'Optimal' Hopfield Network for Combinatorial Optimization ... · C. Combinatorial Optimization...

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 9, NO. 6, NOVEMBER 1998 1319

“Optimal” Hopfield Network for CombinatorialOptimization with Linear Cost Function

Satoshi Matsuda,Member, IEEE

Abstract—An “optimal” Hopfield network is presented formany of combinatorial optimization problems with linear costfunction. It is proved that a vertex of the network state hypercubeis asymptotically stable if and only if it is an optimal solution tothe problem. That is, one can always obtain an optimal solutionwhenever the network converges to a vertex. In this sense, thisnetwork can be called the “optimal” Hopfield network. It isalso shown through simulations of assignment problems thatthis network obtains optimal or nearly optimal solutions morefrequently than other familiar Hopfield networks.

Index Terms—Assignment problems, combinatorial optimiza-tion, Hopfield network, knapsack problems, linear cost function,optimal neural representation, stability of state hypercube vertex.

I. INTRODUCTION

M ANY ATTEMPTS have been made to solve combina-torial optimization problems using Hopfield networks.

Most of them, however, have lacked any theoretical principle.That is, the fine-tuning of the network coefficients has beenaccomplished by trial and error, and the neural representa-tion of the problem has been made heuristically or withoutcare. Accordingly, Hopfield networks have failed in manycombinatorial optimization problems [1], [2].

At the same time, there have been some theoretical workswhich analyze the network dynamics. Analyzing the eigen-values of the network connection matrix, Aiyeret al. [3]have theoretically explained the dynamics of the network fortraveling salesman problems (TSP’s). However, since their aimis only to make the network converge to a feasible solutionrather than to an optimal or nearly optimal one, their treatmentof the distance term of the energy function is theoreticallyinsufficient, consequently, they have not shown the theoreticalrelationship between the values of the network coefficients andthe qualities of the feasible solutions the network convergesto. Abe [4] has shown such theoretical relationship by derivingthe stability condition of the feasible solutions. However,since these are made by comparing the values of the energyat vertices of the network hypercube, these are true onlyfor the networks without diagonal elements of connectionmatrix. On the other hand, by analyzing the eigenvalues ofJacobian matrix of the network dynamic equation, not only forthe networks without diagonal elements of connection matrix

Manuscript received September 16, 1996; revised November 2, 1996, March16, 1998, and September 14, 1998.

The author is with the Computer and Communication Research Center,Tokyo Electric Power Company, 4-1, Egasaki-cho, Tsurumi-ku, Yokohama,230-8510 Japan.

Publisher Item Identifier S 1045-9227(98)08808-0.

but also for those with diagonal elements, Matsuda [5]–[8]has also theoretically derived such relationship between thenetwork coefficients and the feasible solutions qualities, andthe stability and instability conditions of the feasible andinfeasible solutions. Furthermore, based on these theoreticalconclusions, he has also pointed out the importance of the di-agonal elements of connection matrix to obtain good solutions.Actually, the diagonal elements of connection matrix play acrucial role in this paper. By tuning the coefficients to satisfythese conditions, one can obtain good solutions frequently.However, it is also theoretically shown that one cannot alwaysachieve optimal solutions even with well-tuned coefficients.An inherent limitation arises from the neural representationitself. We must represent the problems very carefully, on thebasis of the theoretical principle, rather than on heuristics.

These theoretical works, however, are mainly on the net-work dynamics or tuning of the network coefficients, but donot give theoretical basis to neural representation of opti-mization problems. Not only are the neural representationsmade heuristically, but also their comparisons are made onlyby network simulations of some problem instances ratherthan on a theoretical basis. Since the network performance isdeeply dependent on the nature of each problem instance—forexample, the city locations in TSP’s—it is inappropriate tocompare networks’ performances solely by simulations. Andyet, despite the need for theoretical comparison of the neuralrepresentations or neural networks, few such treatments havebeen made. Matsuda [9], [10], using the stability theory of thefeasible solutions and infeasible solutions mentioned above,has also proposed a theoretical method of comparing theperformances of the neural networks or neural representationsof the same problem. This method is based on the relativeability of networks to distinguish optimal solutions from otherfeasible solutions and infeasible solutions. He has applied thismethod to two familiar networks for TSP’s, and establisheda theoretical justification of the superiority of one networkto the other although this superiority was already observedthrough simulations [11]. Thus, one network can be regardedasmore sharplydistinguishing optimal solutions to TSP’s thanthe other. He, however, has also shown that even this superiornetwork cannotcompletelydistinguish optimal solutions; therecan be nonoptimal solutions which are indistinguishable fromoptimal solutions even by this superior network.

Then, a question rises as to whether there is still an-other network which, in terms of this theoretical comparison,distinguishes optimal solutionsmore sharplythan these twonetworks, or whether there is an “optimal” network which

1045–9227/98$10.00 1998 IEEE

1320 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 9, NO. 6, NOVEMBER 1998

distinguishes optimal solutionscompletelyor most sharply(or completely) of all the networks. We can expect to ob-tain optimal solutions most frequently by such an “optimal”network.

