Research Article Semistrict -Preinvexity and Optimality in Nonlinear Programming · 2019. 7....

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 103864 7 pageshttpdxdoiorg1011552013103864

Research ArticleSemistrict 119866-Preinvexity and Optimality inNonlinear Programming

Z Y Peng12 Z Lin1 and X B Li1

1 College of Science Chongqing JiaoTong University Chongqing 400074 China2Department of Mathematics Inner Mongolia University Hohhot 010021 China

Correspondence should be addressed to Z Y Peng pengzaiyun126com

Received 26 October 2012 Accepted 10 April 2013

Academic Editor Natig M Atakishiyev

Copyright copy 2013 Z Y Peng et alThis is an open access article distributed under theCreativeCommonsAttributionLicense whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A class of semistrictly 119866-preinvex functions and optimality in nonlinear programming are further discussed Firstly the relation-ships between semistrictly 119866-preinvex functions and 119866-preinvex functions are further discussed Then two interesting propertiesof semistrictly119866-preinvexity are given Finally two optimality results for nonlinear programming problems are obtained under theassumption of semistrict 119866-preinvexity The obtained results are new and different from the corresponding ones in the literatureSome examples are given to illustrate our results

1 Introduction

It is well known that convexity and generalized convexityhave been playing a central role in mathematical program-ming economics engineering and optimization theory Theresearch convexity and generalized convexity are one of themost important aspects in mathematical programming andoptimization theory in [1ndash4] Various kinds of generalizedconvexity have been introduced by many authors (see eg[5ndash21] and the references therein) In 1981 Hanson [5] intro-duced the concept of invexity which is extension of differen-tiable convex functions and proved the sufficiency of Kuhn-Tucker condition LaterWeir andMond [6] considered func-tions (not necessarily differentiable) for which there exists avector function 120578 119877119899 times119877119899 rarr 119877

119899 such that for all 119909 119910 isin 119877119899120582 isin [0 1] one hasthe following

119891 (119910 + 120582120578 (119909 119910)) le 120582119891 (119909) + (1 minus 120582) 119891 (119910) (1)

which has been named as preinvex functions with respect tovector-valued function 120578 In 2001 Yang and Li [8] obtainedsome properties of preinvex function At the same timeYang and Li [9] introduced the concept of semistrictly prein-vex functions and investigated the relationships betweensemistrictly preinvex functions and preinvex functions It isworth mentioning that many properties and applications in

mathematical programming for invex functions and preinvexfunctions are discussed by many authors (see eg [6ndash11 21]and the references therein)

On the other hand Avriel et al [12] introduced a classof 119866-convex functions which is another generalization ofconvex functions and obtained some relations with othergeneralization of convex functions In [13] Antczak intro-duced the concept of a class of 119866-invex fuctions which isa generalization of 119866-convex functions and invex functionsRecently Antczak [14] introduced a class of 119866-preinvexfunctions which is a generalization of 119866-invex [13] preinvexfunctions [8] and derived some optimality results for con-strained optimization problems under 119866-preinvexity Veryrecently Luo and Wu introduced a new class of func-tions called semistrictly 119866-preinvex functions in [15] whichinclude semistrictly preinvex functions [9] as a special caseThey investigated the relationships between semistrictly 119866-preinvex functions and 119866-preinvex functions and gave a cri-terion for semistrict 119866-preinvexity Moreover they also pro-posed three open questions (just as they said ldquoan interestingtopic for our future research is tordquo (1) investigate 119866-preinvexfunctions and semicontinuity (2) explore some propertiesof semistrictly 119866-preinvex functions (3) research into someapplications in optimization problems under semistrictly 119866-preinvexity [15])

2 Abstract and Applied Analysis

However as far as we know there are few papers dealingwith the properties and applications of the semistrictly 119866-preinvex functions [16] The questions above in [15] have notbeen solved and one condition in [15] is notmild in research-ing of relationships between 119866-preinvexity and semistrict 119866-preinvexity So in this paper we further discuss semistrictly119866-preinvex functions The rest of the paper is organizedas follows Firstly we investigate 119866-preinvex functions andsemicontinuity and obtain a criterion of 119866-preinvexity undersemicontinuity Then we give new relationships between 119866-preinvexity and semistrictly 119866-preinvexity which are differ-ent from the recent ones in the literature [15] Finally weget two optimality results under semistrict 119866-preinvexity fornonlinear programming The obtained results in this paperimprove and extend the existing ones in the literature (eg[8 9 11 15 16])

2 Preliminaries and Definitions

Throughout this paper let119870 be a nonempty subset of 119877119899 Let119891 119870 rarr 119877 be a real-valued function and 120578 119870 times 119870 rarr 119877

119899

a vector-valued function Let 119868119891(119870) be the range of 119891 that is

the image of119870 under 119891Now we recall some definitions

Definition 1 (see [5 6]) A set 119870 is said to be invex if thereexists a vector-valued function 120578 119870 times 119870 rarr 119877

119899 (120578 = 0) suchthat

119909 119910 isin 119870 120582 isin [0 1] 997904rArr 119910 + 120582120578 (119909 119910) isin 119870 (2)

Definition 2 (see [6 8]) Let 119870 sube 119877119899 be an invex set with

respect to 120578 119877119899times119877119899 rarr 119877119899 and let119891 119870 rarr 119877 be amapping

