904 + 120583
2120583if |119904| le 120583
0 if 119904 lt minus120583
nabla119904120579 (119904 120583) =
1 if |119904| ge 120583119904
120583if |119904| lt 120583
(36)
To implement (34) by circuits we can use the reformu-lated form of (34) as follows
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
120576 (119905) = minus1205762
(119905)
(119905) = minus120583 (119905)
(37)
Equation (34) can be seen as a network with three input andthree output variables that are 119909(119905) 120583(119905) and 120576(119905) A simpleblock structure of the proposed network (37) implementedby circuits is presented in Figure 1
42 Theoretical Analysis In this subsection we study somenecessary dynamical and optimality properties of proposednetwork (34) for solving (24)
The global existence of the solutions of (34) is a necessarycondition for its usability in optimization With an initial
point 1199090isin Ω the solution of (34) is global existentMoreover
the uniqueness of the solution of (34) with 1199090isin Ω is proved
under some conditions The proposed network (34) can beimplemented in circuits and mathematical software Thenthe feasibility and optimality of network (34) for optimizationproblem (24) are proved theoretically
We call 119909 [0 119879) with T gt 0 a solution of (34) if 119909 isabsolutely continuous on [0 119879) and satisfies (34) everywhere
Proposition 6 For any initial point 1199090isin Ω there is a global
solution of (34) defined on [0 +infin) and it satisfies
119909 (119905) isin Ω forall119905 isin [0 +infin) (38)
Proof Since the right hand function in network (34) iscontinuous about 119909 and 119905 there are 119879 gt 0 and an absolutecontinuous function 119909 [0 119879) rarr 119877
119899 such that 119909(119905) satisfies(34) for all 119905 isin [0 119879)
Denote 120579(119905) = 119875Ω[119909(119905) minus nabla
119909(119909(119905) 120583(119905)) minus 120576(119905)nabla
119909(119909(119905)
120583(119905))] Then (34) can be rewritten as
(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)
A simple integration procedure of the above equation gives
119909 (119905) = 119890minus119905
1199090+ 119890
minus119905
int
119905
0
120579 (119904) 119890119904
119889119904 (40)
which can be rewritten as
119909 (119905) = 119890minus119905
1199090+ (1 minus 119890
minus119905
)int
119905
0
120579 (119904)119890119904
119890119905 minus 1119889119904 (41)
By int1199050
119890119904
(119890119905
minus 1)119889119904 = 1 120579(119904) isin Ω forall0 le 119904 le 119905 and 1199090isin Ω we
obtain that 119909(119905) isin Ω forall119905 isin [0 119879)By the boundedness of Ω and the extension theory this
solution of (34) can be extended Thus the solution of (34)with initial point 119909
0isin Ω is globally existent Similarly we can
obtain that
119909 (119905) isin Ω forall119905 isin [0 +infin) (42)
Some Lipschitz condition is often used to guaranteethe uniqueness of the solution of a neural network Inwhat follows we give a sufficient condition to ensure theuniqueness of the solution of (34) with initial point 119909
0isin Ω
Proposition 7 For any initial point 1199090isin Ω if nabla
119909(sdot 120583) and
nabla119909(sdot 120583) are locally Lipschitz continuous for any fixed 120583 isin
(0 1] then the solution of neural network (34) is unique
Proof By Proposition 6 the solutions of (34) with initialpoint119909
0isin Ω exist globally and satisfy119909(119905) isin Ωforall119905 isin [0 +infin)
Suppose that there exist two solutions 119909 [0 +infin) rarr 119877119899
and 119910 [0 +infin) rarr 119877119899 of (34) with initial point 119909
0isin Ω
and suppose there exists such that = inf119905ge0119909(119905) =119910(119905)
119905 By theboundedness of Ω there is 119877 gt 0 such that 119909(119905) le 119877 and119910(119905) le 119877 forall119905 isin [0 +infin)
8 Mathematical Problems in Engineering
Then there is 119871 gt 0 such that10038161003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla
119909 (119910 (119905) 120583 (119905))
10038161003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
1003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla119909 (119910 (119905) 120583 (119905))1003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
(43)
Differentiating (12)119909(119905)minus119910(119905)2 along the two solutionsof (34) by the Lipschitz continuity of 119875
Ωand (43) we have
119889
119889119905
1
2
1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817
2
= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩
le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817
2
(44)
Applying Gronwallrsquos inequality into the integration of theabove inequality it gives 119909(119905) = 119910(119905) forall119905 isin [119905 + 1] whichleads to a contradiction Therefore the solution of (34) withinitial point 119909
0isin Ω is unique
Lyapunov method is employed to analyze the perfor-mance of (34) Here we introduce the following two Lya-punov energy functions
119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
119866 (119909 119905) =1
2dist2 (119909M) + (119909 120583 (119905))
+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
(45)
The above two Lyapunov functions satisfy the followingestimations along the solutions of (34)
Lemma 8 (i) The derivative of 119864(119909 119905) along the solution of(34) can be calculated by
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
(46)
(ii) The derivative of 119866(119909 119905) along the solution of (34) can becalculated by
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(47)
Proof (i) Differentiating119864(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119864 (119909 (119905) 119905) = ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(48)
Equation (34) can be rewritten as
(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(49)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119909(119905) using (49) and (10) we have
⟨119909 (119905) minus nabla119909 (119909 (119905) 120583 (119905)) minus 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
minus (119905) minus 119909 (119905) (119905) + 119909 (119905) minus 119909 (119905)⟩ ge 0
(50)
which follows the fact that
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
le minus (119905)2
forall119905 ge 0
(51)
Combining (48) and (51) we get that
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
(52)
From Lemmas 4 and 5 (119905) = minus119890minus119905 and 120576(119905) lt 0 we obtainthe estimation in (i)
(ii) Differentiating 119866(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩
+ ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(53)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119875M(119909(119905)) using (10) and (49) we have
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0
(54)
which can be rewritten as
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
minus (119905)2
(55)
Mathematical Problems in Engineering 9
Combining (53) and (55) we get
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905) (119891 (119909 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(56)
Similar to the analysis in (i) we obtain the estimation in(ii)
Next we prove the efficiency of proposed network (34) forsolving optimization problem (24) where the convergencefeasibility of the proposed network is a basic property
Theorem 9 For initial point 1199090isin Ω any solution 119909 [0 +infin)
rarr 119877119899 of (34) satisfies
lim119905rarr+infin
dist (119909 (119905) F) = 0 (57)
Proof Denote 119909 [0 +infin) rarr 119877119899 as a global solution of (34)
with initial point 1199090isin Ω
From 119909(119905) isin Ω forall119905 isin [0 +infin) and Lemma 8 119864(119909(119905) 119905) isnonincreasing along the solution of (34) Using119864(119909(119905) 119905) ge 0forall119905 isin [0 +infin) we confirm that
lim119905rarr+infin
119864 (119909 (119905) 119905) exists (58)
From 119909(119905) isin Ω forall119905 isin [0 +infin) and lim119905rarr+infin
120576(119905) = 0 wehave
lim119905rarr+infin
120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ
119891) = 0 (59)
which implies
lim119905rarr+infin
119864 (119909 (119905) 119905) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) (60)
Inwhat follows wewill prove that lim119905rarr+infin
(119909(119905) 120583(119905)) =
0 Arguing by contradiction we assume that
lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 119888 = 0 (61)
By (119909(119905) 120583(119905)) ge 119901(119909(119905)) ge 0 forall119905 isin [0 +infin) we have 119888 gt 0which follows the fact that there is 119879
1ge 0 such that
(119909 (119905) 120583 (119905)) ge119888
2 forall119905 isin [119879
1 +infin) (62)
From Lemma 4 we obtain
1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)
which implies that there exists 1198792gt 119879
1such that
(119875M (119909 (119905)) 120583 (119905)) le119888
8 (64)
Since Ω is bounded and 119909(119905) isin Ω forall119905 isin [0 +infin) byLemma 5 there is 119877 gt 0 such that10038161003816100381610038161003816⟨nabla
119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
10038161003816100381610038161003816le 119877
forall119905 isin [0 +infin)
(65)
Then
lim119905rarr+infin
120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)
which implies that there is 1198793ge 119879
2such that
10038161003816100381610038161003816120576 (119905) ⟨nabla
119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩
10038161003816100381610038161003816le119888
8
forall119905 isin [1198792 +infin)
(67)
From Lemma 8 and the convexity of (sdot 120583) for any fixed120583 gt 0 we obtain that
119889
119889119905119866 (119909 (119905) 119905) le minus ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)
sdot ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(68)
By (62) (67) and (68) we obtain
119889
119889119905119866 (119909 (119905) 119905) le minus
119888
4 forall119905 isin [119879
3 +infin) (69)
Integrating the above inequality from 1198793to 119905 (gt119879
3) we
have
119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879
3) minus
119888
4(119905 minus 119879
3)
forall119905 isin [1198793 +infin)
(70)
Thus
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (71)
which leads to a contradiction with 119866(119909 119905) ge 0 for all 119909 isin 119877119899and 119905 isin [0 +infin) Therefore
lim119905rarr+infin
119901 (119909 (119905)) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 0 (72)
which guarantees that
lim119905rarr+infin
dist (119909 (119905) F) = 0 (73)
The following theorem indicates that any accumulationpoint of the solutions of (34) is just an optimal solution of(34)
10 Mathematical Problems in Engineering
Theorem 10 For initial point 1199090isin Ω any solution 119909 of (34)
is convergent to the optimal solution setM that is
lim119905rarr+infin
dist (119909 (119905) M) = 0 (74)
Proof From (63) (68) and (119909(119905) 120583(119905)) ge 0 forall119905 isin [0 +infin)we have
119889
119889119905119866 (119909 (119905) 119905)
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
+ 119898120583 (119905)
(75)
by (119905) = minus120583(119905) which can be rewritten as
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(76)
Denote
119868 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le 0
119869 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
gt 0
(77)
Owning to the continuity of ⟨nabla119909(119909(119905) 120583(119905)) 119909(119905)minus119875M(119909(119905))⟩
on [0 +infin) 119868 and 119869 are closed and open in [0 +infin) respec-tively
Case 1 In this case we assume that there exists 119879 ge 0 suchthat 119905 isin 119868 forall119905 isin [119879 +infin)
From the definition on 119868 and the pseudoconvexity of(sdot 120583) on 119877119899 we have
(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))
forall119905 isin [119879 +infin)
(78)
which implies
lim sup119905rarr+infin
119891 (119909 (119905)) le min119909isinF
119891 (119909) (79)
ByTheorem 9 we confirm that
lim119905rarr+infin
dist (119909 (119905) M) = 0 (80)
Case 2 In this case we assume that there exists 119879 ge 0 such
that 119905 isin 119869 forall119905 isin [119879 +infin) which means that
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] lt 0 forall119905 isin [119879 +infin) (81)
Then
lim119905rarr+infin
[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)
Since lim119905rarr+infin
119864(119909(119905) 119905) exists and lim119905rarr+infin
120583(119905) = 0 weobtain that
lim119905rarr+infin
dist (119909 (119905) M) exists (83)
Suppose
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889
gt 0
(84)
Then there is 1198791ge such that
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge
119889
2
forall119905 isin [1198791 +infin)
(85)
Then from (76) we have
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus
119889
2120576 (119905)
forall119905 isin [1198791 +infin)
(86)
Integrating the above inequality from 1198791to 119905 (gt119879
1) we
have
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879
1) + 119898120583 (119879
1)
+ int
119905
1198791
minus119889
2120576 (119904) 119889119904
(87)
Let 119905 rarr +infin in the above inequality then we have
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (88)
which leads to a contraction with the boundedness frombelow 119866(119909(119905) 119905) on [0 +infin) Thus
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)
which follows the fact that there is an increasing sequence 119905119899
such that 119905119899rarr +infin and
lim119899rarr+infin
⟨nabla119909 (119909 (119905
119899) 120583 (119905
119899)) 119909 (119905
119899) minus 119875M (119909 (119905
119899))⟩
= 0
(90)
By 119909(119905119899) sube Ω and Ω which is bounded there are 119909lowastlowast isin
Ω and a subsequence of 119909(119905119899) (denoted as 119909(119905
119899119896)) such that
lim119896rarr+infin
119905119899119896= +infin and lim
119896rarr+infin119909(119905
119899119896) = 119909
lowastlowast By Lemma 5there is 120585(119909lowastlowast) isin 120597119891(119909lowastlowast) such that
⟨120585 (119909lowastlowast
) 119909lowastlowast
minus 119875M (119909lowastlowast
)⟩ = 0 (91)
From the pseudoconvexity of 119891 and the above inequalitywe have
119891 (119909lowastlowast
) le 119891 (119875M (119909lowastlowast
)) (92)
Mathematical Problems in Engineering 11
Table 1 Experimental data set in MDF hot-pressing
Regression parameter 120574 120590 119862 MSE DF SEMOR 0001 5 30 00307 09543 minus08299
MOE 0001 6 40 00642 09063 minus17060
IBS 0001 5 40 00451 09256 minus06388
By Theorem 9 we have 119909lowastlowast isin F Therefore 119909lowastlowast isin Mwhich implies that
lim119896rarr+infin
dist (119909 (119905119899119896) M) = 0 (93)
Combining the above results with (83) we conclude thatlim
119905rarr+infindist(119909(119905)M) = 0
Case 3 In this case we assume that both 119868 and 119869 areunbounded
For 119905 isin 119868 similar to the analysis in Case 1 we have
lim119905rarr+infin119905isin119868
dist (119909 (119905) M) = 0 (94)
For 119905 isin 119869 define 120591(119905) = sup119904le119905119904isin119868