In this paper, such an “optimal” Hopfield network is pre-sented for many of combinatorial optimization problems withlinear cost function. A vertex of the network state hyper-cube is asymptotically stable if and only if it is an optimalsolution to the problem. In Section II, the basic definitionsand concepts of the Hopfield network and combinatorialoptimization problems are introduced. Sections III and IVgive theoretical characterizations on two conventional neuralrepresentations for these optimization problems, in order toreveal their theoretical limitation in deriving optimal solutions.The concepts of a theoretical network comparison and an“optimal” network are also given in Section IV. In Section V,a new representation is presented and theoretically shown to be“optimal,” and Section VI gives such an “optimal” network forthe maximization version of these optimization problems. Thesimulation results given in Section VII illustrate our theoreticalfindings. Last, concluding remarks appear in Section VIII.

II. HOPFIELD NETWORKS AND

COMBINATORIAL OPTIMIZATION

A. Hopfield Networks

In a Hopfield network, any two neurons are connected inboth directions with the same synaptic weights. The dynamicsof a Hopfield network are defined by [12]

(1)

(2)

where , , and are an output value, an internalvalue, and a bias of neuron, respectively, and is a synapticweight from neuron to neuron .

Energy of the network is also given by [12]

(3)

It is well known that

(4)

holds. This shows that the value of a neuron changes inorder to decrease energy(energy minimization), and finallyconverges to equilibrium where : that is, to zero,one, or to somewhere which satisfies within thehypercube. Note that the dimensional space [0, 1]is calleda network state hypercube, or, simply, a hypercube, and that

is called a vertex of the hypercube, whereisthe number of neurons in the network.

By representing combinatorial optimization problems in theform of the energy shown above, and constructing a Hopfieldnetwork with this energy, we can expect the network toconverge to a good or optimal solution to the problem by

energy minimization. Hopfield and Tank [13] have shown thatnearly optimal solutions to TSP’s, which are NP-completecombinatorial optimization problems, can be obtained quickly.We herein use the words,energy, neural representation,andnetwork interchangeably.

B. Stability of Vertices of Network State Hypercube

As shown in (4) and mentioned above, at equilibrium,neuron takes a value 0, 1, or a value between zero andone that satisfies . However, since the Hopfieldnetwork converges to asymptotically stable equilibrium andcannot converge to unstable one, the stability of the equi-librium is an important property which makes the networkdynamics theoretically clear. In this paper, however, we focuson the stability of the vertices of the hypercube for two mainreasons. First, strictly speaking, the meaning of the neuron tothe combinatorial optimization problem is given only when ittakes a value of zero or one. Second, analysis of the stabilityof the vertices provides an approximation of the networkbehavior when the network converges inside the hypercube [5],[9], [10], [14]. Actually, in Section VII, we illustrate throughsimulations that our theoretical results based on the stability ofthe vertices also work well in the case of convergence withinthe hypercube. Thus it is crucial to explore the stability of thevertices of the state hypercube.

It is known [15] that a vertex of the network statehypercube is asymptotically stable if

(5)

holds for any neuron, and unstable if

(6)

holds for some neuron. These conditions are helpful to ourtheory. The proof can be found in [15], and is also shown inthe Appendix, for convenience.

C. Combinatorial Optimization Problemswith Linear Cost Function

Combinatorial optimization problems we attack in this paperhave linear cost functions to minimize, and are formally statedas follows:

minimize

subject to for any

for any

for any and

(7)

where is a two dimensional variablematrix, is an integer representing a unit cost, isan integer representing some quantities which is related to,and is also an integer representing some total quantities.

MATSUDA: “OPTIMAL” HOPFIELD NETWORK FOR COMBINATORIAL OPTIMIZATION WITH LINEAR COST FUNCTION 1321

Thus, these problems have two sets of constraints such that onewinner for each column,, and quantities as the sum of allwinners’ quantities, ’s, for each line . Note that

(8)

holds.We call a feasible solution or an infeasible

solution if it does or does not satisfy all the constraints,respectively. Furthermore, a feasible solution which minimizesthe objective or cost function is also called anoptimal solution, and feasible solutions other than optimalone are sometimes called nonoptimal solutions. The goal ofcombinatorial optimization problems is to find an optimalsolution or sometimes a nearly optimal solution.

Note that some of the optimization problems have inequalityconstraint such as

subject to for any (9)

rather than equality one shown in (7), however, by introducingsome slack variables (slack neurons) , we canexpress the inequality constraint as an equality constraint,

subject to

for any (10)

Thus, by letting and for some of ’s,combinatorial optimization problems with inequality constraintcan also be stated as (7). This requires a large number of slackvariables (neurons) and seems to make the neural approachimpractical, however, quantized neurons [16], each of whichtakes an integer value, overcome this difficulty. One quantizedslack neuron, , serves as all slack neurons, ’s, for each ;

. However, in this paper we employfor simplicity. Note that the second constraint in (7) does notcontain slack variables.