One says that 119891 is preinvex if

119891 (119910 + 120582120578 (119909 119910))

le 120582119891 (119909) + (1 minus 120582) 119891 (119910) forall119909 119910 isin 119870 120582 isin [0 1]

(3)

Remark 3 Any convex function is a preinvex function with120578(119909 119910) = 119909 minus 119910

Definition 4 (see [10]) Let119870 sube 119877119899 be an invex setwith respect

to 120578 119877119899

times 119877119899

rarr 119877119899 Let 119891 119870 rarr 119877 One says that 119891 is

prequasi-invex if

119891 (119910 + 120582120578 (119909 119910))

le max 119891 (119909) 119891 (119910) forall119909 119910 isin 119870 120582 isin [0 1]

(4)

Definition 5 (see [14]) Let 119870 be a nonempty invex (withrespect to 120578) subset 119877119899 A function 119891 119870 rarr 119877 is said to be119866-preinvex at 119906 on 119870 if there exists a continuous real-valuedincreasing function 119866 119868

119891(119870) rarr 119877 such that for all 119909 isin 119870

and 120582 isin [0 1] (120582 isin (0 1))

119891 (119906 + 120582120578 (119909 119906)) le 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119906)))

(5)

If (5) is satisfied for any 119906 isin 119870 then 119891 is 119866-preinvex on 119870with respect to 120578

Definition 6 (see [15]) Let 119870 be a nonempty invex (withrespect to 120578) subset 119877119899 A function 119891 119870 rarr 119877 is said to besemistrictly 119866-preinvex at 119906 on119870 if there exists a continuousreal-valued increasing function 119866 119868

119891(119870) rarr 119877 such that for

all 119909 isin 119870 (119891(119909) = 119891(119906)) and 120582 isin (0 1)

119891 (119906 + 120582120578 (119909 119906)) lt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119906)))

(6)

If (6) is satisfied for any 119906 isin 119870 then 119891 is semistrictly 119866-pre-invex on119870 with respect to 120578

Remark 7 In order to define an analogous class of semistrictly119866-preincave functions with respect to 120578 the direction of theinequality in Definition 6 should be changed to the oppositeone

Remark 8 Every semistrictly preinvex function [8 9] issemistrictly 119866-preinvex with respect to the same function 120578where 119866 119868

119891(119870) rarr 119877 is defined by 119866(119909) = 119909

In order to prove our main result we need Condition Cas follows

Condition C (see [9]) The vector-valued function 120578 119883 times

119883 rarr 119883 is said to satisfy Condition C if for any 119909 119910 isin 119883 and120582 isin [0 1]

120578 (119910 119910 + 120582120578 (119909 119910)) = minus120582120578 (119909 119910)

120578 (119909 119910 + 120582120578 (119909 119910)) = (1 minus 120582) 120578 (119909 119910)

(7)

Example 9 Let

120578 (119909 119910) =

119909 minus 119910 if 119909 ge 0 119910 ge 0

119909 minus 119910 if 119909 lt 0 119910 lt 0

minus5

2minus 119910 if 119909 gt 0 119910 le 0

5

2minus 119910 if 119909 le 0 119910 gt 0

(8)

It can be verified that 120578 satisfies the Condition C

3 Relationships withSemistrictly 119866-Preinvexity

In [15] Luo and Wu obtained a sufficient condition of 119866-preinvex functions under the condition of intermediate-point119866-preinvexity Now we investigate 119866-preinvex functions andsemicontinuity without the condition of intermediate-point119866-preinvexity

Theorem 10 Let 119870 be a nonempty invex set with respect to120578 where 120578 satisfies Condition C Let 119891 119870 rarr 119877 be lowersemicontinuous and semistrictly 119866-preinvex for the same 120578 on119870 and let 119891(119910 + 120578(119909 119910)) le 119891(119909) (for all 119909 isin 119870) Then 119891 is119866-preinvex function on 119870

Abstract and Applied Analysis 3

Proof Let 119909 119910 isin 119870 From the assumption of 119891(119910+120578(119909 119910)) le119891(119909) when 120582 = 0 1 we can know that

119891 (119910 + 120582120578 (119909 119910)) le 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(9)

Then there are two cases to be considered

(i) If119891(119909) = 119891(119910) then by the semistrict119866-preinvexity of119891 we have the following

119891 (119910 + 120582120578 (119909 119910))

lt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910))) forall120582 isin (0 1)

(10)

(ii) If 119891(119909) = 119891(119910) to show that 119891 is a 119866-preinvex func-tion we need to show that

119891 (119910 + 120582120578 (119909 119910))

le 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910))) = 119891 (119909)

(11)

By contradiction suppose that there exists an 120572 isin (0 1) suchthat

119891 (119910 + 120572120578 (119909 119910)) gt 119891 (119909) (12)

Let 119911120572= 119910+120572120578(119909 119910) Since 119891 is lower semicontinuous there

exists 120573 120572 lt 120573 lt 1 such that

119891 (119911120573) = 119891 (119910 + 120573120578 (119909 119910)) gt 119891 (119909) = 119891 (119910) (13)

From Condition C

119911120573= 119911120572+120573 minus 120572

1 minus 120572120578 (119909 119911

120572) (14)