119904 Then 120591(119905) isin 119868 and(120591(119905) 119905] sube 119869 By the unboundedness of 119868 and 119869 we have
lim119905isin119869119905rarr+infin
120591 (119905) = +infin (95)
From (81) and the continuity of 119866(119909(119905) 119905) on [0 +infin) wehave
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))
+ 119898120583 (120591 (119905)) forall119905 isin 119869
(96)
By 120591(119905) isin 119868 (94) and (95) we have
lim sup119905rarr+infin119905isin119869
[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le lim119905rarr+infin119905isin119869
[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0
(97)
which gives
lim119905rarr+infin119905isin119869
119866 (119909 (119905) 119905) = 0 (98)
From the definition of 119866(119909 119905) and the above result wehave
lim119905rarr+infin119905isin119869
dist (119909 (119905) M) = 0 (99)
Therefore from (94) and (99) we obtain
lim119905rarr+infin
dist (119909 (119905) M) = 0 (100)
5 Numerical Experiments
In this section we test the proposed neural network (34)for solving problem (24) which is brought forward from
the MDF continuous hot-pressing process Based on theexisting data set we use the established theories and proposedneural network (34) to build the relationships between themain qualities of the hot-pressing plate and some relativetechnology parameters from optimization problem (18)Then based on optimization problem (22) we will useproposed network (34) to solve the optimal values of thetechnology parameters in hot-pressing system for optimizingthe qualities of the hot-pressing plate All these numericalexperiments validate the good performance of the proposednetwork in this paper
The numerical testing was carried out on a Lenovo PC(300GHz 200GB of RAM) with the use of Matlab 74 Andwe use ode23 to realize the neural network (34) in Matlab
51 Construction Relation Models in MDF Continuous Hot-Pressing Process In this part by considered optimizationproblem (18) and network (34) we build the relation modelwhich takes the hot-pressing temperature (TE) hot-pressingpressure (PR) hot-pressing time (TI) and moisture content(MC) of slab as the argument variables and the MOR MOEand IBS indexes of MDF as the dependent variables Thenumerical results show the good fitting of the built modelsfor the data set where the data set is given in Table 4
In order to use the data in Table 4 we first normalize theminto [0 1] And we use the mean square error (MSE) degreeof fitting (DF) and sufficient evaluation (SE) to evaluate thenumerical results where
MSE MSE = radicsum119899
119894=1(119909
119894minus 119909
lowast
119894)2
119899
DF DF = 1 minus radic(sum119899
119894=1(119909
119894minus 119909
lowast
119894)2
sum119899
119894=11199092
119894
)
SE SE = 12119909119879
119876119909 minus 119910119879
119909
(101)
where 119909 and 119909lowast indicate the actual value and predicted valueand 119876 and 119910 are defined as in (18) The smaller the MSE andthe SE and the closer the DF to 1 the better the regressionresult Moreover the SE function is the objective function in(15)
Based on the Gaussian radial basis function kernel thevalues of the initial parameters in problem (18) are given inTable 1 With a random initial point 119909
0isin Ω the numerical
results with respect to the obtained solution are also listed inTable 1 Figures 2ndash4 illustrate the fitting effect of the MORMOE and IBS values by using the proposed network (34) forsolving problem (18) And the parameters of the regressionfunctions in (19) are shown in Table 5
12 Mathematical Problems in Engineering
Table 2 Criterion of physical and mechanical performance indexes
Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45
IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045
MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700
2520151050
OriginalPredicted
0
05
1
15
2
25
3
35
4
5
45
Figure 2 Normalized sample fitting results of network (34) for theregression of MOR
Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5
52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2
521 Case 1
Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 3 Normalized sample fitting results of network (34) for theregression of MOE
By the information in Table 2 we let ]1= 200 ]
2= 4
]3= 7 and ]
4= 12 and choose 119910lowast
1= 22 119910lowast
2= 2500 120580
1= 8
1205761= 8 120580
2= 500 and 120576
2= 500 in optimization problem (21)
Then Ω = 119909 isin 1198774
0 le 1199091le ]
1 0 le 119909
2le ]
2 0 le 119909
3le
]3 0 le 119909
4le ]
4 and we can let
1198921(119909) = minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) =
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
1198923(119909) = minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) =
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(102)
By the proposed network (34) for solving (21) we obtainthe optimal solution
119909lowast
= (1594725 31022 64866 60061)119879
(103)
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
where 120574 120590 ge 0 119886 119887 isin 119877119899 with 119886 lt 119887 Θ 119877
119899
rarr 119877
is continuously differentiable and pseudoconvex on 119877119899 and119892119894 119877
119899
rarr 119877 (119894 = 1 2 119898) is continuously differentiableand convex on 119877119899
On the one hand when Θ(119909) = (12)119909119879119876119909 minus 119910119879119909 119886 flminus119862119890 119887 fl 119862119890 and 120574 and 120590 are defined as in (18) then problem(24) without 119892
119894reduces to problem (18) On the other hand
if we let
Θ (119909) fl minus
119899
sum
119894=1
120581lowast
119894exp(minus
1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817
2
21205902) minus 119887
3
119886 fl 0
119887 fl (]1 ]2 ]3 ]4)119879
(25)
1198921(119909) fl minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) fl
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
(26)
1198923(119909) fl minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) fl
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(27)
then problem (24) reduces to problem (22) Similar reformu-lation can be done for problem (23) by (24)
Therefore problem (24) considered in this paper includesthe optimizationmodels for building up the relationships andoptimizing the relative parameters in MDF continuous hot-pressing process
Inwhat follows we denoteF as the feasible region of (24)that is
F fl 119909 isin Ω 119892119894(119909) le 0 119894 = 1 2 119898 (28)
andM is the optimal solution set of (24)
4 Main Results
41 Proposed Neural Network In this subsection we proposea one-layer recurrent neural network for solving problem(24) where we combine the penalty function and projectionmethods to solve the constraints and use the smoothingtechniques to overcome the nonsmoothness of the objectivefunction and penalty function
Define the penalty function
119901 (119909) =
119898
sum
119894=1
max 0 119892119894(119909) (29)
Then 119909 isin Ω 119901(119909) le 0 = FFrom the smoothing functions in (25) for the plus
function we define the smoothing function of 119901 as
(119909 120583) =
119898
sum
119894=1
120601 (119892119894(119909) 120583) (30)
where
120601 (119904 120583) =
max 0 119904 if |119904| gt 120583
(119904 + 120583)2
4120583if |119904| le 120583
(31)
Form the results in [23] 120601(119904 120583) owns the followingproperties
Lemma 3 (see [23]) (i) For any 119904 isin 119877 120601(119904 sdot) is continuouslydifferentiable and 120601(sdot 120583) is also continuously differentiable forany fixed 120583 gt 0
(ii) 0 le nabla120583120601(119904 120583) le 1 forall119904 isin 119877 forall120583 isin (0 +infin)
(iii) 0 le 120601(119904 120583) minus |119904| le 1205834 forall119904 isin 119877 forall120583 isin (0 +infin)(iv) 120601(sdot 120583) is convex for any fixed 120583 gt 0 and
lim119904rarr119905120583darr0
nabla119904120601(119904 120583) sube 120597max0 119905
Then has the following properties
Lemma 4 is a smoothing function 119901 and satisfies thefollowing
(i) (sdot 120583) is convex for any fixed 120583 gt 0(ii) lim
119911rarr119909120583darr0nabla119911(119911 120583) sube 120597119901(119909)
(iii) 0 le nabla120583(119909 120583) le 119898 forall119909 isin 119877119899 120583 isin (0 +infin)
Next by the smoothing function for the absolute valuefunction | sdot |
120579 (119904 120583) =
|119904| if |119904| ge 120583
1199042
2120583+120583
2if |119904| lt 120583
(32)
we define
(119909 120583) fl Θ (119909) + 120574
119899
sum
119894=1
120579 (119909119894 120583) + 120590120579 (119890
119879
119909 120583) (33)
Since 120579(119904 120583) = 120601(119904 120583) + 120601(minus119904 120583) 120579(119904 120583) owns allproperties in Lemma 8 and the following results hold
Lemma 5 is a smoothing function 119891 in (24) with thefollowing properties
(i) (sdot 120583) is pseudoconvex for any fixed 120583 gt 0
(ii) lim119911rarr119909120583darr0
nabla119909(119909 120583) sube 120597119891(119909)
(iii) 0 le nabla120583(119909 120583) le 120574119899 + 120590 forall119909 isin 119877119899 120583 isin (0 +infin)
From the projected gradient method and the viscosityregularization method we introduce the following neuralnetwork to solve (24)
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(34)
where 120583(119905) = 119890minus119905 and 120576(119905) = 1(119905 + 1) with 1205760gt 0
Mathematical Problems in Engineering 7
120583intminus1
minus1
xintminus
sumsum PΩminus
minus
++
120576intab
bba
aab
pnablax
xqnabla
Figure 1 Simple block structure of proposed network (37)
By the definitions for (34) can be expressed as
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)
minus
119903
sum
119894=1
nabla119904120601 (119904 120583 (119905))
119904=119892119894(119909(119905))nabla119892
119894(119909 (119905)) minus 120576 (119905)
sdot (nablaΘ (119909 (119905)) + 120574
119899
sum
119894=1
nabla119904120579 (119904 120583)
119904=119909119894(119905)119890119894
+ 120590nabla119904120579 (119904 120583)
119904=119890119879119909(119905)
119890)]
(35)
where
nabla119904120601 (119904 120583) =
1 if 119904 gt 120583119904 + 120583
2120583if |119904| le 120583
0 if 119904 lt minus120583
nabla119904120579 (119904 120583) =
1 if |119904| ge 120583119904
120583if |119904| lt 120583
(36)
To implement (34) by circuits we can use the reformu-lated form of (34) as follows
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
120576 (119905) = minus1205762
(119905)
(119905) = minus120583 (119905)
(37)
Equation (34) can be seen as a network with three input andthree output variables that are 119909(119905) 120583(119905) and 120576(119905) A simpleblock structure of the proposed network (37) implementedby circuits is presented in Figure 1
42 Theoretical Analysis In this subsection we study somenecessary dynamical and optimality properties of proposednetwork (34) for solving (24)
The global existence of the solutions of (34) is a necessarycondition for its usability in optimization With an initial
point 1199090isin Ω the solution of (34) is global existentMoreover
the uniqueness of the solution of (34) with 1199090isin Ω is proved
under some conditions The proposed network (34) can beimplemented in circuits and mathematical software Thenthe feasibility and optimality of network (34) for optimizationproblem (24) are proved theoretically
We call 119909 [0 119879) with T gt 0 a solution of (34) if 119909 isabsolutely continuous on [0 119879) and satisfies (34) everywhere
Proposition 6 For any initial point 1199090isin Ω there is a global
solution of (34) defined on [0 +infin) and it satisfies
119909 (119905) isin Ω forall119905 isin [0 +infin) (38)
Proof Since the right hand function in network (34) iscontinuous about 119909 and 119905 there are 119879 gt 0 and an absolutecontinuous function 119909 [0 119879) rarr 119877
119899 such that 119909(119905) satisfies(34) for all 119905 isin [0 119879)
Denote 120579(119905) = 119875Ω[119909(119905) minus nabla
119909(119909(119905) 120583(119905)) minus 120576(119905)nabla
119909(119909(119905)
120583(119905))] Then (34) can be rewritten as
(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)
A simple integration procedure of the above equation gives
119909 (119905) = 119890minus119905
1199090+ 119890
minus119905
int
119905
0
120579 (119904) 119890119904
119889119904 (40)
which can be rewritten as
119909 (119905) = 119890minus119905
1199090+ (1 minus 119890
minus119905
)int
119905
0
120579 (119904)119890119904
119890119905 minus 1119889119904 (41)
By int1199050
119890119904
(119890119905
minus 1)119889119904 = 1 120579(119904) isin Ω forall0 le 119904 le 119905 and 1199090isin Ω we
obtain that 119909(119905) isin Ω forall119905 isin [0 119879)By the boundedness of Ω and the extension theory this
solution of (34) can be extended Thus the solution of (34)with initial point 119909
0isin Ω is globally existent Similarly we can
obtain that
119909 (119905) isin Ω forall119905 isin [0 +infin) (42)
Some Lipschitz condition is often used to guaranteethe uniqueness of the solution of a neural network Inwhat follows we give a sufficient condition to ensure theuniqueness of the solution of (34) with initial point 119909
0isin Ω
Proposition 7 For any initial point 1199090isin Ω if nabla
119909(sdot 120583) and
nabla119909(sdot 120583) are locally Lipschitz continuous for any fixed 120583 isin
(0 1] then the solution of neural network (34) is unique
Proof By Proposition 6 the solutions of (34) with initialpoint119909
0isin Ω exist globally and satisfy119909(119905) isin Ωforall119905 isin [0 +infin)
Suppose that there exist two solutions 119909 [0 +infin) rarr 119877119899
and 119910 [0 +infin) rarr 119877119899 of (34) with initial point 119909
0isin Ω
and suppose there exists such that = inf119905ge0119909(119905) =119910(119905)
119905 By theboundedness of Ω there is 119877 gt 0 such that 119909(119905) le 119877 and119910(119905) le 119877 forall119905 isin [0 +infin)
8 Mathematical Problems in Engineering
Then there is 119871 gt 0 such that10038161003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla
119909 (119910 (119905) 120583 (119905))
10038161003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
1003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla119909 (119910 (119905) 120583 (119905))1003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
(43)
Differentiating (12)119909(119905)minus119910(119905)2 along the two solutionsof (34) by the Lipschitz continuity of 119875
Ωand (43) we have
119889
119889119905
1
2
1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817
2
= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩
le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817
2
(44)
Applying Gronwallrsquos inequality into the integration of theabove inequality it gives 119909(119905) = 119910(119905) forall119905 isin [119905 + 1] whichleads to a contradiction Therefore the solution of (34) withinitial point 119909
0isin Ω is unique
Lyapunov method is employed to analyze the perfor-mance of (34) Here we introduce the following two Lya-punov energy functions