Many of the combinatorial optimization problems can bestated as (7). For example, minimization version of the gen-eralized assignment problem, which is NP-hard, is such prob-lems [17]. Since the assignment problems, which are notNP-hard, are special cases of the generalized assignmentproblems, they can be also stated as (7) of course. In this paper,for simplicity, by taking assignment problems as examples, wewill explain and illustrate our theory. The assignment problemsare stated as follows. Given a set oftasks, people, andcost to carry out task by person , the goal of theassignment problem is to find an assignmentwhich minimizes total cost, , for carrying out allthe tasks under the constraints that each task must be carriedout once and only once by some person, and that each personmust carry out one and only one task, where a value ofmeans that persondoes or does not carry out task. So, the

assignment problems are formulated as

minimize

subject to for any

for any

for any and

(11)

Thus, one can immediately see that assignment problems arestated as (7) with , , for any and ,and .

In this paper, we present an “optimal” Hopfield networkfor the combinatorial optimization problem with linear costfunction given as (7). In the next two sections we theoreticallyexplore two conventional neural representations for theseproblems to compare with our “optimal” representation.

III. A N ETWORK THAT MINIMIZES THE TOTAL COST

A. Neural Representation

The first neural representation for the combinatorial op-timization problem given as (7) is a familiar one whichminimizes the total cost, and given by

(12)

We can easily see that the first and second terms areconstraint conditions explained above, and that the last termdenotes the total cost to minimize. The third term, whichmoves values of neurons toward zero or one, i.e., to a vertexof the hypercube, is called a binary value constraint because ittakes minimum value when all the neurons take the value zeroor one. Note that the existence of the binary value constraintterm does not change the value of energy at each vertex ofthe hypercube. Thus, a vertex is also calleda feasible or an infeasible solution if the values of the firstthree terms in (12) are all zero or not, respectively. Constants

, , , and are coefficients or weights of eachcondition. We must tune the values of these coefficients inorder for the network to converge to an optimal or goodsolution. As noted before, by we denote, interchangeably,the energy, the neural representation, and the network with thisenergy. First we will theoretically characterize the networkby the set of all the asymptotically stable vertices of its statehypercube.

B. Stability of Feasible and Infeasible Solutions

So, following [5]–[7], we show the asymptotical stabilityand instability conditions of the feasible and infeasible solu-tions in the network . First, as the next theorem shows, allthe infeasible solutions can be unstable.


Theorem 1: In the Hopfield network , all the infeasiblesolutions are unstable if

hold, where , , and.

Proof: For any infeasible solution, at least one of the following cases holds:

1) for some ;2) for some ;3) for some ;4) for some .

In the case of 1), for , since , we have

Hence, if

we have , thus, from (6), is unstable.For the case of 2), for , since

holds, we similarly have

Hence, if

we have , so that is unstable.For the case of 3), there exists . Since we can

assume that 2) does not hold, holds for any. So, we have

Hence, if

we have , so that is unstable.For the case of 4), since we can assume that neither 1) nor 3)

holds, holds for any . Hence, similarly,for , we have

Therefore, if

we have , so that is unstable.

Note that, in the instability conditions shown in Theorem1, the value of depends on that of , however, since

does not depend on that of , these conditions can besatisfied without any contradiction.

For the stability and instability of feasible solutions we canalso derive the next theorem.

Theorem 2: In the Hopfield network , a feasible solution,, is asymptotically stable if

and unstable if

Proof: For any and , since

we have

.

For , always holds. For , if

holds, we have . Hence, from (5), isasymptotically stable if

holds.Conversely, if , we have

for some , and from (6), isunstable.

Theorem 2 shows that, by tuning the value of coefficientor , we can make some particular feasible solution, say, theoptimal one, asymptotically stable or unstable. Furthermore,since the conditions in Theorem 2 do not depend on the valueof or , it is possible to satisfy the instability conditionof infeasible solutions in Theorem 1 simultaneously. Since wedo not want infeasible solutions to be asymptotically stable,we hereafter assume that the values of and satisfythe instability conditions of infeasible solutions in Theorem 1.Hence the set of all the asymptotically stable vertices of thisnetwork hypercube contains all the feasible solutions whichsatisfy the former condition in Theorem 2, and does not containall the infeasible solutions and feasible solutions which satisfythe latter condition in Theorem 2.