By the semistrict119866-preinvexity of119891 and (12) we have the fol-lowing

119891 (119911120573)

= 119891(119911120572+120573 minus 120572

1 minus 120572120578 (119909 119911

120572))

lt 119866minus1

[120573 minus 120572

1 minus 120572119866 (119891 (119909)) + (1 minus

120573 minus 120572

1 minus 120572)119866 (119891 (119911

120572))]

lt 119866minus1

[120573 minus 120572

1 minus 120572119866 (119891 (119911

120572)) + (1 minus

120573 minus 120572

1 minus 120572)119866 (119891 (119911

120572))]

= 119891 (119911120572)

(15)

On the other hand fromCondition C one can obtain the fol-lowing

119911120572= 119911120573+ (1 minus

120572

120573) 120578 (119910 119911

120573) (16)

According to (13) and the semistrictly 119866-preinvexity of 119891 wehave the following

119891 (119911120572)

= 119891(119911120573+ (1 minus

120572

120573) 120578 (119910 119911

120573))

lt 119866minus1

[(1 minus120572

120573)119866 (119891 (119910)) +

120572

120573119866 (119891 (119911

120573))]

lt 119866minus1

[(1 minus120572

120573)119866 (119891 (119911

120573)) +

120572

120573119866 (119891 (119911

120573))]

= 119891 (119911120573)

(17)

which contradicts (15) This completes the proof

Remark 11 In Theorem 10 we investigate 119866-preinvex func-tions and lower semicontinuity and establish a new suffi-cient condition for 119866-preinvexity without the condition ofintermediate-point 119866-preinvexity Therefore we also answerthe open question (1) which proposed in [15] (ldquo(1) investigate119866-preinvex functions and semicontinuityrdquo [15])

Now we give a new sufficient condition for semistrictly119866-preinvexity

Theorem 12 Let 119870 be a nonempty invex set with respect to 120578where 120578 satisfies Condition C Let 119891 119870 rarr 119877 be a 119866-preinvexfunction for the same 120578 on 119870 Suppose that for any 119909 119910 isin 119870

with 119891(119909) = 119891(119910) there exists an 120572 isin (0 1) such that

119891 (119910 + 120572120578 (119909 119910)) lt 119866minus1

(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))

(18)

Then 119891 is semistrictly 119866-preinvex function on 119870

Proof For any 119909 119910 isin 119870 with 119891(119909) = 119891(119910) and 120582 isin (0 1) byassumption we have the following

119891 (119910 + 120582120578 (119909 119910)) le 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(19)

(i) Let 120582 le 120572 From Condition C we can obtain the fol-lowing

119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910)

= 119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910 + 120572120578 (119909 119910) minus 120572120578 (119909 119910))

= 119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910 + 120572120578 (119909 119910)

+120578 (119910 119910 + 120572120578 (119909 119910)))

= 119910 minus120582

120572120578 (119910 119910 + 120572120578 (119909 119910))

= 119910 + 120582120578 (119909 119910)

(20)


Using (18) and the 119866-preinvexity of 119891 we have the following

119891 (119910 + 120582120578 (119909 119910))

= 119891 [119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910)]

le 119866minus1

[120582

120572119866 ( 119891 (119910 + 120572120578 (119909 119910))

+ (1 minus120582

120572)119866 (119891 (119910))]

lt 119866minus1

[120582

120572119866 (119866minus1

(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910))))

+(1 minus120582

120572)119866 (119891 (119910))]

= 119866minus1

[120582

120572(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))

+ (1 minus120582

120572)119866 (119891 (119910))]

= 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(21)

(ii) Let 120572 lt 120582 that is

0 lt1 minus 120582

1 minus 120572lt 1 (22)

From Condition C we have the following

119910 + 120572120578 (119909 119910) + (1 minus1 minus 120582

1 minus 120572) 120578 (119909 119910 + 120572120578 (119909 119910))

= 119910 + [120572 + (1 minus1 minus 120582

1 minus 120572) (1 minus 120572)] 120578 (119909 119910)

= 119910 + 120582120578 (119909 119910)

(23)

According to (18) and the 119866-preinvexity of 119891 we get the fol-lowing

119891 (119910 + 120582120578 (119909 119910))

= 119891 [119910 + 120572120578 (119909 119910) + (1 minus1 minus 120582

1 minus 120572) 120578 (119909 119910 + 120572120578 (119909 119910))]

le 119866minus1

[(1 minus1 minus 120582

1 minus 120572)119866 (119891 (119909))

+1 minus 120582

1 minus 120572119866 (119891 (119910 + 120572120578 (119909 119910))) ]

lt 119866minus1


1 minus 120572)119866 (119891 (119909))

+1 minus 120582

1 minus 120572119866 (119866minus1

(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))) ]

= 119866minus1


1 minus 120572)119866 (119891 (119909))

+1 minus 120582

1 minus 120572(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))]

= 119866minus1

( 120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(24)

(21) and (24) imply that 119891 is a semistrictly 119866-preinvex func-tion on119870

Remark 13 Theorem 12 extends and improves Theorem 1 in[15] In Theorem 1 of [15] a uniform 120572 isin (0 1) is neededwhile in assumption (18) of Theorem 12 this condition hasbeen weakened where a uniform 120572 isin (0 1) is not necessaryMoreover our proof is also different from the correspondingresult of [15]