119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
119866 (119909 119905) =1
2dist2 (119909M) + (119909 120583 (119905))
+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
(45)
The above two Lyapunov functions satisfy the followingestimations along the solutions of (34)
Lemma 8 (i) The derivative of 119864(119909 119905) along the solution of(34) can be calculated by
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
(46)
(ii) The derivative of 119866(119909 119905) along the solution of (34) can becalculated by
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(47)
Proof (i) Differentiating119864(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119864 (119909 (119905) 119905) = ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(48)
Equation (34) can be rewritten as
(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(49)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119909(119905) using (49) and (10) we have
⟨119909 (119905) minus nabla119909 (119909 (119905) 120583 (119905)) minus 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
minus (119905) minus 119909 (119905) (119905) + 119909 (119905) minus 119909 (119905)⟩ ge 0
(50)
which follows the fact that
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
le minus (119905)2
forall119905 ge 0
(51)
Combining (48) and (51) we get that
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
(52)
From Lemmas 4 and 5 (119905) = minus119890minus119905 and 120576(119905) lt 0 we obtainthe estimation in (i)
(ii) Differentiating 119866(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩
+ ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(53)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119875M(119909(119905)) using (10) and (49) we have
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0
(54)
which can be rewritten as
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
minus (119905)2
(55)
Mathematical Problems in Engineering 9
Combining (53) and (55) we get
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905) (119891 (119909 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(56)
Similar to the analysis in (i) we obtain the estimation in(ii)
Next we prove the efficiency of proposed network (34) forsolving optimization problem (24) where the convergencefeasibility of the proposed network is a basic property
Theorem 9 For initial point 1199090isin Ω any solution 119909 [0 +infin)
rarr 119877119899 of (34) satisfies
lim119905rarr+infin
dist (119909 (119905) F) = 0 (57)
Proof Denote 119909 [0 +infin) rarr 119877119899 as a global solution of (34)
with initial point 1199090isin Ω
From 119909(119905) isin Ω forall119905 isin [0 +infin) and Lemma 8 119864(119909(119905) 119905) isnonincreasing along the solution of (34) Using119864(119909(119905) 119905) ge 0forall119905 isin [0 +infin) we confirm that
lim119905rarr+infin
119864 (119909 (119905) 119905) exists (58)
From 119909(119905) isin Ω forall119905 isin [0 +infin) and lim119905rarr+infin
120576(119905) = 0 wehave
lim119905rarr+infin
120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ
119891) = 0 (59)
which implies
lim119905rarr+infin
119864 (119909 (119905) 119905) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) (60)
Inwhat follows wewill prove that lim119905rarr+infin
(119909(119905) 120583(119905)) =
0 Arguing by contradiction we assume that
lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 119888 = 0 (61)
By (119909(119905) 120583(119905)) ge 119901(119909(119905)) ge 0 forall119905 isin [0 +infin) we have 119888 gt 0which follows the fact that there is 119879
1ge 0 such that
(119909 (119905) 120583 (119905)) ge119888
2 forall119905 isin [119879
1 +infin) (62)
From Lemma 4 we obtain
1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)
which implies that there exists 1198792gt 119879
1such that
(119875M (119909 (119905)) 120583 (119905)) le119888
8 (64)
Since Ω is bounded and 119909(119905) isin Ω forall119905 isin [0 +infin) byLemma 5 there is 119877 gt 0 such that10038161003816100381610038161003816⟨nabla
119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
10038161003816100381610038161003816le 119877
forall119905 isin [0 +infin)
(65)
Then
lim119905rarr+infin
120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)
which implies that there is 1198793ge 119879
2such that
10038161003816100381610038161003816120576 (119905) ⟨nabla
119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩
10038161003816100381610038161003816le119888
8
forall119905 isin [1198792 +infin)
(67)
From Lemma 8 and the convexity of (sdot 120583) for any fixed120583 gt 0 we obtain that
119889
119889119905119866 (119909 (119905) 119905) le minus ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)
sdot ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(68)
By (62) (67) and (68) we obtain
119889
119889119905119866 (119909 (119905) 119905) le minus
119888
4 forall119905 isin [119879
3 +infin) (69)
Integrating the above inequality from 1198793to 119905 (gt119879
3) we
have
119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879
3) minus
119888
4(119905 minus 119879
3)
forall119905 isin [1198793 +infin)
(70)
Thus
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (71)
which leads to a contradiction with 119866(119909 119905) ge 0 for all 119909 isin 119877119899and 119905 isin [0 +infin) Therefore
lim119905rarr+infin
119901 (119909 (119905)) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 0 (72)
which guarantees that
lim119905rarr+infin
dist (119909 (119905) F) = 0 (73)
The following theorem indicates that any accumulationpoint of the solutions of (34) is just an optimal solution of(34)
10 Mathematical Problems in Engineering
Theorem 10 For initial point 1199090isin Ω any solution 119909 of (34)
is convergent to the optimal solution setM that is
lim119905rarr+infin
dist (119909 (119905) M) = 0 (74)
Proof From (63) (68) and (119909(119905) 120583(119905)) ge 0 forall119905 isin [0 +infin)we have
119889
119889119905119866 (119909 (119905) 119905)
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
+ 119898120583 (119905)
(75)
by (119905) = minus120583(119905) which can be rewritten as
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(76)
Denote
119868 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le 0
119869 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
gt 0
(77)
Owning to the continuity of ⟨nabla119909(119909(119905) 120583(119905)) 119909(119905)minus119875M(119909(119905))⟩
on [0 +infin) 119868 and 119869 are closed and open in [0 +infin) respec-tively
Case 1 In this case we assume that there exists 119879 ge 0 suchthat 119905 isin 119868 forall119905 isin [119879 +infin)
From the definition on 119868 and the pseudoconvexity of(sdot 120583) on 119877119899 we have
(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))
forall119905 isin [119879 +infin)
(78)
which implies
lim sup119905rarr+infin
119891 (119909 (119905)) le min119909isinF
119891 (119909) (79)
ByTheorem 9 we confirm that
lim119905rarr+infin
dist (119909 (119905) M) = 0 (80)
Case 2 In this case we assume that there exists 119879 ge 0 such
that 119905 isin 119869 forall119905 isin [119879 +infin) which means that
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] lt 0 forall119905 isin [119879 +infin) (81)
Then
lim119905rarr+infin
[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)
Since lim119905rarr+infin
119864(119909(119905) 119905) exists and lim119905rarr+infin
120583(119905) = 0 weobtain that
lim119905rarr+infin
dist (119909 (119905) M) exists (83)
Suppose
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889
gt 0
(84)
Then there is 1198791ge such that
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge
119889
2
forall119905 isin [1198791 +infin)
(85)
Then from (76) we have
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus
119889
2120576 (119905)
forall119905 isin [1198791 +infin)
(86)
Integrating the above inequality from 1198791to 119905 (gt119879
1) we
have
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879
1) + 119898120583 (119879
1)
+ int
119905
1198791
minus119889
2120576 (119904) 119889119904
(87)
Let 119905 rarr +infin in the above inequality then we have
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (88)
which leads to a contraction with the boundedness frombelow 119866(119909(119905) 119905) on [0 +infin) Thus
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)
which follows the fact that there is an increasing sequence 119905119899
such that 119905119899rarr +infin and
lim119899rarr+infin
⟨nabla119909 (119909 (119905
119899) 120583 (119905
119899)) 119909 (119905
119899) minus 119875M (119909 (119905
119899))⟩
= 0
(90)
By 119909(119905119899) sube Ω and Ω which is bounded there are 119909lowastlowast isin
Ω and a subsequence of 119909(119905119899) (denoted as 119909(119905
119899119896)) such that
lim119896rarr+infin
119905119899119896= +infin and lim
119896rarr+infin119909(119905
119899119896) = 119909
lowastlowast By Lemma 5there is 120585(119909lowastlowast) isin 120597119891(119909lowastlowast) such that
⟨120585 (119909lowastlowast
) 119909lowastlowast
minus 119875M (119909lowastlowast
)⟩ = 0 (91)
From the pseudoconvexity of 119891 and the above inequalitywe have
119891 (119909lowastlowast
) le 119891 (119875M (119909lowastlowast
)) (92)
Mathematical Problems in Engineering 11
Table 1 Experimental data set in MDF hot-pressing
Regression parameter 120574 120590 119862 MSE DF SEMOR 0001 5 30 00307 09543 minus08299
MOE 0001 6 40 00642 09063 minus17060
IBS 0001 5 40 00451 09256 minus06388
By Theorem 9 we have 119909lowastlowast isin F Therefore 119909lowastlowast isin Mwhich implies that
lim119896rarr+infin
dist (119909 (119905119899119896) M) = 0 (93)
Combining the above results with (83) we conclude thatlim
119905rarr+infindist(119909(119905)M) = 0
Case 3 In this case we assume that both 119868 and 119869 areunbounded
For 119905 isin 119868 similar to the analysis in Case 1 we have
lim119905rarr+infin119905isin119868
dist (119909 (119905) M) = 0 (94)
For 119905 isin 119869 define 120591(119905) = sup119904le119905119904isin119868
119904 Then 120591(119905) isin 119868 and(120591(119905) 119905] sube 119869 By the unboundedness of 119868 and 119869 we have
lim119905isin119869119905rarr+infin
120591 (119905) = +infin (95)
From (81) and the continuity of 119866(119909(119905) 119905) on [0 +infin) wehave
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))
+ 119898120583 (120591 (119905)) forall119905 isin 119869
(96)
By 120591(119905) isin 119868 (94) and (95) we have
lim sup119905rarr+infin119905isin119869
[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le lim119905rarr+infin119905isin119869
[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0
(97)
which gives
lim119905rarr+infin119905isin119869
119866 (119909 (119905) 119905) = 0 (98)
From the definition of 119866(119909 119905) and the above result wehave
lim119905rarr+infin119905isin119869
dist (119909 (119905) M) = 0 (99)
Therefore from (94) and (99) we obtain
lim119905rarr+infin
dist (119909 (119905) M) = 0 (100)
5 Numerical Experiments
In this section we test the proposed neural network (34)for solving problem (24) which is brought forward from
the MDF continuous hot-pressing process Based on theexisting data set we use the established theories and proposedneural network (34) to build the relationships between themain qualities of the hot-pressing plate and some relativetechnology parameters from optimization problem (18)Then based on optimization problem (22) we will useproposed network (34) to solve the optimal values of thetechnology parameters in hot-pressing system for optimizingthe qualities of the hot-pressing plate All these numericalexperiments validate the good performance of the proposednetwork in this paper
The numerical testing was carried out on a Lenovo PC(300GHz 200GB of RAM) with the use of Matlab 74 Andwe use ode23 to realize the neural network (34) in Matlab
51 Construction Relation Models in MDF Continuous Hot-Pressing Process In this part by considered optimizationproblem (18) and network (34) we build the relation modelwhich takes the hot-pressing temperature (TE) hot-pressingpressure (PR) hot-pressing time (TI) and moisture content(MC) of slab as the argument variables and the MOR MOEand IBS indexes of MDF as the dependent variables Thenumerical results show the good fitting of the built modelsfor the data set where the data set is given in Table 4
In order to use the data in Table 4 we first normalize theminto [0 1] And we use the mean square error (MSE) degreeof fitting (DF) and sufficient evaluation (SE) to evaluate thenumerical results where
MSE MSE = radicsum119899
119894=1(119909
119894minus 119909
lowast
119894)2
119899
DF DF = 1 minus radic(sum119899
119894=1(119909
119894minus 119909
lowast
119894)2
sum119899
119894=11199092
119894
)
SE SE = 12119909119879
119876119909 minus 119910119879
119909
(101)
where 119909 and 119909lowast indicate the actual value and predicted valueand 119876 and 119910 are defined as in (18) The smaller the MSE andthe SE and the closer the DF to 1 the better the regressionresult Moreover the SE function is the objective function in(15)
Based on the Gaussian radial basis function kernel thevalues of the initial parameters in problem (18) are given inTable 1 With a random initial point 119909
0isin Ω the numerical
results with respect to the obtained solution are also listed inTable 1 Figures 2ndash4 illustrate the fitting effect of the MORMOE and IBS values by using the proposed network (34) forsolving problem (18) And the parameters of the regressionfunctions in (19) are shown in Table 5
12 Mathematical Problems in Engineering
Table 2 Criterion of physical and mechanical performance indexes
Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45
IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045
MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700
2520151050
OriginalPredicted
0
05
1
15
2
25
3
35
4
5
45
Figure 2 Normalized sample fitting results of network (34) for theregression of MOR
Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5
52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2
521 Case 1
Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 3 Normalized sample fitting results of network (34) for theregression of MOE
By the information in Table 2 we let ]1= 200 ]
2= 4
]3= 7 and ]
4= 12 and choose 119910lowast
1= 22 119910lowast
2= 2500 120580
1= 8
1205761= 8 120580
2= 500 and 120576
2= 500 in optimization problem (21)
Then Ω = 119909 isin 1198774
0 le 1199091le ]
1 0 le 119909
2le ]
2 0 le 119909
3le
]3 0 le 119909
4le ]
4 and we can let
1198921(119909) = minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) =
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
1198923(119909) = minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) =
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(102)
By the proposed network (34) for solving (21) we obtainthe optimal solution
119909lowast