Note that, for assignment problems, Theorems 1 and 2 alsohold, of course, however, since , the instabilityconditions of all the infeasible solutions shown in Theorem 1can be simply expressed as


TABLE IAN INSTANCE OF AN ASSIGNMENT PROBLEM [COST MATRIX (cij )] AND ITS THREE FEASIBLE SOLUTIONS. SYMBOL � MEANS THAT THE VALUE OF

cij IS TOO LARGE TO BE TAKEN INTO CONSIDERATION. THE OPTIMAL SOLUTION VVV 0 (TOTAL COST= 14), THE NONOPTIMAL

SOLUTION VVV 1 (TOTAL COST= 25), AND THE NONOPTIMAL SOLUTION VVV 2 (TOTAL COST = 16) ARE SHOWN FROM LEFT

TO RIGHT, RESPECTIVELY, WHEREckij WITH BRACKETS, h i, INDICATES V kij = 1 FOR EACH FEASIBLE SOLUTION VVV k = (V k

ij)

C. Limitations

As Theorem 2 shows, the stability of a feasible solution,,depends on only the maximum unit cost which is included inthe total cost of this feasible solution, that is,

Hence, in any network where an optimal solution,, is asymptotically stable, any feasible solution, say, is also asymptotically stable and can be converged to if

Consider the assignment problem instance given in Table I.The total cost of its optimal solution or nonoptimal solution

is 14 or 25, respectively. Since

holds, no matter how well the network is tuned, thenonoptimal solution is asymptotically stable and can beconverged to if the optimal solution is asymptoticallystable. Thus, in general, we cannot always obtain an optimalsolution, even if the network converges to a vertex. Thenetwork cannotsharplydistinguish optimal solutions fromother feasible solutions such as , and we cannot expectthe average quality of the feasible solutions obtained by thenetwork to be good. This is a theoretical limitation of thenetwork .

IV. A N ETWORK THAT MINIMIZES

THE SQUARE OF THE TOTAL COST


We can solve the problem by minimizing the square of thecost function [18], as well as by minimizing the cost functionitself as . Thus, we sometimes represent the problem (7)as follows:

(13)

The neural representation is the same as except forthe last term, which minimizes the square of the total cost,whereas minimizes the total cost itself. Both are valid

representations of the problems (7), of course. In this section,by exploring the set of all the asymptotically stable verticesof this network hypercube, we show that the networkdistinguishes optimal solutionsmore sharplythan , and thatit greatly overcomes many of the weakness of the network.However, it is also shown that, in general, the networkcannot always obtain an optimal solution even if it convergesto a vertex.


As the next theorem shows, all the infeasible solutions arepotentially unstable, just as in the previous case.


holds.Proof: For any infeasible solution

, at least one of the following cases holds:

1) for some ,2) for some ,3) for some ,4) for some .

For the cases of 1) and 2), just the same as those forTheorem 1, we can show that is unstable if

and

hold, respectively. Then, for the case of 3), there exists. Since we can assume that 2) does not hold,

holds for any , and . Hence, we have


Hence, if

we have , so that is unstable.For the case of 4), since we can assume that neither 1) nor

3) holds, holds for any . And, since wecan also assume that 2) does not hold, for any, and . Hence, for , we similarly have

Hence, if

holds, so that is unstable.For the stability and instability of feasible solutions we can

also have the next theorem.Theorem 4: In the Hopfield network , a feasible solution,

, is asymptotically stable if

and unstable if

where is the total cost of , that is

Proof: For any and , we have

.

For , always holds. For , if

holds, we have . Hence, is asymptoticallystable if

holds.Conversely, if , we have

for some , and is unstable.Just the same as , the instability conditions of all the

infeasible solutions to assignment problems can be simplyrestated as

Also the same as , we can assume that coefficientsand satisfy the instability conditions of infeasible solutions

shown in Theorem 3. Hence, the set of all the asymptoticallystable vertices of this network hypercube contains all the fea-sible solutions which satisfy the former condition in Theorem4, and does not contain all the infeasible solutions and feasiblesolutions which satisfy this theorem’s latter condition.

As Theorem 4 shows, the stability of a feasible solution,, depends on the product of its total cost and

maximum unit cost, which is included in the total cost ofthis solution, that is,

In the previous section, the limitation of the network wastheoretically shown. That is, in the network , the feasiblesolution shown in Table I is asymptotically stableif the optimal solution shown in Table I isasymptotically stable. However, since

holds, by selecting and to satisfy

we can make the nonoptimal solution unstable and theoptimal solution asymptotically stable in the network

. This suggests that the network distinguishes optimalsolutionsmore sharplythan , and that the average qualityof the solutions obtained is improved. Actually, we willtheoretically confirm this expectation soon. Thus we canexpect to obtain optimal or good solutions more frequentlyby the network than by .

C. Networks’ Partial Ordering

Many neural representations have been proposed for eachof combinatorial optimization problems, and their perfor-mances have been evaluated and compared with each othersolely through network simulations for some problem in-stances rather than on a theoretical basis. Since the networkperformance deeply depends on the nature of each of probleminstances—for example, city locations in TSP’s—it is inap-propriate to compare networks’ performances solely throughsimulations. Therefore, we need some theoretical comparisonmethods for network performances.