The following example illustrates that assumption (18) inTheorem 12 is essential

Example 14 Let

119891 (119909) = ln(32|119909| +

7

4)

120578 (119909 119910) =

119909 minus 119910 if 119909 ge 0 119910 ge 0119909 minus 119910 if 119909 le 0 119910 le 0minus119909 minus 119910 if 119909 gt 0 119910 lt 0minus119909 minus 119910 if 119909 lt 0 119910 gt 0

(25)

It is obvious that 119891 is a119866-preinvex function with respect to 120578where 119866(119905) = 119890119905119870 = 119877

However from Definition 6 we can verify that 119891 isa semistrictly 119866-preinvex function The reason is that theassumption (18) is violated Indeed there exist 119909 = 12 119910 = 4(with 119891(119909) = 119891(119910)) for any 120582 isin (0 1) we have the following

119891 (119910 + 120582120578 (119909 119910))

= 119891(4 minus7

2120582) = ln(3

2

10038161003816100381610038161003816100381610038164 minus

7

2120582

1003816100381610038161003816100381610038161003816+7

4)

= ln(314minus21

4120582)

= 119866minus1

(3

2(120582 |119909| + (1 minus 120582)

10038161003816100381610038161199101003816100381610038161003816) +

7

4)

= 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(26)

Therefore (18) is essential

Lemma 15 (see [16]) Let 119870 be a nonempty invex set withrespect to 120578 Suppose that 119891 is a semistrictly 119866

1-preinvex func-

tion with respect to 120578 and 1198662is a continuous and strictly

increasing function on 119868119891(119883) If 119892(119905) = 119866

2119866minus1

1is convex on

the image under1198661of the range of 119891 then 119891 is also semistrictly

1198662-preinvex function with respect to the same 120578 on 119870

Theorem 16 Let K be an invex set with respect to 120578 and 119891

119870 rarr 119877 be a semistrictly 119866-preinvex function on 119870 If 119866 is


concave on 119868119891(119883) then 119891 is a preinvex function with respect to

the same 120578 on 119870

Proof Let 119910 and 119911 be two points in 119868119891(119883) Because 119866 is con-

cave on 119868119891(119883) the following inequality

119866 (120582119910 + (1 minus 120582) 119911) ge 120582119866 (119910) + (1 minus 120582)119866 (119911) (27)

holds for any 120582 isin [0 1] Let 119866(119910) = 119909 and 119866(119911) = 119906 Thenfor each pair of points 119909 and 119906 in image 119866 of 119868

119891(119883) that is

119866minus1

(119909) = 119910 and 119866minus1(119906) = 119911 we have the following

119866(120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906))

ge 120582119866 (119866minus1

(119909)) + (1 minus 120582)119866 (119866minus1

(119906))

= 120582119909 + (1 minus 120582) 119906

(28)

It follows from (28) and the increasing property of 119866minus1 that

119866minus1

119866(120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906))

ge 119866minus1

(120582119866 (119866minus1

(119909)) + (1 minus 120582)119866 (119866minus1

(119906)))

= 119866minus1

(120582119909 + (1 minus 120582) 119906)

(29)

Thus

120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906) ge 119866minus1

(120582119909 + (1 minus 120582) 119906)

(30)

Thismeans that119866minus1 is convex Letting1198661= 119866119866

2(119905) = 119905 then

119892(119905) = 1198662119866minus1

1(119905) is convex Hence by virtue of Lemma 15119891 is

a semistrictly1198662-preinvex functionwith respect to 120578 Because

1198662(119905) = 119905 is identity function 119891 is a preinvex function with

respect to the same 120578 on119870

FromTheorem 16 and Definitions 2-4 we can obtain thefollowing Corollary easily

Corollary 17 Let K be an invex set with respect to 120578 and119891 119870 rarr 119877 is a semistrictly 119866-preinvex function on 119870 If 119866is concave on 119868

119891(119883) then 119891 be a prequasi-invex function with

respect to the same 120578 on 119870

4 Semistrictly 119866-Preinvexity and Optimality

In order to solve the open question (3) proposed in [15] (seethe part of Introduction) in this section we consider nonlin-ear programming problems with constraint and obtain twooptimality results under semistrict 119866-preinvexity

We consider the following nonlinear programming Prob-lem (P) with inequality constraint

min119891 (119909) (P)

119892119894(119909) le 0 119894 isin 119869 = 1 119898 (31)

where 119891 119870 rarr 119877 119892119894 119870 rarr 119877 119894 isin 119869 and 119870 is a nonempty

subset of 119877119899 We denote the set of all feasible solutions in (P)by the following

119863 = 119909 isin 119870 119892119894(119909) le 0 119894 isin 119869 (32)

Theorem 18 Suppose the set of all feasible solutions 119863 ofproblem (P) is an invex set with respect to 120578 and 119863 atleast contains two points with nonempty interior Let 119891 be anonconstant semistrictly 119866-preincave function with respect to120578 on 119863 Then no interior of 119863 is an optimal solution of (P) orequivalently any optimal solution 119909 in problem (P) if existsmust be a boundary point of119863