= (1594725 31022 64866 60061)119879
(103)
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
120583intminus1
minus1
xintminus
sumsum PΩminus
minus
++
120576intab
bba
aab
pnablax
xqnabla
Figure 1 Simple block structure of proposed network (37)
By the definitions for (34) can be expressed as
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)
minus
119903
sum
119894=1
nabla119904120601 (119904 120583 (119905))
119904=119892119894(119909(119905))nabla119892
119894(119909 (119905)) minus 120576 (119905)
sdot (nablaΘ (119909 (119905)) + 120574
119899
sum
119894=1
nabla119904120579 (119904 120583)
119904=119909119894(119905)119890119894
+ 120590nabla119904120579 (119904 120583)
119904=119890119879119909(119905)
119890)]
(35)
where
nabla119904120601 (119904 120583) =
1 if 119904 gt 120583119904 + 120583
2120583if |119904| le 120583
0 if 119904 lt minus120583
nabla119904120579 (119904 120583) =
1 if |119904| ge 120583119904
120583if |119904| lt 120583
(36)
To implement (34) by circuits we can use the reformu-lated form of (34) as follows
(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
120576 (119905) = minus1205762
(119905)
(119905) = minus120583 (119905)
(37)
Equation (34) can be seen as a network with three input andthree output variables that are 119909(119905) 120583(119905) and 120576(119905) A simpleblock structure of the proposed network (37) implementedby circuits is presented in Figure 1
42 Theoretical Analysis In this subsection we study somenecessary dynamical and optimality properties of proposednetwork (34) for solving (24)
The global existence of the solutions of (34) is a necessarycondition for its usability in optimization With an initial
point 1199090isin Ω the solution of (34) is global existentMoreover
the uniqueness of the solution of (34) with 1199090isin Ω is proved
under some conditions The proposed network (34) can beimplemented in circuits and mathematical software Thenthe feasibility and optimality of network (34) for optimizationproblem (24) are proved theoretically
We call 119909 [0 119879) with T gt 0 a solution of (34) if 119909 isabsolutely continuous on [0 119879) and satisfies (34) everywhere
Proposition 6 For any initial point 1199090isin Ω there is a global
solution of (34) defined on [0 +infin) and it satisfies
119909 (119905) isin Ω forall119905 isin [0 +infin) (38)
Proof Since the right hand function in network (34) iscontinuous about 119909 and 119905 there are 119879 gt 0 and an absolutecontinuous function 119909 [0 119879) rarr 119877
119899 such that 119909(119905) satisfies(34) for all 119905 isin [0 119879)
Denote 120579(119905) = 119875Ω[119909(119905) minus nabla
119909(119909(119905) 120583(119905)) minus 120576(119905)nabla
119909(119909(119905)
120583(119905))] Then (34) can be rewritten as
(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)
A simple integration procedure of the above equation gives
119909 (119905) = 119890minus119905
1199090+ 119890
minus119905
int
119905
0
120579 (119904) 119890119904
119889119904 (40)
which can be rewritten as
119909 (119905) = 119890minus119905
1199090+ (1 minus 119890
minus119905
)int
119905
0
120579 (119904)119890119904
119890119905 minus 1119889119904 (41)
By int1199050
119890119904
(119890119905
minus 1)119889119904 = 1 120579(119904) isin Ω forall0 le 119904 le 119905 and 1199090isin Ω we
obtain that 119909(119905) isin Ω forall119905 isin [0 119879)By the boundedness of Ω and the extension theory this
solution of (34) can be extended Thus the solution of (34)with initial point 119909
0isin Ω is globally existent Similarly we can
obtain that
119909 (119905) isin Ω forall119905 isin [0 +infin) (42)
Some Lipschitz condition is often used to guaranteethe uniqueness of the solution of a neural network Inwhat follows we give a sufficient condition to ensure theuniqueness of the solution of (34) with initial point 119909
0isin Ω
Proposition 7 For any initial point 1199090isin Ω if nabla
119909(sdot 120583) and
nabla119909(sdot 120583) are locally Lipschitz continuous for any fixed 120583 isin
(0 1] then the solution of neural network (34) is unique
Proof By Proposition 6 the solutions of (34) with initialpoint119909
0isin Ω exist globally and satisfy119909(119905) isin Ωforall119905 isin [0 +infin)
Suppose that there exist two solutions 119909 [0 +infin) rarr 119877119899
and 119910 [0 +infin) rarr 119877119899 of (34) with initial point 119909
0isin Ω
and suppose there exists such that = inf119905ge0119909(119905) =119910(119905)
119905 By theboundedness of Ω there is 119877 gt 0 such that 119909(119905) le 119877 and119910(119905) le 119877 forall119905 isin [0 +infin)
8 Mathematical Problems in Engineering
Then there is 119871 gt 0 such that10038161003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla
119909 (119910 (119905) 120583 (119905))
10038161003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
1003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla119909 (119910 (119905) 120583 (119905))1003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
(43)
Differentiating (12)119909(119905)minus119910(119905)2 along the two solutionsof (34) by the Lipschitz continuity of 119875
Ωand (43) we have
119889
119889119905
1
2
1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817
2
= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩
le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817
2
(44)
Applying Gronwallrsquos inequality into the integration of theabove inequality it gives 119909(119905) = 119910(119905) forall119905 isin [119905 + 1] whichleads to a contradiction Therefore the solution of (34) withinitial point 119909
0isin Ω is unique
Lyapunov method is employed to analyze the perfor-mance of (34) Here we introduce the following two Lya-punov energy functions
119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
119866 (119909 119905) =1
2dist2 (119909M) + (119909 120583 (119905))
+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
(45)
The above two Lyapunov functions satisfy the followingestimations along the solutions of (34)
Lemma 8 (i) The derivative of 119864(119909 119905) along the solution of(34) can be calculated by
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
(46)
(ii) The derivative of 119866(119909 119905) along the solution of (34) can becalculated by
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(47)
Proof (i) Differentiating119864(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119864 (119909 (119905) 119905) = ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(48)
Equation (34) can be rewritten as
(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(49)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119909(119905) using (49) and (10) we have
⟨119909 (119905) minus nabla119909 (119909 (119905) 120583 (119905)) minus 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
minus (119905) minus 119909 (119905) (119905) + 119909 (119905) minus 119909 (119905)⟩ ge 0
(50)
which follows the fact that
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
le minus (119905)2
forall119905 ge 0
(51)
Combining (48) and (51) we get that
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
(52)
From Lemmas 4 and 5 (119905) = minus119890minus119905 and 120576(119905) lt 0 we obtainthe estimation in (i)
(ii) Differentiating 119866(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩
+ ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(53)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119875M(119909(119905)) using (10) and (49) we have
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0
(54)
which can be rewritten as
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
minus (119905)2
(55)
Mathematical Problems in Engineering 9
Combining (53) and (55) we get
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905) (119891 (119909 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(56)
Similar to the analysis in (i) we obtain the estimation in(ii)
Next we prove the efficiency of proposed network (34) forsolving optimization problem (24) where the convergencefeasibility of the proposed network is a basic property
Theorem 9 For initial point 1199090isin Ω any solution 119909 [0 +infin)
rarr 119877119899 of (34) satisfies
lim119905rarr+infin
dist (119909 (119905) F) = 0 (57)
Proof Denote 119909 [0 +infin) rarr 119877119899 as a global solution of (34)
with initial point 1199090isin Ω
From 119909(119905) isin Ω forall119905 isin [0 +infin) and Lemma 8 119864(119909(119905) 119905) isnonincreasing along the solution of (34) Using119864(119909(119905) 119905) ge 0forall119905 isin [0 +infin) we confirm that
lim119905rarr+infin
119864 (119909 (119905) 119905) exists (58)
From 119909(119905) isin Ω forall119905 isin [0 +infin) and lim119905rarr+infin
120576(119905) = 0 wehave
lim119905rarr+infin
120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ
119891) = 0 (59)
which implies
lim119905rarr+infin
119864 (119909 (119905) 119905) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) (60)
Inwhat follows wewill prove that lim119905rarr+infin
(119909(119905) 120583(119905)) =
0 Arguing by contradiction we assume that
lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 119888 = 0 (61)
By (119909(119905) 120583(119905)) ge 119901(119909(119905)) ge 0 forall119905 isin [0 +infin) we have 119888 gt 0which follows the fact that there is 119879
1ge 0 such that
(119909 (119905) 120583 (119905)) ge119888
2 forall119905 isin [119879
1 +infin) (62)
From Lemma 4 we obtain
1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)
which implies that there exists 1198792gt 119879
1such that
(119875M (119909 (119905)) 120583 (119905)) le119888
8 (64)
Since Ω is bounded and 119909(119905) isin Ω forall119905 isin [0 +infin) byLemma 5 there is 119877 gt 0 such that10038161003816100381610038161003816⟨nabla
119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
10038161003816100381610038161003816le 119877
forall119905 isin [0 +infin)
(65)
Then
lim119905rarr+infin
120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)
which implies that there is 1198793ge 119879
2such that
10038161003816100381610038161003816120576 (119905) ⟨nabla
119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩
10038161003816100381610038161003816le119888
8
forall119905 isin [1198792 +infin)
(67)
From Lemma 8 and the convexity of (sdot 120583) for any fixed120583 gt 0 we obtain that
119889
119889119905119866 (119909 (119905) 119905) le minus ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)
sdot ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(68)
By (62) (67) and (68) we obtain
119889
119889119905119866 (119909 (119905) 119905) le minus
119888
4 forall119905 isin [119879
3 +infin) (69)
Integrating the above inequality from 1198793to 119905 (gt119879
3) we
have
119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879
3) minus
119888
4(119905 minus 119879
3)
forall119905 isin [1198793 +infin)
(70)
Thus
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (71)
which leads to a contradiction with 119866(119909 119905) ge 0 for all 119909 isin 119877119899and 119905 isin [0 +infin) Therefore
lim119905rarr+infin
119901 (119909 (119905)) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 0 (72)
which guarantees that
lim119905rarr+infin
dist (119909 (119905) F) = 0 (73)
The following theorem indicates that any accumulationpoint of the solutions of (34) is just an optimal solution of(34)
10 Mathematical Problems in Engineering
Theorem 10 For initial point 1199090isin Ω any solution 119909 of (34)
is convergent to the optimal solution setM that is
lim119905rarr+infin
dist (119909 (119905) M) = 0 (74)
Proof From (63) (68) and (119909(119905) 120583(119905)) ge 0 forall119905 isin [0 +infin)we have
119889
119889119905119866 (119909 (119905) 119905)
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
+ 119898120583 (119905)
(75)
by (119905) = minus120583(119905) which can be rewritten as
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(76)
Denote
119868 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le 0
119869 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
gt 0
(77)
Owning to the continuity of ⟨nabla119909(119909(119905) 120583(119905)) 119909(119905)minus119875M(119909(119905))⟩
on [0 +infin) 119868 and 119869 are closed and open in [0 +infin) respec-tively
Case 1 In this case we assume that there exists 119879 ge 0 suchthat 119905 isin 119868 forall119905 isin [119879 +infin)
From the definition on 119868 and the pseudoconvexity of(sdot 120583) on 119877119899 we have
(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))
forall119905 isin [119879 +infin)
(78)
which implies
lim sup119905rarr+infin
119891 (119909 (119905)) le min119909isinF
119891 (119909) (79)
ByTheorem 9 we confirm that
lim119905rarr+infin
dist (119909 (119905) M) = 0 (80)
Case 2 In this case we assume that there exists 119879 ge 0 such
that 119905 isin 119869 forall119905 isin [119879 +infin) which means that
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] lt 0 forall119905 isin [119879 +infin) (81)
Then
lim119905rarr+infin
[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)
Since lim119905rarr+infin
119864(119909(119905) 119905) exists and lim119905rarr+infin
120583(119905) = 0 weobtain that
lim119905rarr+infin
dist (119909 (119905) M) exists (83)
Suppose
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889
gt 0
(84)
Then there is 1198791ge such that
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge
119889
2
forall119905 isin [1198791 +infin)
(85)
Then from (76) we have
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus
119889
2120576 (119905)
forall119905 isin [1198791 +infin)
(86)
Integrating the above inequality from 1198791to 119905 (gt119879
1) we
have
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879
1) + 119898120583 (119879
1)
+ int
119905
1198791
minus119889
2120576 (119904) 119889119904
(87)
Let 119905 rarr +infin in the above inequality then we have
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (88)
which leads to a contraction with the boundedness frombelow 119866(119909(119905) 119905) on [0 +infin) Thus
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)
which follows the fact that there is an increasing sequence 119905119899
such that 119905119899rarr +infin and
lim119899rarr+infin
⟨nabla119909 (119909 (119905
119899) 120583 (119905
119899)) 119909 (119905
119899) minus 119875M (119909 (119905
119899))⟩
= 0
(90)
By 119909(119905119899) sube Ω and Ω which is bounded there are 119909lowastlowast isin
Ω and a subsequence of 119909(119905119899) (denoted as 119909(119905
119899119896)) such that
lim119896rarr+infin
119905119899119896= +infin and lim
119896rarr+infin119909(119905
119899119896) = 119909
lowastlowast By Lemma 5there is 120585(119909lowastlowast) isin 120597119891(119909lowastlowast) such that
⟨120585 (119909lowastlowast
) 119909lowastlowast
minus 119875M (119909lowastlowast
)⟩ = 0 (91)
From the pseudoconvexity of 119891 and the above inequalitywe have
119891 (119909lowastlowast
) le 119891 (119875M (119909lowastlowast