Matsuda [9], [10], using the stability theory of feasibleand infeasible solutions shown above, has also proposed atheoretical method of comparing the performances of networksfor each of the combinatorial optimization problems. Thismethod is based on the relative abilities of networks todistinguish optimal solutions from other feasible and infeasiblesolutions. More precisely, the network is characterized bya set of all the asymptotically stable vertices of its statehypercube, and then, of two networks, one is regarded asmoresharply distinguishing optimal solutions than the other if itscharacterizing set is set-theoretically included in that of theother. Thus, based on this set-theoretical inclusion relation, apartial ordering is given to the set of all the networks for each


of the combinatorial optimization problems, and it representsthe relative performance of the network. This is formally statedas follows.

Definition 1: For any two Hopfield networks or neuralrepresentations, say and , for the same particular probleminstance, let and be sets of asymptotically stablevertices of the state hypercube of the networksand ,respectively. Note that the values of all coefficients of theseneural representation, for example, , , , and in ,are fixed. We assume that and contain all theoptimal solutions. Then, we call the networkbetter,or moresharply distinguishing optimal solutions,than the network ,denoted as , if . In addition, we denote

if and , that is,holds. If neither nor holds, wecall and incomparable. Thus, a partial ordering of allneural representations or Hopfield networks for each probleminstance is given.

Although this theoretical method concerns solely withasymptotically stable vertices, the performance of networkdepends not only on the asymptotically stable vertices butalso on asymptotically stable points inside the hypercube andthe width of basin of each of the asymptotically stable points.However, the stability of vertices have a close relation tothe behavior of the network converging inside the hypercube.Furthermore, by applying this theoretical comparison methodto two familiar networks for TSP’s, Matsuda [9], [10] hasestablished a theoretical justification of the superiority ofone network to the other. The superior network is one withrelative distances between cities, and the inferior is one withabsolute distances. That is, for all TSP instances, the formernetwork is better than the latter in the above partial orderingof all the networks for TSP’s, and greatly overcomes thetheoretical limitation of the latter network. This superiorityhad already been observed through simulations [11], andthen has been theoretically justified. Thus, this theoreticalcomparison method is appropriate for comparing networks’capabilities.

D. Superiority and Limitation

Based on the above partial ordering of the networks, we cannow theoretically show that distinguishes optimal solutionsmore sharplythan .

Theorem 5: Let be an optimal solution. In the Hopfieldnetwork , if

and is small enough, then

holds for any satisfying , that is, the followinghold:

1) For any , the optimal solution is asymptoticallystable in the above network , that is, .

2) holds if is small enoughand , that is, the optimal solution isasymptotically stable in .

Proof: 1) is obvious from Theorem 4. We will next show2). Let be a feasible solution. Then, by Theorem4, we have

Then, by letting , we have

Hence, we have

So, since , by Theorem 2, we have .Thus, .

Thus, no matter how well the network is tuned, thenetwork tuned to satisfy the conditions shown in Theorem5 can distinguish optimal solutionsmore sharplythan any .

However, for the nearly optimal but nonoptimal solutionof cost 16 shown in Table I, since

holds, by Theorem 4, the nonoptimal solution is asymp-totically stable in the network if the optimal solution

is asymptotically stable. Thus, although the networkdistinguishes optimal solutionsmore sharplythan , it cannotalways yield an optimal solution even if it converges toa vertex, and so cannot completely overcome theoreticallimitation mentioned before. This is a theoretical limitationof the network .

E. Notion of “Optimal” Network

Then, we have an interest in presenting thebest or “op-timal” network in the above partial ordering, if any, ratherthan presenting just abetter network than or . Thus,the “optimal” network distinguishes optimal solutionsmostsharplyof all the neural networks, and so we can expect thatsuch a network obtains optimal or nearly optimal solutionsmost frequently. This concept is formally stated as follows.

Definition 2: A Hopfield network or neural representation,, for some particular problem instance is called “optimal,”

or most sharply distinguishing optimal solutions, if ,equivalently, for any Hopfield network forthis problem instance.

Note that, however, this definition does not mean that onecan always obtain an optimal solution whenever the “optimal”network converges to a vertex. That is, the “optimal” networkcannot always distinguish optimal solutionscompletely. Thus,even the “optimal” network cannot completely overcome


the theoretical limitation mentioned above although it distin-guishes optimal solutionsmost sharplyof all the networks. Sowe have the next definition.

Definition 3: A Hopfield network is called “complete”for some particular problem instance if is the set of alloptimal solutions.

Then, we immediately have the next theorem on the rela-tionship between the “complete” network and the “optimal”one.

Theorem 6: For any problem instance, a “complete” net-work is “optimal,” and can always obtain an optimal solutionwhenever it converges to a vertex.

Thus, the “complete” network completely overcomes thetheoretical limitation mentioned above, however, it is uncertainwhether the complete network exists or not. In the next section,for each instance of the combinatorial optimization problemsgiven by (7), we design thecomplete,and so “optimal,”network. Thus, the “optimal” networks for (7) are shown tobe “complete.”