Proof If problem (P) has no solution the theorem is triviallytrue Let 119909 be an optimal solution in problem (P) By assump-tion 119891 is a nonconstant on 119863 Then there exists a feasiblepoint 119909lowast isin 119863 such that

119891 (119909lowast

) gt 119891 (119909) (33)

Let 119911 (119911 = 119909lowast) be an interior point of 119863 By assumption 119863 is

an invex set with respect to 120578 It follows from the definitionof invex set 119863 that there exists 119910 isin 119863 such that for some 120582 isin(0 1)

119911 = 119909lowast

+ 120582120578 (119910 119909lowast

) (34)

By assumption119891 is semistrictly119866-preincave with respectto 120578 on119863 Then we have the following

119891 (119911) = 119891 (119909lowast

+ 120582120578 (119910 119909lowast

))

gt 119866minus1

(120582119866 (119891 (119910)) + (1 minus 120582)119866 (119891 (119909lowast

)))

gt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119909))) = 119891 (119909)

(35)

where 119911 is an interior point of119863From the inequality above we conclude that no interior of

119863 is an optimal solution of (P) that is any optimal solution 119909in problem (P) if exists must be a boundary point of119863 Thiscompletes the proof

Theorem 19 Let 119909 isin 119863 be local optimal in problem (P) More-over we assume that 119891 is semistrictly 119866-preinvex with respectto 120578 at 119909 on 119863 and the constraint functions 119892

119894 119894 isin 119869 are 119866-

preinvex with respect to 120578 at 119909 on119863Then 119909 is a global optimalsolution in problem (P)

Proof Assume that 119909 isin 119870 is a local optimal solution in (P)Hence there is an neighborhood119873

120598(119909) around 119909 such that

119891 (119909) le 119891 (119909) forall119909 isin 119870 cap 119873120598(119909) (36)

Suppose to the contrary 119909 is not a global minimum in (P)there exists an 119909lowast isin 119863 such that

119891 (119909lowast

) lt 119891 (119909) (37)

By assumption the constraint functions119892119894 119894 isin 119869 are119866-prein-

vexwith respect to the same 120578 at119909 on119863 By usingDefinition 5together with 119909lowast 119909 isin 119863 then for all 119894 isin 119869 and any 120582 isin [0 1]we have the following

119892119894(119909 + 120582120578 (119909

lowast

119909)) le 119866minus1

119894(120582119866119894(119892119894(119909lowast

)) + (1 minus 120582)119866119894(119892119894(119909)))

le 119866minus1

119894(120582119866119894(0) + (1 minus 120582)119866

119894(0)) = 0

(38)


Thus for all 119894 isin 119869 and any 120582 isin [0 1]

119909 + 120582120578 (119909lowast

119909) isin 119863 (39)

By assumption 119891 is semistrictly 119866-preinvex with respectto the same 120578 at 119909 on 119863 Therefore by Definition 6 we havethe following

119891 (119909 + 120582120578 (119909lowast

119909))

lt 119866minus1

(120582119866 (119891 (119909lowast

)) + (1 minus 120582)119866 (119891 (119909))) forall120582 isin (0 1)

(40)

From (37) and (40) we have the following

119891 (119909 + 120582120578 (119909lowast

119909))

lt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119909)))

= 119891 (119909) forall120582 isin (0 1)

(41)

For a sufficiently small 120582 gt 0 it follows that

119909 + 120582120578 (119909lowast

119909) isin 119870 cap 119873120598(119909) (42)

which contradicts that 119909 is local optimal in problem (P) Thiscompletes the proof

Now we give an example to illustrate Theorem 19

Example 20 Let 119891 119870 rarr 119877 be defined as follows

119891 (119909) = ln (5 minus 4 |119909|) if |119909| le 10 if |119909| ge 1

(43)

and let119892119894(119909) = 3119909 minus 1 119894 isin 119869 = 1 119898

120578 (119909 119910)=

3

4minus 119910

if 119909 gt 1 0 le 119910 lt 1

or 119909 lt minus1 minus1 lt 119910 le 0

1 if 119910 gt 1 0 le 119909 lt 1119909 minus 119910 if 119910 lt minus1 minus1 lt 119909 le 0minus6119909 minus 119910 if 0 le 119909 lt 1 0 lt 119910 lt 11 minus 119910 if 0 le 119909 lt 1 119910 = 1119909 minus 119910 if |119909| ge 1 1003816100381610038161003816119910

1003816100381610038161003816 ge 1

119910 minus 119909 minus 2 if 119909 gt 1 minus1 lt 119910 lt 0

119910 minus 119909if 119909 gt 1 minus1 lt 119910 lt 0or 119910 gt 1 minus1 lt 119909 lt 0

119910 minus 119909 if 119910 lt minus1 0 lt 119909 lt 1

minus5 minus1

2119910 if |119909| lt 1 minus1 le 119910 le 0

minus4119909 +3

2119910 if 119909 = 1 0 lt 119910 lt 1

3119909 +3

2119910 if 119909 = minus1 minus1 lt 119910 lt 0

minus119910 minus 1199092

minus 119909 minus15

2 if minus 1 lt 119909 lt 0 0 lt 119910 le 1

119910 minus 2119909 if 119909 = 1 minus1 lt 119910 le 0119910 minus 2119909 minus 6 if 119909 = minus1 0 le 119910 lt 1