)) (92)
Mathematical Problems in Engineering 11
Table 1 Experimental data set in MDF hot-pressing
Regression parameter 120574 120590 119862 MSE DF SEMOR 0001 5 30 00307 09543 minus08299
MOE 0001 6 40 00642 09063 minus17060
IBS 0001 5 40 00451 09256 minus06388
By Theorem 9 we have 119909lowastlowast isin F Therefore 119909lowastlowast isin Mwhich implies that
lim119896rarr+infin
dist (119909 (119905119899119896) M) = 0 (93)
Combining the above results with (83) we conclude thatlim
119905rarr+infindist(119909(119905)M) = 0
Case 3 In this case we assume that both 119868 and 119869 areunbounded
For 119905 isin 119868 similar to the analysis in Case 1 we have
lim119905rarr+infin119905isin119868
dist (119909 (119905) M) = 0 (94)
For 119905 isin 119869 define 120591(119905) = sup119904le119905119904isin119868
119904 Then 120591(119905) isin 119868 and(120591(119905) 119905] sube 119869 By the unboundedness of 119868 and 119869 we have
lim119905isin119869119905rarr+infin
120591 (119905) = +infin (95)
From (81) and the continuity of 119866(119909(119905) 119905) on [0 +infin) wehave
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))
+ 119898120583 (120591 (119905)) forall119905 isin 119869
(96)
By 120591(119905) isin 119868 (94) and (95) we have
lim sup119905rarr+infin119905isin119869
[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le lim119905rarr+infin119905isin119869
[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0
(97)
which gives
lim119905rarr+infin119905isin119869
119866 (119909 (119905) 119905) = 0 (98)
From the definition of 119866(119909 119905) and the above result wehave
lim119905rarr+infin119905isin119869
dist (119909 (119905) M) = 0 (99)
Therefore from (94) and (99) we obtain
lim119905rarr+infin
dist (119909 (119905) M) = 0 (100)
5 Numerical Experiments
In this section we test the proposed neural network (34)for solving problem (24) which is brought forward from
the MDF continuous hot-pressing process Based on theexisting data set we use the established theories and proposedneural network (34) to build the relationships between themain qualities of the hot-pressing plate and some relativetechnology parameters from optimization problem (18)Then based on optimization problem (22) we will useproposed network (34) to solve the optimal values of thetechnology parameters in hot-pressing system for optimizingthe qualities of the hot-pressing plate All these numericalexperiments validate the good performance of the proposednetwork in this paper
The numerical testing was carried out on a Lenovo PC(300GHz 200GB of RAM) with the use of Matlab 74 Andwe use ode23 to realize the neural network (34) in Matlab
51 Construction Relation Models in MDF Continuous Hot-Pressing Process In this part by considered optimizationproblem (18) and network (34) we build the relation modelwhich takes the hot-pressing temperature (TE) hot-pressingpressure (PR) hot-pressing time (TI) and moisture content(MC) of slab as the argument variables and the MOR MOEand IBS indexes of MDF as the dependent variables Thenumerical results show the good fitting of the built modelsfor the data set where the data set is given in Table 4
In order to use the data in Table 4 we first normalize theminto [0 1] And we use the mean square error (MSE) degreeof fitting (DF) and sufficient evaluation (SE) to evaluate thenumerical results where
MSE MSE = radicsum119899
119894=1(119909
119894minus 119909
lowast
119894)2
119899
DF DF = 1 minus radic(sum119899
119894=1(119909
119894minus 119909
lowast
119894)2
sum119899
119894=11199092
119894
)
SE SE = 12119909119879
119876119909 minus 119910119879
119909
(101)
where 119909 and 119909lowast indicate the actual value and predicted valueand 119876 and 119910 are defined as in (18) The smaller the MSE andthe SE and the closer the DF to 1 the better the regressionresult Moreover the SE function is the objective function in(15)
Based on the Gaussian radial basis function kernel thevalues of the initial parameters in problem (18) are given inTable 1 With a random initial point 119909
0isin Ω the numerical
results with respect to the obtained solution are also listed inTable 1 Figures 2ndash4 illustrate the fitting effect of the MORMOE and IBS values by using the proposed network (34) forsolving problem (18) And the parameters of the regressionfunctions in (19) are shown in Table 5
12 Mathematical Problems in Engineering
Table 2 Criterion of physical and mechanical performance indexes
Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45
IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045
MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700
2520151050
OriginalPredicted
0
05
1
15
2
25
3
35
4
5
45
Figure 2 Normalized sample fitting results of network (34) for theregression of MOR
Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5
52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2
521 Case 1
Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 3 Normalized sample fitting results of network (34) for theregression of MOE
By the information in Table 2 we let ]1= 200 ]
2= 4
]3= 7 and ]
4= 12 and choose 119910lowast
1= 22 119910lowast
2= 2500 120580
1= 8
1205761= 8 120580
2= 500 and 120576
2= 500 in optimization problem (21)
Then Ω = 119909 isin 1198774
0 le 1199091le ]
1 0 le 119909
2le ]
2 0 le 119909
3le
]3 0 le 119909
4le ]
4 and we can let
1198921(119909) = minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) =
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
1198923(119909) = minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) =
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(102)
By the proposed network (34) for solving (21) we obtainthe optimal solution
119909lowast
= (1594725 31022 64866 60061)119879
(103)
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Then there is 119871 gt 0 such that10038161003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla
119909 (119910 (119905) 120583 (119905))
10038161003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
1003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla119909 (119910 (119905) 120583 (119905))1003816100381610038161003816
le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817 forall119905 isin [119905 + 1]
(43)
Differentiating (12)119909(119905)minus119910(119905)2 along the two solutionsof (34) by the Lipschitz continuity of 119875
Ωand (43) we have
119889
119889119905
1
2
1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817
2
= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩
le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)
1003817100381710038171003817
2
(44)
Applying Gronwallrsquos inequality into the integration of theabove inequality it gives 119909(119905) = 119910(119905) forall119905 isin [119905 + 1] whichleads to a contradiction Therefore the solution of (34) withinitial point 119909
0isin Ω is unique
Lyapunov method is employed to analyze the perfor-mance of (34) Here we introduce the following two Lya-punov energy functions
119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
119866 (119909 119905) =1
2dist2 (119909M) + (119909 120583 (119905))
+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ
119891)
(45)
The above two Lyapunov functions satisfy the followingestimations along the solutions of (34)
Lemma 8 (i) The derivative of 119864(119909 119905) along the solution of(34) can be calculated by
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
(46)
(ii) The derivative of 119866(119909 119905) along the solution of (34) can becalculated by
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(47)
Proof (i) Differentiating119864(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119864 (119909 (119905) 119905) = ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(48)
Equation (34) can be rewritten as
(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla
119909 (119909 (119905) 120583 (119905))
minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]
(49)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119909(119905) using (49) and (10) we have
⟨119909 (119905) minus nabla119909 (119909 (119905) 120583 (119905)) minus 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
minus (119905) minus 119909 (119905) (119905) + 119909 (119905) minus 119909 (119905)⟩ ge 0
(50)
which follows the fact that
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
le minus (119905)2
forall119905 ge 0
(51)
Combining (48) and (51) we get that
119889
119889119905119864 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
(52)
From Lemmas 4 and 5 (119905) = minus119890minus119905 and 120576(119905) lt 0 we obtainthe estimation in (i)
(ii) Differentiating 119866(119909(119905) 119905) along the solutions of (34)we have
119889
119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩
+ ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)
sdot ( (119909 (119905) 120583 (119905)) minus infΩ
119891) + (nabla120583 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)
(53)
Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla
119909(119909(119905) 120583(119905)) and
119906 = 119875M(119909(119905)) using (10) and (49) we have
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905))
+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0
(54)
which can be rewritten as
⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
119909 (119909 (119905) 120583 (119905)) (119905)⟩
+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
minus (119905)2
(55)
Mathematical Problems in Engineering 9
Combining (53) and (55) we get
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905) (119891 (119909 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(56)
Similar to the analysis in (i) we obtain the estimation in(ii)
Next we prove the efficiency of proposed network (34) forsolving optimization problem (24) where the convergencefeasibility of the proposed network is a basic property
Theorem 9 For initial point 1199090isin Ω any solution 119909 [0 +infin)
rarr 119877119899 of (34) satisfies
lim119905rarr+infin
dist (119909 (119905) F) = 0 (57)
Proof Denote 119909 [0 +infin) rarr 119877119899 as a global solution of (34)
with initial point 1199090isin Ω
From 119909(119905) isin Ω forall119905 isin [0 +infin) and Lemma 8 119864(119909(119905) 119905) isnonincreasing along the solution of (34) Using119864(119909(119905) 119905) ge 0forall119905 isin [0 +infin) we confirm that
lim119905rarr+infin
119864 (119909 (119905) 119905) exists (58)
From 119909(119905) isin Ω forall119905 isin [0 +infin) and lim119905rarr+infin
120576(119905) = 0 wehave
lim119905rarr+infin
120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ
119891) = 0 (59)
which implies
lim119905rarr+infin
119864 (119909 (119905) 119905) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) (60)
Inwhat follows wewill prove that lim119905rarr+infin
(119909(119905) 120583(119905)) =
0 Arguing by contradiction we assume that
lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 119888 = 0 (61)
By (119909(119905) 120583(119905)) ge 119901(119909(119905)) ge 0 forall119905 isin [0 +infin) we have 119888 gt 0which follows the fact that there is 119879
1ge 0 such that
(119909 (119905) 120583 (119905)) ge119888
2 forall119905 isin [119879
1 +infin) (62)
From Lemma 4 we obtain
1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)
which implies that there exists 1198792gt 119879
1such that
(119875M (119909 (119905)) 120583 (119905)) le119888
8 (64)
Since Ω is bounded and 119909(119905) isin Ω forall119905 isin [0 +infin) byLemma 5 there is 119877 gt 0 such that10038161003816100381610038161003816⟨nabla
119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
10038161003816100381610038161003816le 119877
forall119905 isin [0 +infin)
(65)
Then
lim119905rarr+infin
120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)
which implies that there is 1198793ge 119879
2such that
10038161003816100381610038161003816120576 (119905) ⟨nabla
119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩
10038161003816100381610038161003816le119888
8
forall119905 isin [1198792 +infin)
(67)
From Lemma 8 and the convexity of (sdot 120583) for any fixed120583 gt 0 we obtain that
119889
119889119905119866 (119909 (119905) 119905) le minus ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)
sdot ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(68)
By (62) (67) and (68) we obtain
119889
119889119905119866 (119909 (119905) 119905) le minus
119888
4 forall119905 isin [119879
3 +infin) (69)
Integrating the above inequality from 1198793to 119905 (gt119879
3) we
have
119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879
3) minus
119888
4(119905 minus 119879
3)
forall119905 isin [1198793 +infin)
(70)
Thus
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (71)
which leads to a contradiction with 119866(119909 119905) ge 0 for all 119909 isin 119877119899and 119905 isin [0 +infin) Therefore
lim119905rarr+infin
119901 (119909 (119905)) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 0 (72)
which guarantees that
lim119905rarr+infin
dist (119909 (119905) F) = 0 (73)
The following theorem indicates that any accumulationpoint of the solutions of (34) is just an optimal solution of(34)
10 Mathematical Problems in Engineering
Theorem 10 For initial point 1199090isin Ω any solution 119909 of (34)
is convergent to the optimal solution setM that is
lim119905rarr+infin
dist (119909 (119905) M) = 0 (74)
Proof From (63) (68) and (119909(119905) 120583(119905)) ge 0 forall119905 isin [0 +infin)we have
119889
119889119905119866 (119909 (119905) 119905)
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
+ 119898120583 (119905)
(75)
by (119905) = minus120583(119905) which can be rewritten as
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(76)
Denote
119868 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le 0
119869 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
gt 0
(77)
Owning to the continuity of ⟨nabla119909(119909(119905) 120583(119905)) 119909(119905)minus119875M(119909(119905))⟩
on [0 +infin) 119868 and 119869 are closed and open in [0 +infin) respec-tively
Case 1 In this case we assume that there exists 119879 ge 0 suchthat 119905 isin 119868 forall119905 isin [119879 +infin)
From the definition on 119868 and the pseudoconvexity of(sdot 120583) on 119877119899 we have
(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))
forall119905 isin [119879 +infin)
(78)
which implies
lim sup119905rarr+infin
119891 (119909 (119905)) le min119909isinF
119891 (119909) (79)
ByTheorem 9 we confirm that
lim119905rarr+infin
dist (119909 (119905) M) = 0 (80)
Case 2 In this case we assume that there exists 119879 ge 0 such
that 119905 isin 119869 forall119905 isin [119879 +infin) which means that
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] lt 0 forall119905 isin [119879 +infin) (81)
Then
lim119905rarr+infin
[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)
Since lim119905rarr+infin
119864(119909(119905) 119905) exists and lim119905rarr+infin
120583(119905) = 0 weobtain that
lim119905rarr+infin