V. “OPTIMAL” NETWORK


Here we present a new neural representation, which isthe same as except for the third term

(14)

where

.

Thus the value of the coefficient of the binary value con-straint is proportional to the cost . Note that theexistence of this binary value constraint also does not changethe value of energy at each vertex of the hypercube. This isalso a valid representation of the problems given by (7). Inthis section we theoretically show that, of all the Hopfieldnetworks, the network can distinguish optimal solutionsmost sharply.


As before, we show the asymptotical stability and instabilityconditions of feasible and infeasible solutions for the network

. Here again, all the infeasible solutions are potentiallyunstable, as the next theorem shows.


holds.

Proof: Similar to that for Theorem 3.Just the same as and , the instability conditions of all

the infeasible solutions to assignment problems can be simplystated as

For the stability and instability of feasible solutions we canalso give the following theorem.

Theorem 8: In the Hopfield network , a feasible solution,, is asymptotically stable if

and unstable if

where is the total cost of , that is,

Proof: First, for any and such that , we have

.

For , always holds since .For , if

holds, we have .Also, for any and such that

.

Then, we always have for , andfor . Hence, is asymptotically

stable if

Conversely, if , we havefor any . Thus, is unstable.

Just as in the previous networks, if coefficients andsatisfy the instability condition of infeasible solutions shownin Theorem 7, the set of all the asymptotically stable verticesof this network hypercube contains all the feasible solutionswhich satisfy the former condition in Theorem 8, and doesnot contain all the infeasible solutions and feasible solutionswhich satisfy the latter condition in Theorem 8.


C. “Optimal” Network

Theorem 8 shows that, if the values ofand are properlyselected, the network can make all the optimal solutionsasymptotically stable and all the nonoptimal solutions unstable.Thus we can immediately obtain the next theorem.

Theorem 9: Let the Hopfield network satisfy

where is an optimal solution. Then, if is small enough,is “complete,” consequently, “optimal.”

Thus, in the network which satisfies the conditions inTheorem 9, a vertex is asymptotically stable if and only if itis an optimal solution. As a result, one can always obtain anoptimal solution whenever the network converges to a vertex.In order to construct this optimal network , however, theknowledge of the minimum cost of the problem is required inadvance. Since our objective is to find an optimal solution, this“optimal” network seems meaningless. However, andalso needa priori knowledge of an optimal solution in orderto be best-tuned. Hence, as far as we employ these Hopfieldnetworks, we cannot completely escape from trial and errorto find the appropriate value of the network coefficient,or . On this point there is no difference between andother networks. However, differently from , , and manyother networks, achieves the above novel performance ifit is well tuned. This is why we are to employ . We areexpected to find such appropriate or nearly appropriate valueof or for shown in Theorem 9 by trial and errorwithout knowing the minimum cost or optimal solution.

For the quadratic combinatorial optimization problems suchas quadratic assignment problems and TSP’s, we can designsuch “optimal” network by employing higher order Hopfieldnetworks [19].

VI. “O PTIMAL” NETWORK FOR MAXIMIZATION PROBLEMS

In the previous section we have shown the optimal networkfor the combinatorial optimization problems given by

(7), which minimize the value of the cost function. Someoptimization problems, however, maximize the value of theobjective function rather than minimize. In this section optimalnetwork for these problems are shown.

These maximization problems are formally stated as fol-lows.

maximize

subject to for any

for any

for any and

(15)

Many of the NP complete combinatorial optimization prob-lems, for example, multiple knapsack problems and knapsack

problems, can be expressed as (15). Knapsack problems arestated as follows. A set of items is available to be packedinto a knapsack, the capacity of which is ofunit size. Item

is of unit size, and is a unit profit given by item. Thegoal is to find the subsetof items which should be packed inorder to maximize their total profit. Thus, it can be formallystated as

maximize

subject to

for any

(16)

where

if item is packedotherwise.

As mentioned in Section II, by introducing some slack neurons, we can express knapsack problems given by (16)

as (15).In order to solve the optimization problems (15), however,

we must restate them as minimizing the objective functionrather than maximizing, since the Hopfield network approachrelies on the energy minimization. So, we express the objectivefunction in (15) as

minimize (17)

Thus, we can construct an “optimal” Hopfield network for(15) as follows:

(18)

where

.

Then, we can similarly show that is actually “optimal.”Theorem 10:Let the Hopfield network satisfy

where is an optimal solution and . Then, ifis small enough, is “complete,” consequently, “optimal.”