(44)

From Definitions 5-6 we can get that 119891 is semistrictly 119866-preinvex with respect to 120578 and the constraint functions 119892

119894 119894 isin

119869 are119866-preinvex with respect to 120578 respectively where119866(119905) =119890119905119870 = 119877 Obviously 119909 = minus1 is a local optimal solution of (P)Then all the conditions inTheorem 19 are satisfied By virtueofTheorem 19 119909 = minus1 is a global optimal solution in problem(P)

Acknowledgments

The authors would like to express their thanks to ProfessorX M Yang and the anonymous referees for their valuablecomments and suggestions which helped to improve thepaper This work was supported by the Natural ScienceFoundation of China (nos 11271389 11201509 and 71271226)the Natural Science Foundation Project of ChongQing (nosCSTC 2012jjA00016 and 2011AC6104) and the ResearchGrant of Chongqing Key Laboratory of Operations andSystem Engineering

References

[1] O LMangasarianNonlinear ProgrammingMcGraw-Hill NewYork NY USA 1969

[2] M S Bazaraa H D Sherali and C M Shetty Nonlinear Pro-gramming Theory and Algorithms John Wiley amp Sons NewYork NY USA 1979

[3] S Schaible and W T Ziemba Generalized Concavity in Opti-mization and Economics Academic Press Inc London UK1981

[4] S KMishra andG Giorgi Invexity and Optimization vol 88 ofNonconvex Optimization and Its Applications Springer BerlinGermany 2008

[5] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981

[6] TWeir and BMond ldquoPre-invex functions inmultiple objectiveoptimizationrdquo Journal of Mathematical Analysis and Applica-tions vol 136 no 1 pp 29ndash38 1988

[7] T Weir and V Jeyakumar ldquoA class of nonconvex functions andmathematical programmingrdquo Bulletin of the Australian Mathe-matical Society vol 38 no 2 pp 177ndash189 1988

[8] X M Yang and D Li ldquoOn properties of preinvex functionsrdquoJournal of Mathematical Analysis and Applications vol 256 no1 pp 229ndash241 2001

[9] X M Yang and D Li ldquoSemistrictly preinvex functionsrdquo Journalof Mathematical Analysis and Applications vol 258 no 1 pp287ndash308 2001

[10] X M Yang X Q Yang and K L Teo ldquoCharacterizations andapplications of prequasi-invex functionsrdquo Journal of Optimiza-tion Theory and Applications vol 110 no 3 pp 645ndash668 2001

[11] J W Peng and X M Yang ldquoTwo properties of strictly preinvexfunctionsrdquo Operations Research Transactions vol 9 no 1 pp37ndash42 2005

[12] M Avriel W E Diewert S Schaible and I Zang GeneralizedConcavity Plenum Press New York NY USA 1975

[13] T Antczak ldquoOn 119866-invex multiobjective programming I Opti-malityrdquo Journal of Global Optimization vol 43 no 1 pp 97ndash1092009


[14] T Antczak ldquo119866-pre-invex functions in mathematical program-mingrdquo Journal of Computational and Applied Mathematics vol217 no 1 pp 212ndash226 2008

[15] H Z Luo and H X Wu ldquoOn the relationships between119866-preinvex functions and semistrictly 119866-preinvex functionsrdquoJournal of Computational andAppliedMathematics vol 222 no2 pp 372ndash380 2008

[16] Z Y Peng ldquoSemistrict G-preinvexity and its applicationrdquo Jour-nal of Inequalities and Applications vol 2012 article 198 2012

[17] X J Long ldquoOptimality conditions and duality for nondif-ferentiable multiobjective fractional programming problemswith (119862 120572 120588 119889)-convexityrdquo Journal of OptimizationTheory andApplications vol 148 no 1 pp 197ndash208 2011

[18] X-J Long and N-J Huang ldquoLipschitz 119861-preinvex functionsand nonsmooth multiobjective programmingrdquo Pacific Journalof Optimization vol 7 no 1 pp 97ndash107 2011

[19] S K Mishra Topics in Nonconvex Optimization Theory andApplications vol 50 of Springer Optimization and Its Applica-tions Springer New York NY USA 2011

[20] S K Mishra and N G Rueda ldquoGeneralized invexity-type con-ditions in constrained optimizationrdquo Journal of Systems Scienceamp Complexity vol 24 no 2 pp 394ndash400 2011

[21] X Liu and D H Yuan ldquoOn semi-(119861 119866)-preinvex functionsrdquoAbstract and Applied Analysis vol 2012 Article ID 530468 13pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


However as far as we know there are few papers dealingwith the properties and applications of the semistrictly 119866-preinvex functions [16] The questions above in [15] have notbeen solved and one condition in [15] is notmild in research-ing of relationships between 119866-preinvexity and semistrict 119866-preinvexity So in this paper we further discuss semistrictly119866-preinvex functions The rest of the paper is organizedas follows Firstly we investigate 119866-preinvex functions andsemicontinuity and obtain a criterion of 119866-preinvexity undersemicontinuity Then we give new relationships between 119866-preinvexity and semistrictly 119866-preinvexity which are differ-ent from the recent ones in the literature [15] Finally weget two optimality results under semistrict 119866-preinvexity fornonlinear programming The obtained results in this paperimprove and extend the existing ones in the literature (eg[8 9 11 15 16])