dist (119909 (119905) M) exists (83)
Suppose
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889
gt 0
(84)
Then there is 1198791ge such that
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge
119889
2
forall119905 isin [1198791 +infin)
(85)
Then from (76) we have
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus
119889
2120576 (119905)
forall119905 isin [1198791 +infin)
(86)
Integrating the above inequality from 1198791to 119905 (gt119879
1) we
have
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879
1) + 119898120583 (119879
1)
+ int
119905
1198791
minus119889
2120576 (119904) 119889119904
(87)
Let 119905 rarr +infin in the above inequality then we have
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (88)
which leads to a contraction with the boundedness frombelow 119866(119909(119905) 119905) on [0 +infin) Thus
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)
which follows the fact that there is an increasing sequence 119905119899
such that 119905119899rarr +infin and
lim119899rarr+infin
⟨nabla119909 (119909 (119905
119899) 120583 (119905
119899)) 119909 (119905
119899) minus 119875M (119909 (119905
119899))⟩
= 0
(90)
By 119909(119905119899) sube Ω and Ω which is bounded there are 119909lowastlowast isin
Ω and a subsequence of 119909(119905119899) (denoted as 119909(119905
119899119896)) such that
lim119896rarr+infin
119905119899119896= +infin and lim
119896rarr+infin119909(119905
119899119896) = 119909
lowastlowast By Lemma 5there is 120585(119909lowastlowast) isin 120597119891(119909lowastlowast) such that
⟨120585 (119909lowastlowast
) 119909lowastlowast
minus 119875M (119909lowastlowast
)⟩ = 0 (91)
From the pseudoconvexity of 119891 and the above inequalitywe have
119891 (119909lowastlowast
) le 119891 (119875M (119909lowastlowast
)) (92)
Mathematical Problems in Engineering 11
Table 1 Experimental data set in MDF hot-pressing
Regression parameter 120574 120590 119862 MSE DF SEMOR 0001 5 30 00307 09543 minus08299
MOE 0001 6 40 00642 09063 minus17060
IBS 0001 5 40 00451 09256 minus06388
By Theorem 9 we have 119909lowastlowast isin F Therefore 119909lowastlowast isin Mwhich implies that
lim119896rarr+infin
dist (119909 (119905119899119896) M) = 0 (93)
Combining the above results with (83) we conclude thatlim
119905rarr+infindist(119909(119905)M) = 0
Case 3 In this case we assume that both 119868 and 119869 areunbounded
For 119905 isin 119868 similar to the analysis in Case 1 we have
lim119905rarr+infin119905isin119868
dist (119909 (119905) M) = 0 (94)
For 119905 isin 119869 define 120591(119905) = sup119904le119905119904isin119868
119904 Then 120591(119905) isin 119868 and(120591(119905) 119905] sube 119869 By the unboundedness of 119868 and 119869 we have
lim119905isin119869119905rarr+infin
120591 (119905) = +infin (95)
From (81) and the continuity of 119866(119909(119905) 119905) on [0 +infin) wehave
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))
+ 119898120583 (120591 (119905)) forall119905 isin 119869
(96)
By 120591(119905) isin 119868 (94) and (95) we have
lim sup119905rarr+infin119905isin119869
[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le lim119905rarr+infin119905isin119869
[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0
(97)
which gives
lim119905rarr+infin119905isin119869
119866 (119909 (119905) 119905) = 0 (98)
From the definition of 119866(119909 119905) and the above result wehave
lim119905rarr+infin119905isin119869
dist (119909 (119905) M) = 0 (99)
Therefore from (94) and (99) we obtain
lim119905rarr+infin
dist (119909 (119905) M) = 0 (100)
5 Numerical Experiments
In this section we test the proposed neural network (34)for solving problem (24) which is brought forward from
the MDF continuous hot-pressing process Based on theexisting data set we use the established theories and proposedneural network (34) to build the relationships between themain qualities of the hot-pressing plate and some relativetechnology parameters from optimization problem (18)Then based on optimization problem (22) we will useproposed network (34) to solve the optimal values of thetechnology parameters in hot-pressing system for optimizingthe qualities of the hot-pressing plate All these numericalexperiments validate the good performance of the proposednetwork in this paper
The numerical testing was carried out on a Lenovo PC(300GHz 200GB of RAM) with the use of Matlab 74 Andwe use ode23 to realize the neural network (34) in Matlab
51 Construction Relation Models in MDF Continuous Hot-Pressing Process In this part by considered optimizationproblem (18) and network (34) we build the relation modelwhich takes the hot-pressing temperature (TE) hot-pressingpressure (PR) hot-pressing time (TI) and moisture content(MC) of slab as the argument variables and the MOR MOEand IBS indexes of MDF as the dependent variables Thenumerical results show the good fitting of the built modelsfor the data set where the data set is given in Table 4
In order to use the data in Table 4 we first normalize theminto [0 1] And we use the mean square error (MSE) degreeof fitting (DF) and sufficient evaluation (SE) to evaluate thenumerical results where
MSE MSE = radicsum119899
119894=1(119909
119894minus 119909
lowast
119894)2
119899
DF DF = 1 minus radic(sum119899
119894=1(119909
119894minus 119909
lowast
119894)2
sum119899
119894=11199092
119894
)
SE SE = 12119909119879
119876119909 minus 119910119879
119909
(101)
where 119909 and 119909lowast indicate the actual value and predicted valueand 119876 and 119910 are defined as in (18) The smaller the MSE andthe SE and the closer the DF to 1 the better the regressionresult Moreover the SE function is the objective function in(15)
Based on the Gaussian radial basis function kernel thevalues of the initial parameters in problem (18) are given inTable 1 With a random initial point 119909
0isin Ω the numerical
results with respect to the obtained solution are also listed inTable 1 Figures 2ndash4 illustrate the fitting effect of the MORMOE and IBS values by using the proposed network (34) forsolving problem (18) And the parameters of the regressionfunctions in (19) are shown in Table 5
12 Mathematical Problems in Engineering
Table 2 Criterion of physical and mechanical performance indexes
Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45
IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045
MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700
2520151050
OriginalPredicted
0
05
1
15
2
25
3
35
4
5
45
Figure 2 Normalized sample fitting results of network (34) for theregression of MOR
Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5
52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2
521 Case 1
Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 3 Normalized sample fitting results of network (34) for theregression of MOE
By the information in Table 2 we let ]1= 200 ]
2= 4
]3= 7 and ]
4= 12 and choose 119910lowast
1= 22 119910lowast
2= 2500 120580
1= 8
1205761= 8 120580
2= 500 and 120576
2= 500 in optimization problem (21)
Then Ω = 119909 isin 1198774
0 le 1199091le ]
1 0 le 119909
2le ]
2 0 le 119909
3le
]3 0 le 119909
4le ]
4 and we can let
1198921(119909) = minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) =
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
1198923(119909) = minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) =
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(102)
By the proposed network (34) for solving (21) we obtainthe optimal solution
119909lowast
= (1594725 31022 64866 60061)119879
(103)
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Combining (53) and (55) we get
119889
119889119905119866 (119909 (119905) 119905) le minus (119905)
2
+ 120576 (119905) (119891 (119909 (119905)) minus infΩ
119891)
+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla
120583 (119909 (119905) 120583 (119905))) (119905)
minus ⟨nabla119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(56)
Similar to the analysis in (i) we obtain the estimation in(ii)
Next we prove the efficiency of proposed network (34) forsolving optimization problem (24) where the convergencefeasibility of the proposed network is a basic property
Theorem 9 For initial point 1199090isin Ω any solution 119909 [0 +infin)
rarr 119877119899 of (34) satisfies
lim119905rarr+infin
dist (119909 (119905) F) = 0 (57)
Proof Denote 119909 [0 +infin) rarr 119877119899 as a global solution of (34)
with initial point 1199090isin Ω
From 119909(119905) isin Ω forall119905 isin [0 +infin) and Lemma 8 119864(119909(119905) 119905) isnonincreasing along the solution of (34) Using119864(119909(119905) 119905) ge 0forall119905 isin [0 +infin) we confirm that
lim119905rarr+infin
119864 (119909 (119905) 119905) exists (58)
From 119909(119905) isin Ω forall119905 isin [0 +infin) and lim119905rarr+infin
120576(119905) = 0 wehave
lim119905rarr+infin
120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ
119891) = 0 (59)
which implies
lim119905rarr+infin
119864 (119909 (119905) 119905) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) (60)
Inwhat follows wewill prove that lim119905rarr+infin
(119909(119905) 120583(119905)) =
0 Arguing by contradiction we assume that
lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 119888 = 0 (61)
By (119909(119905) 120583(119905)) ge 119901(119909(119905)) ge 0 forall119905 isin [0 +infin) we have 119888 gt 0which follows the fact that there is 119879
1ge 0 such that
(119909 (119905) 120583 (119905)) ge119888
2 forall119905 isin [119879
1 +infin) (62)
From Lemma 4 we obtain
1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)
which implies that there exists 1198792gt 119879
1such that
(119875M (119909 (119905)) 120583 (119905)) le119888
8 (64)
Since Ω is bounded and 119909(119905) isin Ω forall119905 isin [0 +infin) byLemma 5 there is 119877 gt 0 such that10038161003816100381610038161003816⟨nabla
119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
10038161003816100381610038161003816le 119877
forall119905 isin [0 +infin)
(65)
Then
lim119905rarr+infin
120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)
which implies that there is 1198793ge 119879
2such that
10038161003816100381610038161003816120576 (119905) ⟨nabla
119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩
10038161003816100381610038161003816le119888
8
forall119905 isin [1198792 +infin)
(67)
From Lemma 8 and the convexity of (sdot 120583) for any fixed120583 gt 0 we obtain that
119889
119889119905119866 (119909 (119905) 119905) le minus ⟨nabla
119909 (119909 (119905) 120583 (119905))
+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)
sdot ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(68)
By (62) (67) and (68) we obtain
119889
119889119905119866 (119909 (119905) 119905) le minus
119888
4 forall119905 isin [119879
3 +infin) (69)
Integrating the above inequality from 1198793to 119905 (gt119879
3) we
have
119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879
3) minus
119888
4(119905 minus 119879
3)
forall119905 isin [1198793 +infin)
(70)
Thus
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (71)
which leads to a contradiction with 119866(119909 119905) ge 0 for all 119909 isin 119877119899and 119905 isin [0 +infin) Therefore
lim119905rarr+infin
119901 (119909 (119905)) = lim119905rarr+infin
(119909 (119905) 120583 (119905)) = 0 (72)
which guarantees that
lim119905rarr+infin
dist (119909 (119905) F) = 0 (73)
The following theorem indicates that any accumulationpoint of the solutions of (34) is just an optimal solution of(34)
10 Mathematical Problems in Engineering
Theorem 10 For initial point 1199090isin Ω any solution 119909 of (34)
is convergent to the optimal solution setM that is
lim119905rarr+infin
dist (119909 (119905) M) = 0 (74)
Proof From (63) (68) and (119909(119905) 120583(119905)) ge 0 forall119905 isin [0 +infin)we have
119889
119889119905119866 (119909 (119905) 119905)
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
+ 119898120583 (119905)
(75)
by (119905) = minus120583(119905) which can be rewritten as
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(76)
Denote
119868 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le 0
119869 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
gt 0
(77)
Owning to the continuity of ⟨nabla119909(119909(119905) 120583(119905)) 119909(119905)minus119875M(119909(119905))⟩
on [0 +infin) 119868 and 119869 are closed and open in [0 +infin) respec-tively
Case 1 In this case we assume that there exists 119879 ge 0 suchthat 119905 isin 119868 forall119905 isin [119879 +infin)
From the definition on 119868 and the pseudoconvexity of(sdot 120583) on 119877119899 we have
(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))
forall119905 isin [119879 +infin)
(78)
which implies
lim sup119905rarr+infin
119891 (119909 (119905)) le min119909isinF
119891 (119909) (79)
ByTheorem 9 we confirm that
lim119905rarr+infin
dist (119909 (119905) M) = 0 (80)
Case 2 In this case we assume that there exists 119879 ge 0 such
that 119905 isin 119869 forall119905 isin [119879 +infin) which means that
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] lt 0 forall119905 isin [119879 +infin) (81)
Then
lim119905rarr+infin
[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)
Since lim119905rarr+infin
119864(119909(119905) 119905) exists and lim119905rarr+infin
120583(119905) = 0 weobtain that
lim119905rarr+infin
dist (119909 (119905) M) exists (83)
Suppose
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889
gt 0
(84)
Then there is 1198791ge such that
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge
119889
2
forall119905 isin [1198791 +infin)
(85)
Then from (76) we have
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus
119889
2120576 (119905)
forall119905 isin [1198791 +infin)
(86)
Integrating the above inequality from 1198791to 119905 (gt119879
1) we
have
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879
1) + 119898120583 (119879
1)
+ int
119905
1198791
minus119889
2120576 (119904) 119889119904
(87)
Let 119905 rarr +infin in the above inequality then we have
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (88)
which leads to a contraction with the boundedness frombelow 119866(119909(119905) 119905) on [0 +infin) Thus
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)
which follows the fact that there is an increasing sequence 119905119899
such that 119905119899rarr +infin and
lim119899rarr+infin
⟨nabla119909 (119909 (119905
119899) 120583 (119905
119899)) 119909 (119905
119899) minus 119875M (119909 (119905
119899))⟩
= 0
(90)
By 119909(119905119899) sube Ω and Ω which is bounded there are 119909lowastlowast isin
Ω and a subsequence of 119909(119905119899) (denoted as 119909(119905
119899119896)) such that
lim119896rarr+infin
119905119899119896= +infin and lim
119896rarr+infin119909(119905
119899119896) = 119909
lowastlowast By Lemma 5there is 120585(119909lowastlowast) isin 120597119891(119909lowastlowast) such that
⟨120585 (119909lowastlowast
) 119909lowastlowast
minus 119875M (119909lowastlowast
)⟩ = 0 (91)