TABLE IIRESULTS OF SIMULATIONS CONVERGING

TO A VERTEX OF THE NETWORK HYPERCUBE

VII. SIMULATIONS

A. Objectives and Network Tuning

In this section we illustrate, through simulations of thenetwork , , and , the theoretical characterizationsderived above. Simulations are made for ten instances of20 20 assignment problems, each unit cost of which israndomly generated from one to 100. Three networks aretuned to distinguish optimal solutions from other feasible andinfeasible solutionsas sharply aspossible. For these networks,we can obtain an optimal solution at a vertex of the hypercube.Such is the “optimal” network given by Theorem 9. Wecan also construct such and by making the coefficient

slightly larger than the values of the right-hand side ofthe asymptotical stability conditions for an optimal solutionshown in Theorems 2 and 4, respectively, and also by making

satisfy the instability conditions of infeasible solutions inTheorems 1 and 3, respectively. For each network, we maketen simulations by changing the initial states of the networksrandomly near the center of the hypercubes. Thus, 300 (“tenproblem instances” “ten initial states” “three networks”)simulations are made.

Update of each neuron value is made asynchronously byemploying the next difference equation rather than differentialequation (1)

(19)

an output value is given by a sigmoid function (2). Thevalues of and are selected by trial and error. Note thatannealing is not employed.

B. Results

Although all the simulations converge, some of them donot converge to a vertex of the hypercube, of course. First,only the results of all the simulations which converge to avertex are shown in Table II. For three networks, the ratesof the convergence to a vertex are shown in the first line.We can see ’s good convergence to vertices. The secondline shows the rates of the convergence to feasible solutions,and, by comparing the first and second lines, we can seethat all the simulations converging to a vertex obtain feasiblesolutions. This is because all the networks satisfy the instability

TABLE IIIRESULTS OF SIMULATIONS CONVERGING TO A

VERTEX OF OR INSIDE THE NETWORK HYPERCUBE

conditions of infeasible solutions shown in Theorems 1, 3,or 7. Most importantly, as the third line shows, there isa significant difference in the rates of simulations whichconverge to optimal solutions. The network converges tooptimal solutions only 1%, however, and obtain optimalsolutions 31 and 58%, respectively. By comparing the firstand third lines, we can see that always obtains an optimalsolution whenever it converges to a vertex, whereas either

or cannot always obtain an optimal solution even ifit converges to a vertex. This is what Theorems 2, 4, and9 mean. The last line shows another important difference inthe average error of the feasible solutions obtained by thesenetworks, where the average error (%) is given by

average error (%)“average cost obtained” “minimum cost”

“minimum cost”

Thus, we can see the great improvements in the networksand , in particular, in .

As mentioned before, one cannot, strictly speaking, obtain asolution to a combinatorial optimization problem if a networkconverges inside the hypercube, because the meaning of eachneuron is given for a value zero or one. However, one usuallyaccepts simulations which converge inside the hypercube ifthere is a correspondence between a converging point and avertex by some criterion. Although our theory focuses onlyon cases in which the networks converge to vertices, it is alsointeresting to notice the qualities of solutions obtained by thesenetworks converging inside the hypercube. By corresponding

to one if , and to 0 if , in Table III,we show the results of all the simulations which convergeeither to a vertex or inside the hypercube. As shown in thefirst line, we always obtain feasible solutions for these 300simulations. By the second and third lines, also in this casewe can see the similar differences in the performance amongthe three networks.

Thus, in either case, we can see the relation ,and that the “optimal” network obtains optimal solutionsmost frequently and achieves the best performance.

VIII. C ONCLUSIONS

We have presented the “optimal” Hopfield network for manyof the combinatorial optimization problems with linear costfunction. It is proved that a vertex of this network statehypercube is asymptotically stable if and only if it is an optimalsolution. That is, one can always obtain an optimal solution


whenever the network converges to a vertex. On the basisof some theoretical measure of the network performance, ithas been shown that this network is “optimal.” It has beenalso shown through simulations of assignment problems thatthis network obtains optimal or nearly optimal solutions morefrequently than other conventional networks, including whenthey converge inside their state hypercubes.

However, some problems Hopfield networks have remainunsolved even for the “optimal” Hopfield network. That is,in order to obtain “optimal” network we need a tuning ofthe network coefficients, probably by trial and error. So,we still cannot escape from trial and error. Furthermore,the “optimal” network may not converge to a vertex of thestate hypercube. However, differently from other Hopfieldnetworks, the “optimal” network achieves the above novelperformance once it is well-tuned and converges to a vertex.

Although the “optimal” network presented here cannotbe applicable to the combinatorial optimization problemswith quadratic cost function, such as quadratic assignmentproblems and TSP’s, we can design such “optimal” networkfor quadratic problems by employing higher order Hopfieldnetworks [19].

APPENDIX

Proof of (5) and (6): The asymptotical stability and in-stability conditions of a vertex of the network state hypercubehave been already given [15], but their proof is repeated again,for convenience.