2 Preliminaries and Definitions

Throughout this paper let119870 be a nonempty subset of 119877119899 Let119891 119870 rarr 119877 be a real-valued function and 120578 119870 times 119870 rarr 119877

119899

a vector-valued function Let 119868119891(119870) be the range of 119891 that is

the image of119870 under 119891Now we recall some definitions

Definition 1 (see [5 6]) A set 119870 is said to be invex if thereexists a vector-valued function 120578 119870 times 119870 rarr 119877

119899 (120578 = 0) suchthat

119909 119910 isin 119870 120582 isin [0 1] 997904rArr 119910 + 120582120578 (119909 119910) isin 119870 (2)

Definition 2 (see [6 8]) Let 119870 sube 119877119899 be an invex set with

respect to 120578 119877119899times119877119899 rarr 119877119899 and let119891 119870 rarr 119877 be amapping

One says that 119891 is preinvex if

119891 (119910 + 120582120578 (119909 119910))

le 120582119891 (119909) + (1 minus 120582) 119891 (119910) forall119909 119910 isin 119870 120582 isin [0 1]

(3)

Remark 3 Any convex function is a preinvex function with120578(119909 119910) = 119909 minus 119910

Definition 4 (see [10]) Let119870 sube 119877119899 be an invex setwith respect

to 120578 119877119899

times 119877119899

rarr 119877119899 Let 119891 119870 rarr 119877 One says that 119891 is

prequasi-invex if

119891 (119910 + 120582120578 (119909 119910))

le max 119891 (119909) 119891 (119910) forall119909 119910 isin 119870 120582 isin [0 1]

(4)

Definition 5 (see [14]) Let 119870 be a nonempty invex (withrespect to 120578) subset 119877119899 A function 119891 119870 rarr 119877 is said to be119866-preinvex at 119906 on 119870 if there exists a continuous real-valuedincreasing function 119866 119868

119891(119870) rarr 119877 such that for all 119909 isin 119870

and 120582 isin [0 1] (120582 isin (0 1))

119891 (119906 + 120582120578 (119909 119906)) le 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119906)))

(5)

If (5) is satisfied for any 119906 isin 119870 then 119891 is 119866-preinvex on 119870with respect to 120578

Definition 6 (see [15]) Let 119870 be a nonempty invex (withrespect to 120578) subset 119877119899 A function 119891 119870 rarr 119877 is said to besemistrictly 119866-preinvex at 119906 on119870 if there exists a continuousreal-valued increasing function 119866 119868

119891(119870) rarr 119877 such that for

all 119909 isin 119870 (119891(119909) = 119891(119906)) and 120582 isin (0 1)

119891 (119906 + 120582120578 (119909 119906)) lt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119906)))

(6)

If (6) is satisfied for any 119906 isin 119870 then 119891 is semistrictly 119866-pre-invex on119870 with respect to 120578

Remark 7 In order to define an analogous class of semistrictly119866-preincave functions with respect to 120578 the direction of theinequality in Definition 6 should be changed to the oppositeone

Remark 8 Every semistrictly preinvex function [8 9] issemistrictly 119866-preinvex with respect to the same function 120578where 119866 119868

119891(119870) rarr 119877 is defined by 119866(119909) = 119909

In order to prove our main result we need Condition Cas follows

Condition C (see [9]) The vector-valued function 120578 119883 times

119883 rarr 119883 is said to satisfy Condition C if for any 119909 119910 isin 119883 and120582 isin [0 1]

120578 (119910 119910 + 120582120578 (119909 119910)) = minus120582120578 (119909 119910)

120578 (119909 119910 + 120582120578 (119909 119910)) = (1 minus 120582) 120578 (119909 119910)

(7)

Example 9 Let

120578 (119909 119910) =

119909 minus 119910 if 119909 ge 0 119910 ge 0

119909 minus 119910 if 119909 lt 0 119910 lt 0

minus5

2minus 119910 if 119909 gt 0 119910 le 0

5

2minus 119910 if 119909 le 0 119910 gt 0

(8)

It can be verified that 120578 satisfies the Condition C

3 Relationships withSemistrictly 119866-Preinvexity

In [15] Luo and Wu obtained a sufficient condition of 119866-preinvex functions under the condition of intermediate-point119866-preinvexity Now we investigate 119866-preinvex functions andsemicontinuity without the condition of intermediate-point119866-preinvexity

Theorem 10 Let 119870 be a nonempty invex set with respect to120578 where 120578 satisfies Condition C Let 119891 119870 rarr 119877 be lowersemicontinuous and semistrictly 119866-preinvex for the same 120578 on119870 and let 119891(119910 + 120578(119909 119910)) le 119891(119909) (for all 119909 isin 119870) Then 119891 is119866-preinvex function on 119870



119891 (119910 + 120582120578 (119909 119910)) le 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(9)



119891 (119910 + 120582120578 (119909 119910))

lt 119866minus1


(10)


119891 (119910 + 120582120578 (119909 119910))

le 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910))) = 119891 (119909)

(11)


119891 (119910 + 120572120578 (119909 119910)) gt 119891 (119909) (12)



119891 (119911120573) = 119891 (119910 + 120573120578 (119909 119910)) gt 119891 (119909) = 119891 (119910) (13)