From the pseudoconvexity of 119891 and the above inequalitywe have
119891 (119909lowastlowast
) le 119891 (119875M (119909lowastlowast
)) (92)
Mathematical Problems in Engineering 11
Table 1 Experimental data set in MDF hot-pressing
Regression parameter 120574 120590 119862 MSE DF SEMOR 0001 5 30 00307 09543 minus08299
MOE 0001 6 40 00642 09063 minus17060
IBS 0001 5 40 00451 09256 minus06388
By Theorem 9 we have 119909lowastlowast isin F Therefore 119909lowastlowast isin Mwhich implies that
lim119896rarr+infin
dist (119909 (119905119899119896) M) = 0 (93)
Combining the above results with (83) we conclude thatlim
119905rarr+infindist(119909(119905)M) = 0
Case 3 In this case we assume that both 119868 and 119869 areunbounded
For 119905 isin 119868 similar to the analysis in Case 1 we have
lim119905rarr+infin119905isin119868
dist (119909 (119905) M) = 0 (94)
For 119905 isin 119869 define 120591(119905) = sup119904le119905119904isin119868
119904 Then 120591(119905) isin 119868 and(120591(119905) 119905] sube 119869 By the unboundedness of 119868 and 119869 we have
lim119905isin119869119905rarr+infin
120591 (119905) = +infin (95)
From (81) and the continuity of 119866(119909(119905) 119905) on [0 +infin) wehave
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))
+ 119898120583 (120591 (119905)) forall119905 isin 119869
(96)
By 120591(119905) isin 119868 (94) and (95) we have
lim sup119905rarr+infin119905isin119869
[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le lim119905rarr+infin119905isin119869
[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0
(97)
which gives
lim119905rarr+infin119905isin119869
119866 (119909 (119905) 119905) = 0 (98)
From the definition of 119866(119909 119905) and the above result wehave
lim119905rarr+infin119905isin119869
dist (119909 (119905) M) = 0 (99)
Therefore from (94) and (99) we obtain
lim119905rarr+infin
dist (119909 (119905) M) = 0 (100)
5 Numerical Experiments
In this section we test the proposed neural network (34)for solving problem (24) which is brought forward from
the MDF continuous hot-pressing process Based on theexisting data set we use the established theories and proposedneural network (34) to build the relationships between themain qualities of the hot-pressing plate and some relativetechnology parameters from optimization problem (18)Then based on optimization problem (22) we will useproposed network (34) to solve the optimal values of thetechnology parameters in hot-pressing system for optimizingthe qualities of the hot-pressing plate All these numericalexperiments validate the good performance of the proposednetwork in this paper
The numerical testing was carried out on a Lenovo PC(300GHz 200GB of RAM) with the use of Matlab 74 Andwe use ode23 to realize the neural network (34) in Matlab
51 Construction Relation Models in MDF Continuous Hot-Pressing Process In this part by considered optimizationproblem (18) and network (34) we build the relation modelwhich takes the hot-pressing temperature (TE) hot-pressingpressure (PR) hot-pressing time (TI) and moisture content(MC) of slab as the argument variables and the MOR MOEand IBS indexes of MDF as the dependent variables Thenumerical results show the good fitting of the built modelsfor the data set where the data set is given in Table 4
In order to use the data in Table 4 we first normalize theminto [0 1] And we use the mean square error (MSE) degreeof fitting (DF) and sufficient evaluation (SE) to evaluate thenumerical results where
MSE MSE = radicsum119899
119894=1(119909
119894minus 119909
lowast
119894)2
119899
DF DF = 1 minus radic(sum119899
119894=1(119909
119894minus 119909
lowast
119894)2
sum119899
119894=11199092
119894
)
SE SE = 12119909119879
119876119909 minus 119910119879
119909
(101)
where 119909 and 119909lowast indicate the actual value and predicted valueand 119876 and 119910 are defined as in (18) The smaller the MSE andthe SE and the closer the DF to 1 the better the regressionresult Moreover the SE function is the objective function in(15)
Based on the Gaussian radial basis function kernel thevalues of the initial parameters in problem (18) are given inTable 1 With a random initial point 119909
0isin Ω the numerical
results with respect to the obtained solution are also listed inTable 1 Figures 2ndash4 illustrate the fitting effect of the MORMOE and IBS values by using the proposed network (34) forsolving problem (18) And the parameters of the regressionfunctions in (19) are shown in Table 5
12 Mathematical Problems in Engineering
Table 2 Criterion of physical and mechanical performance indexes
Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45
IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045
MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700
2520151050
OriginalPredicted
0
05
1
15
2
25
3
35
4
5
45
Figure 2 Normalized sample fitting results of network (34) for theregression of MOR
Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5
52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2
521 Case 1
Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 3 Normalized sample fitting results of network (34) for theregression of MOE
By the information in Table 2 we let ]1= 200 ]
2= 4
]3= 7 and ]
4= 12 and choose 119910lowast
1= 22 119910lowast
2= 2500 120580
1= 8
1205761= 8 120580
2= 500 and 120576
2= 500 in optimization problem (21)
Then Ω = 119909 isin 1198774
0 le 1199091le ]
1 0 le 119909
2le ]
2 0 le 119909
3le
]3 0 le 119909
4le ]
4 and we can let
1198921(119909) = minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) =
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
1198923(119909) = minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) =
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(102)
By the proposed network (34) for solving (21) we obtainthe optimal solution
119909lowast
= (1594725 31022 64866 60061)119879
(103)
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
Theorem 10 For initial point 1199090isin Ω any solution 119909 of (34)
is convergent to the optimal solution setM that is
lim119905rarr+infin
dist (119909 (119905) M) = 0 (74)
Proof From (63) (68) and (119909(119905) 120583(119905)) ge 0 forall119905 isin [0 +infin)we have
119889
119889119905119866 (119909 (119905) 119905)
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
+ 119898120583 (119905)
(75)
by (119905) = minus120583(119905) which can be rewritten as
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
(76)
Denote
119868 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
le 0
119869 = 119905
isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩
gt 0
(77)
Owning to the continuity of ⟨nabla119909(119909(119905) 120583(119905)) 119909(119905)minus119875M(119909(119905))⟩
on [0 +infin) 119868 and 119869 are closed and open in [0 +infin) respec-tively
Case 1 In this case we assume that there exists 119879 ge 0 suchthat 119905 isin 119868 forall119905 isin [119879 +infin)
From the definition on 119868 and the pseudoconvexity of(sdot 120583) on 119877119899 we have
(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))
forall119905 isin [119879 +infin)
(78)
which implies
lim sup119905rarr+infin
119891 (119909 (119905)) le min119909isinF
119891 (119909) (79)
ByTheorem 9 we confirm that
lim119905rarr+infin
dist (119909 (119905) M) = 0 (80)
Case 2 In this case we assume that there exists 119879 ge 0 such
that 119905 isin 119869 forall119905 isin [119879 +infin) which means that
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] lt 0 forall119905 isin [119879 +infin) (81)
Then
lim119905rarr+infin
[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)
Since lim119905rarr+infin
119864(119909(119905) 119905) exists and lim119905rarr+infin
120583(119905) = 0 weobtain that
lim119905rarr+infin
dist (119909 (119905) M) exists (83)
Suppose
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889
gt 0
(84)
Then there is 1198791ge such that
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge
119889
2
forall119905 isin [1198791 +infin)
(85)
Then from (76) we have
119889
119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus
119889
2120576 (119905)
forall119905 isin [1198791 +infin)
(86)
Integrating the above inequality from 1198791to 119905 (gt119879
1) we
have
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879
1) + 119898120583 (119879
1)
+ int
119905
1198791
minus119889
2120576 (119904) 119889119904
(87)
Let 119905 rarr +infin in the above inequality then we have
lim119905rarr+infin
119866 (119909 (119905) 119905) = minusinfin (88)
which leads to a contraction with the boundedness frombelow 119866(119909(119905) 119905) on [0 +infin) Thus
lim inf119905rarr+infin
⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)
which follows the fact that there is an increasing sequence 119905119899
such that 119905119899rarr +infin and
lim119899rarr+infin
⟨nabla119909 (119909 (119905
119899) 120583 (119905
119899)) 119909 (119905
119899) minus 119875M (119909 (119905
119899))⟩
= 0
(90)
By 119909(119905119899) sube Ω and Ω which is bounded there are 119909lowastlowast isin
Ω and a subsequence of 119909(119905119899) (denoted as 119909(119905
119899119896)) such that
lim119896rarr+infin
119905119899119896= +infin and lim
119896rarr+infin119909(119905
119899119896) = 119909
lowastlowast By Lemma 5there is 120585(119909lowastlowast) isin 120597119891(119909lowastlowast) such that
⟨120585 (119909lowastlowast
) 119909lowastlowast
minus 119875M (119909lowastlowast
)⟩ = 0 (91)
From the pseudoconvexity of 119891 and the above inequalitywe have
119891 (119909lowastlowast
) le 119891 (119875M (119909lowastlowast
)) (92)
Mathematical Problems in Engineering 11
Table 1 Experimental data set in MDF hot-pressing
Regression parameter 120574 120590 119862 MSE DF SEMOR 0001 5 30 00307 09543 minus08299
MOE 0001 6 40 00642 09063 minus17060
IBS 0001 5 40 00451 09256 minus06388
By Theorem 9 we have 119909lowastlowast isin F Therefore 119909lowastlowast isin Mwhich implies that
lim119896rarr+infin
dist (119909 (119905119899119896) M) = 0 (93)
Combining the above results with (83) we conclude thatlim
119905rarr+infindist(119909(119905)M) = 0
Case 3 In this case we assume that both 119868 and 119869 areunbounded
For 119905 isin 119868 similar to the analysis in Case 1 we have
lim119905rarr+infin119905isin119868
dist (119909 (119905) M) = 0 (94)
For 119905 isin 119869 define 120591(119905) = sup119904le119905119904isin119868
119904 Then 120591(119905) isin 119868 and(120591(119905) 119905] sube 119869 By the unboundedness of 119868 and 119869 we have
lim119905isin119869119905rarr+infin
120591 (119905) = +infin (95)
From (81) and the continuity of 119866(119909(119905) 119905) on [0 +infin) wehave
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))
+ 119898120583 (120591 (119905)) forall119905 isin 119869
(96)
By 120591(119905) isin 119868 (94) and (95) we have
lim sup119905rarr+infin119905isin119869
[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le lim119905rarr+infin119905isin119869
[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0
(97)
which gives
lim119905rarr+infin119905isin119869
119866 (119909 (119905) 119905) = 0 (98)
From the definition of 119866(119909 119905) and the above result wehave
lim119905rarr+infin119905isin119869
dist (119909 (119905) M) = 0 (99)
Therefore from (94) and (99) we obtain
lim119905rarr+infin
dist (119909 (119905) M) = 0 (100)
5 Numerical Experiments
In this section we test the proposed neural network (34)for solving problem (24) which is brought forward from
the MDF continuous hot-pressing process Based on theexisting data set we use the established theories and proposedneural network (34) to build the relationships between themain qualities of the hot-pressing plate and some relativetechnology parameters from optimization problem (18)Then based on optimization problem (22) we will useproposed network (34) to solve the optimal values of thetechnology parameters in hot-pressing system for optimizingthe qualities of the hot-pressing plate All these numericalexperiments validate the good performance of the proposednetwork in this paper
The numerical testing was carried out on a Lenovo PC(300GHz 200GB of RAM) with the use of Matlab 74 Andwe use ode23 to realize the neural network (34) in Matlab
51 Construction Relation Models in MDF Continuous Hot-Pressing Process In this part by considered optimizationproblem (18) and network (34) we build the relation modelwhich takes the hot-pressing temperature (TE) hot-pressingpressure (PR) hot-pressing time (TI) and moisture content(MC) of slab as the argument variables and the MOR MOEand IBS indexes of MDF as the dependent variables Thenumerical results show the good fitting of the built modelsfor the data set where the data set is given in Table 4
In order to use the data in Table 4 we first normalize theminto [0 1] And we use the mean square error (MSE) degreeof fitting (DF) and sufficient evaluation (SE) to evaluate thenumerical results where
MSE MSE = radicsum119899
119894=1(119909
119894minus 119909
lowast
119894)2
119899
DF DF = 1 minus radic(sum119899
119894=1(119909
119894minus 119909
lowast
119894)2
sum119899
119894=11199092
119894
)
SE SE = 12119909119879
119876119909 minus 119910119879
119909
(101)
where 119909 and 119909lowast indicate the actual value and predicted valueand 119876 and 119910 are defined as in (18) The smaller the MSE andthe SE and the closer the DF to 1 the better the regressionresult Moreover the SE function is the objective function in(15)
Based on the Gaussian radial basis function kernel thevalues of the initial parameters in problem (18) are given inTable 1 With a random initial point 119909
0isin Ω the numerical
results with respect to the obtained solution are also listed inTable 1 Figures 2ndash4 illustrate the fitting effect of the MORMOE and IBS values by using the proposed network (34) forsolving problem (18) And the parameters of the regressionfunctions in (19) are shown in Table 5
12 Mathematical Problems in Engineering
Table 2 Criterion of physical and mechanical performance indexes
Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45
IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045
MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700
2520151050
OriginalPredicted
0
05
1
15
2
25
3
35
4
5
45
Figure 2 Normalized sample fitting results of network (34) for theregression of MOR
Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5
52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2
521 Case 1
Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 3 Normalized sample fitting results of network (34) for theregression of MOE
By the information in Table 2 we let ]1= 200 ]
2= 4
]3= 7 and ]
4= 12 and choose 119910lowast
1= 22 119910lowast
2= 2500 120580
1= 8
1205761= 8 120580
2= 500 and 120576
2= 500 in optimization problem (21)
Then Ω = 119909 isin 1198774
0 le 1199091le ]
1 0 le 119909
2le ]
2 0 le 119909
3le
]3 0 le 119909
4le ]
4 and we can let
1198921(119909) = minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) =