First, since holds, by taking a deriva-tive of (2) w.r.t. time , we have (4)

for any . Thus, any vertex is equilibrium.Then, by letting

and

we can state (4) as

(20)

We now show the stability condition of a vertex, equilibriumof (20), by deriving eigenvalues of Jacobian matrix

of evaluated at . For any vertex , since

holds, we have

where is Kronecker’s delta. Hence, Jacobian matrixis a diagonal matrix. So its eigenvalueis a diagonal element,

, and we have

(21)

It is well known that equilibrium is asymptotically stableif all the eigenvalues of have negative real parts, andunstable if at least one eigenvalue has a positive real part [20].So, a vertex, , is asymptotically stable if

(5)

holds for any neuron, and unstable if

(6)

holds for some neuron.

REFERENCES

[1] G. W. Wilson and G. S. Pawley, “On the stability of the travellingsalesman problem algorithm of Hopfield and Tank,”Biol. Cybern.,vol.58, pp. 63–70, 1988.

[2] B. Kamgar-Parsi and B. Kamgar-Parsi, “On the problem solving withHopfield neural networks,”Biol. Cybern.,vol. 62, pp. 415–423, 1990.

[3] S. V. B. Aiyer, M. Niranjan, and F. Fallside, “A theoretical investigationinto the performance of the Hopfield model,”IEEE Trans. NeuralNetworks,vol. 1, pp. 204–215, June 1990.

[4] S. Abe, “Global convergence and suppression of spurious states of theHopfield neural networks,”IEEE Trans. Circuits Syst. I,vol. 40, pp.246–257, Apr. 1993.

[5] S. Matsuda, “On the stability of the solution in Hopfield neural net-work—The case of travelling salesman problem,” Inst. Electr., Inform.,Commun. Eng., Tech. Rep., vol. NC93-8, 1993, in Japanese.

[6] , “The stability of the solution in Hopfield neural network,” inProc. IEEE Int. Joint Conf. Neural Networks (IJCNN’93),1993, pp.1524–1527.

[7] , “Stability of solutions in Hopfield neural network,”Trans. Inst.Electr., Inform., Commun. Eng.,vol. J77-D-II, pp. 1366–1374, July1994, in Japanese; also translated inSyst. Comput. Japan,vol. 26, pp.67–78, May 1995.

[8] , “Theoretical characterizations of possibilities and impossibilitiesof Hopfield neural networks in solving combinatorial optimizationproblems,” inProc. IEEE Int. Conf. Neural Networks (ICNN’94),1994,pp. 4563–4566.

[9] , “Theoretical comparison of mappings of combinatorial optimiza-tion problems to Hopfield networks (extended abstract),” inProc. IEEEInt. Conf. Neural Networks (ICNN’95),1995, pp. 1674–1677.

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[12] J. J. Hopfield, “Neurons with graded response have collective computa-tional properties like those of two-state neurons,” inProc. Nat. AcademySci. USA,May 1984, vol. 81, pp. 3088–3092.


[13] J. J. Hopfield and D. W. Tank, “‘Neural’ computation of decisions ofoptimization problems,”Biol. Cybern.,vol. 52, pp. 141–152, 1985.

[14] S. Matsuda, “Distribution of asymptotically stable states in Hopfieldnetwork for TSP (extended abstract),” inProc. IEEE Int. Conf. NeuralNetworks (ICNN’96),1996, pp. 529–534.

[15] Y. Uesaka, “On the stability of neural networks with the energy functionof binary variables,” Inst. Electron., Inform., Commun. Eng., Tech. Rep.,vol. PRU88-6, 1988, in Japanese.

[16] S. Matsuda, “Quantized Hopfield networks for integer programming,”Trans. Inst. Electron., Inform., Commun. Eng.,vol. J81-D-II, pp.1354–1364, June 1998, in Japanese; also translated inSyst. Comput.Japan, to be published.

[17] S. Martello and P. Toth,Knapsack Problems. New York: Wiley, 1990.[18] M. Takeda and J. W. Goodman, “Neural network for computation:

Number representation and programming complexity,”Appl. Opt.,vol.25, pp. 3033–3046, Sept. 1986.

[19] S. Matsuda, “‘Optimal’ Hopfield network of higher order for quadraticcombinatorial optimization,” to be published.

[20] M. W. Hirsch and S. Smale,Differential Equations, Dynamical Systems,and Linear Algebra. New York: Academic, 1974.

Satoshi Matsuda (M’95) received the B.E., M.E.,and Ph.D. degrees in 1971, 1973, and 1976, respec-tively, from Waseda University, Tokyo, Japan

He then was with Fujitsu Corp., where he engagedin research and development of operating systems,on-line systems, and artificial intelligence. In 1987,he joined Tokyo Electric Power Company, where heis currently in the Computer and CommunicationsResearch Center. He is also a Lecturer of computerscience at the Graduate School of Chuo University,Tokyo. His research fields include the theory of

artificial intelligence and neural networks, but he is interested in computersciences in general.

Dr. Matsuda is a member of the IEEE Computer Society, ACM, the Instituteof Electronics, Information, and Communication Engineers, Information Pro-cessing Society of Japan, Japan Society for Software Science and Technology,and Japanese Society for Artificial Intelligence.

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