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
1198923(119909) = minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) =
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(102)
By the proposed network (34) for solving (21) we obtainthe optimal solution
119909lowast
= (1594725 31022 64866 60061)119879
(103)
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Table 1 Experimental data set in MDF hot-pressing
Regression parameter 120574 120590 119862 MSE DF SEMOR 0001 5 30 00307 09543 minus08299
MOE 0001 6 40 00642 09063 minus17060
IBS 0001 5 40 00451 09256 minus06388
By Theorem 9 we have 119909lowastlowast isin F Therefore 119909lowastlowast isin Mwhich implies that
lim119896rarr+infin
dist (119909 (119905119899119896) M) = 0 (93)
Combining the above results with (83) we conclude thatlim
119905rarr+infindist(119909(119905)M) = 0
Case 3 In this case we assume that both 119868 and 119869 areunbounded
For 119905 isin 119868 similar to the analysis in Case 1 we have
lim119905rarr+infin119905isin119868
dist (119909 (119905) M) = 0 (94)
For 119905 isin 119869 define 120591(119905) = sup119904le119905119904isin119868
119904 Then 120591(119905) isin 119868 and(120591(119905) 119905] sube 119869 By the unboundedness of 119868 and 119869 we have
lim119905isin119869119905rarr+infin
120591 (119905) = +infin (95)
From (81) and the continuity of 119866(119909(119905) 119905) on [0 +infin) wehave
119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))
+ 119898120583 (120591 (119905)) forall119905 isin 119869
(96)
By 120591(119905) isin 119868 (94) and (95) we have
lim sup119905rarr+infin119905isin119869
[119866 (119909 (119905) 119905) + 119898120583 (119905)]
le lim119905rarr+infin119905isin119869
[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0
(97)
which gives
lim119905rarr+infin119905isin119869
119866 (119909 (119905) 119905) = 0 (98)
From the definition of 119866(119909 119905) and the above result wehave
lim119905rarr+infin119905isin119869
dist (119909 (119905) M) = 0 (99)
Therefore from (94) and (99) we obtain
lim119905rarr+infin
dist (119909 (119905) M) = 0 (100)
5 Numerical Experiments
In this section we test the proposed neural network (34)for solving problem (24) which is brought forward from
the MDF continuous hot-pressing process Based on theexisting data set we use the established theories and proposedneural network (34) to build the relationships between themain qualities of the hot-pressing plate and some relativetechnology parameters from optimization problem (18)Then based on optimization problem (22) we will useproposed network (34) to solve the optimal values of thetechnology parameters in hot-pressing system for optimizingthe qualities of the hot-pressing plate All these numericalexperiments validate the good performance of the proposednetwork in this paper
The numerical testing was carried out on a Lenovo PC(300GHz 200GB of RAM) with the use of Matlab 74 Andwe use ode23 to realize the neural network (34) in Matlab
51 Construction Relation Models in MDF Continuous Hot-Pressing Process In this part by considered optimizationproblem (18) and network (34) we build the relation modelwhich takes the hot-pressing temperature (TE) hot-pressingpressure (PR) hot-pressing time (TI) and moisture content(MC) of slab as the argument variables and the MOR MOEand IBS indexes of MDF as the dependent variables Thenumerical results show the good fitting of the built modelsfor the data set where the data set is given in Table 4
In order to use the data in Table 4 we first normalize theminto [0 1] And we use the mean square error (MSE) degreeof fitting (DF) and sufficient evaluation (SE) to evaluate thenumerical results where
MSE MSE = radicsum119899
119894=1(119909
119894minus 119909
lowast
119894)2
119899
DF DF = 1 minus radic(sum119899
119894=1(119909
119894minus 119909
lowast
119894)2
sum119899
119894=11199092
119894
)
SE SE = 12119909119879
119876119909 minus 119910119879
119909
(101)
where 119909 and 119909lowast indicate the actual value and predicted valueand 119876 and 119910 are defined as in (18) The smaller the MSE andthe SE and the closer the DF to 1 the better the regressionresult Moreover the SE function is the objective function in(15)
Based on the Gaussian radial basis function kernel thevalues of the initial parameters in problem (18) are given inTable 1 With a random initial point 119909
0isin Ω the numerical
results with respect to the obtained solution are also listed inTable 1 Figures 2ndash4 illustrate the fitting effect of the MORMOE and IBS values by using the proposed network (34) forsolving problem (18) And the parameters of the regressionfunctions in (19) are shown in Table 5
12 Mathematical Problems in Engineering
Table 2 Criterion of physical and mechanical performance indexes
Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45
IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045
MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700
2520151050
OriginalPredicted
0
05
1
15
2
25
3
35
4
5
45
Figure 2 Normalized sample fitting results of network (34) for theregression of MOR
Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5
52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2
521 Case 1
Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 3 Normalized sample fitting results of network (34) for theregression of MOE
By the information in Table 2 we let ]1= 200 ]
2= 4
]3= 7 and ]
4= 12 and choose 119910lowast
1= 22 119910lowast
2= 2500 120580
1= 8
1205761= 8 120580
2= 500 and 120576
2= 500 in optimization problem (21)
Then Ω = 119909 isin 1198774
0 le 1199091le ]
1 0 le 119909
2le ]
2 0 le 119909
3le
]3 0 le 119909
4le ]
4 and we can let
1198921(119909) = minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) =
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
1198923(119909) = minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) =
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(102)
By the proposed network (34) for solving (21) we obtainthe optimal solution
119909lowast
= (1594725 31022 64866 60061)119879
(103)
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
Table 2 Criterion of physical and mechanical performance indexes
Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45
IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045
MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700
2520151050
OriginalPredicted
0
05
1
15
2
25
3
35
4
5
45
Figure 2 Normalized sample fitting results of network (34) for theregression of MOR
Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5
52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2
521 Case 1
Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 3 Normalized sample fitting results of network (34) for theregression of MOE
By the information in Table 2 we let ]1= 200 ]
2= 4
]3= 7 and ]
4= 12 and choose 119910lowast
1= 22 119910lowast
2= 2500 120580
1= 8
1205761= 8 120580
2= 500 and 120576
2= 500 in optimization problem (21)
Then Ω = 119909 isin 1198774
0 le 1199091le ]
1 0 le 119909
2le ]
2 0 le 119909
3le
]3 0 le 119909
4le ]
4 and we can let
1198921(119909) = minus
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ minus 119887
1+ 119910
lowast
1minus 120580
1
1198922(119909) =
119899
sum
119894=1
120588lowast
119894⟨119911
119894 119909⟩ + 119887
1minus 119910
lowast
1minus 120576
1
1198923(119909) = minus
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ minus 119887
2+ 119910
lowast
2minus 120580
2
1198924(119909) =
119899
sum
119894=1
120579lowast
119894⟨119911
119894 119909⟩ + 119887
2minus 119910
lowast
2minus 120576
2
(102)
By the proposed network (34) for solving (21) we obtainthe optimal solution
119909lowast
= (1594725 31022 64866 60061)119879
(103)
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
OriginalPredicted
25201510500
05
1
15
2
25
3
35
4
5
45
Figure 4 Normalized sample fitting results of network (34) for theregression of IBS
k
454035302520151050
265
26
255
25
245
24
235
23
225
22
215
Figure 5 Convergence of MOR regression function 1198911(119909) along the
solution of (34)
Table 3 Case 1 performance index values of obtained hot-pressingplate
MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040
This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891
1(119909)MOE
regression function 1198912(119909) and IBS regression function 119891
3(119909)
along the solution of (34) are plotted in Figures 5ndash7
2750
2700
2650
2600
2550
25005 10 15 20 25 30 35 40 450
k
Figure 6 Convergence of MOE regression function 1198912(119909) along the
solution of (34)
062
06
058
056
054
052
05
0485 10 15 20 25 30 35 40 450
k
Figure 7 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
10 20 30 40 50 600
1135
114
1145
115
1155
116
1165
117
1175
t
Figure 8 Convergence of 119891(119909) along the solution of (34)
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
Table 4 Experimental data set in MDF hot-pressing
Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077
Table 5 Parameter values in the regression functions
In the regression functions (19) In the regression functions (20)120588lowast
1minus01385 120579
lowast
1minus00335 120581
lowast
1minus0073 120588
lowast
125815 120579
lowast
1137581 120581
lowast
1minus05275
120588lowast
200139 120579
lowast
2minus00794 120581
lowast
2minus00060 120588
lowast
211358 120579
lowast
2minus07298 120581
lowast
2minus65132
120588lowast
301869 120579
lowast
301332 120581
lowast
300259 120588
lowast
324638 120579
lowast
310424 120581
lowast
397535
120588lowast
400119 120579
lowast
4minus00365 120581
lowast
4minus00000 120588
lowast
4minus10246 120579
lowast
4minus13607 120581
lowast
4minus59500
120588lowast
500948 120579
lowast
500554 120581
lowast
500939 120588
lowast
501880 120579
lowast
5minus06708 120581
lowast
563092
120588lowast
602217 120579
lowast
605229 120581
lowast
600218 120588
lowast
648889 120579
lowast
661594 120581
lowast
6155774
120588lowast
701409 120579
lowast
700001 120581
lowast
700823 120588
lowast
7minus09675 120579
lowast
7minus12267 120581
lowast
7103006
120588lowast
802361 120579
lowast
801338 120581
lowast
800054 120588
lowast
816596 120579
lowast
8minus01886 120581
lowast
8minus239383
120588lowast
9500000 120579
lowast
9500000 120581
lowast
9458063 120588
lowast
912497 120579
lowast
930084 120581
lowast
9308681
120588lowast
10minus01413 120579
lowast
10minus02405 120581
lowast
10minus01362 120588
lowast
1000644 120579
lowast
10minus22099 120581
lowast
10minus259216
120588lowast
11minus03874 120579
lowast
11minus01625 120581
lowast
11minus01190 120588
lowast
1136153 120579
lowast
11minus16600 120581
lowast
11minus270340
120588lowast
12minus00627 120579
lowast
12minus01128 120581
lowast
12minus01673 120588
lowast
12minus08880 120579
lowast
1207098 120581
lowast
12minus178497
120588lowast
1302501 120579
lowast
1300338 120581
lowast
1300488 120588
lowast
1314724 120579
lowast
13minus01911 120581
lowast
13136398
120588lowast
14minus03013 120579
lowast
14minus00673 120581
lowast
1401250 120588
lowast
14minus25732 120579
lowast
1400263 120581
lowast
14285659
120588lowast
15434098 120579
lowast
15500000 120581
lowast
15500000 120588
lowast
15minus09224 120579
lowast
1528766 120581
lowast
15267062
120588lowast
1601186 120579
lowast
1600058 120581
lowast
1600221 120588
lowast
16minus31735 120579
lowast
16minus44727 120581
lowast
16minus174470
120588lowast
17minus00048 120579
lowast
1703582 120581
lowast
1701597 120588
lowast
1700304 120579
lowast
1732063 120581
lowast
17211381
120588lowast
18minus01428 120579
lowast
18minus03514 120581
lowast
18minus01356 120588
lowast
1810775 120579
lowast
18minus08331 120581
lowast
18minus127596
120588lowast
19minus500000 120579
lowast
19minus500000 120581
lowast
19minus458399 120588
lowast
19minus16929 120579
lowast
19minus17854 120581
lowast
19minus119152
120588lowast
20minus434157 120579
lowast
20minus500000 120581
lowast
20minus500000 120588
lowast
20minus24732 120579
lowast
20minus23246 120581
lowast
20minus435023
120588lowast
21minus06476 120579
lowast
21minus07696 120581
lowast
21minus00472 120588
lowast
21minus058057 120579
lowast
21minus63103 120581
lowast
21198815
120588lowast
2201968 120579
lowast
22minus03108 120581
lowast
22minus00101 120588
lowast
2214923 120579
lowast
2233107 120581
lowast
22minus36604
120588lowast
2303606 120579
lowast
2303021 120581
lowast
2301471 120588
lowast
2330559 120579
lowast
2322478 120581
lowast
23142783
1198871
06361 1198872
06554 1198873
06052 1198871
02784 1198872
02693 1198873
04230
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
5 10 15 20 25 30 35 40 45 500
k
294
292
29
288
286
284
282
Figure 9 Convergence ofMOR regression function1198911(119909) along the
solution of (34)
5 10 15 20 25 30 35 40 45 500
2930
2920
2910
2900
2890
2880
2870
2860
2850
2840
Figure 10 Convergence ofMOE regression function1198912(119909) along the
solution of (34)
0608
0606
0604
0602
06
0598
0596
0594
0592
059
05885 10 15 20 25 30 35 40 45 500
k
Figure 11 Convergence of IBS regression function 1198913(119909) along the
solution of (34)
522 Case 2
Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab
In this case we use optimization problem (23) with 1205821=
13 1205822= 13 120582
3= 13 ]
1= 200 ]
2= 4 ]
3= 7 ]
4= 12
119891lowast
1= 22 (Mpa) 119891lowast
2= 2500 (Mpa) and 119891lowast
3= 060 (Mpa)
Using proposed network (34) to solve (23) we obtain theoptimal solution
119909lowast
= (1822461 30991 85167 51801)119879
(104)
The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891
1(119909) MOE regression function 119891
2(119909) and IBS
regression function 1198913(119909) is shown in Figures 9ndash11
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)
References
[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996
[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003
[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985
[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988
[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986
[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009
[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013
[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012
[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992
[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007
[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010
[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004
[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011
[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004
[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006
[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008
[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012
[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983
[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998
[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002
[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012
[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012
[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996
[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of