From Condition C

119911120573= 119911120572+120573 minus 120572

1 minus 120572120578 (119909 119911

120572) (14)


119891 (119911120573)

= 119891(119911120572+120573 minus 120572

1 minus 120572120578 (119909 119911

120572))

lt 119866minus1

[120573 minus 120572

1 minus 120572119866 (119891 (119909)) + (1 minus

120573 minus 120572

1 minus 120572)119866 (119891 (119911

120572))]

lt 119866minus1

[120573 minus 120572

1 minus 120572119866 (119891 (119911

120572)) + (1 minus

120573 minus 120572

1 minus 120572)119866 (119891 (119911

120572))]

= 119891 (119911120572)

(15)


119911120572= 119911120573+ (1 minus

120572

120573) 120578 (119910 119911

120573) (16)


119891 (119911120572)

= 119891(119911120573+ (1 minus

120572

120573) 120578 (119910 119911

120573))

lt 119866minus1

[(1 minus120572

120573)119866 (119891 (119910)) +

120572

120573119866 (119891 (119911

120573))]

lt 119866minus1

[(1 minus120572

120573)119866 (119891 (119911

120573)) +

120572

120573119866 (119891 (119911

120573))]

= 119891 (119911120573)

(17)






119891 (119910 + 120572120578 (119909 119910)) lt 119866minus1

(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))

(18)



119891 (119910 + 120582120578 (119909 119910)) le 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(19)


119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910)

= 119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910 + 120572120578 (119909 119910) minus 120572120578 (119909 119910))

= 119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910 + 120572120578 (119909 119910)

+120578 (119910 119910 + 120572120578 (119909 119910)))

= 119910 minus120582

120572120578 (119910 119910 + 120572120578 (119909 119910))

= 119910 + 120582120578 (119909 119910)

(20)



119891 (119910 + 120582120578 (119909 119910))

= 119891 [119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910)]

le 119866minus1

[120582

120572119866 ( 119891 (119910 + 120572120578 (119909 119910))

+ (1 minus120582

120572)119866 (119891 (119910))]

lt 119866minus1

[120582

120572119866 (119866minus1

(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910))))

+(1 minus120582

120572)119866 (119891 (119910))]

= 119866minus1

[120582

120572(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))

+ (1 minus120582

120572)119866 (119891 (119910))]

= 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(21)

(ii) Let 120572 lt 120582 that is

0 lt1 minus 120582

1 minus 120572lt 1 (22)


119910 + 120572120578 (119909 119910) + (1 minus1 minus 120582

1 minus 120572) 120578 (119909 119910 + 120572120578 (119909 119910))

= 119910 + [120572 + (1 minus1 minus 120582

1 minus 120572) (1 minus 120572)] 120578 (119909 119910)

= 119910 + 120582120578 (119909 119910)

(23)


119891 (119910 + 120582120578 (119909 119910))

= 119891 [119910 + 120572120578 (119909 119910) + (1 minus1 minus 120582

1 minus 120572) 120578 (119909 119910 + 120572120578 (119909 119910))]

le 119866minus1


1 minus 120572)119866 (119891 (119909))

+1 minus 120582

1 minus 120572119866 (119891 (119910 + 120572120578 (119909 119910))) ]

lt 119866minus1


1 minus 120572)119866 (119891 (119909))

+1 minus 120582

1 minus 120572119866 (119866minus1

(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))) ]

= 119866minus1


1 minus 120572)119866 (119891 (119909))

+1 minus 120582

1 minus 120572(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))]

= 119866minus1

( 120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(24)




Example 14 Let

119891 (119909) = ln(32|119909| +

7

4)

120578 (119909 119910) =


(25)



119891 (119910 + 120582120578 (119909 119910))

= 119891(4 minus7

2120582) = ln(3

2

10038161003816100381610038161003816100381610038164 minus

7

2120582

1003816100381610038161003816100381610038161003816+7

4)

= ln(314minus21

4120582)

= 119866minus1

(3

2(120582 |119909| + (1 minus 120582)

10038161003816100381610038161199101003816100381610038161003816) +

7

4)

= 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(26)



1-preinvex func-



2119866minus1

1is convex on







the same 120578 on 119870



119866 (120582119910 + (1 minus 120582) 119911) ge 120582119866 (119910) + (1 minus 120582)119866 (119911) (27)


119891(119883) that is

119866minus1


119866(120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906))

ge 120582119866 (119866minus1

(119909)) + (1 minus 120582)119866 (119866minus1

(119906))

= 120582119909 + (1 minus 120582) 119906

(28)


119866minus1

119866(120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906))

ge 119866minus1

(120582119866 (119866minus1

(119909)) + (1 minus 120582)119866 (119866minus1

(119906)))

= 119866minus1

(120582119909 + (1 minus 120582) 119906)

(29)

Thus

120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906) ge 119866minus1

(120582119909 + (1 minus 120582) 119906)

(30)


2(119905) = 119905 then

119892(119905) = 1198662119866minus1












min119891 (119909) (P)

119892119894(119909) le 0 119894 isin 119869 = 1 119898 (31)



119863 = 119909 isin 119870 119892119894(119909) le 0 119894 isin 119869 (32)



119891 (119909lowast

) gt 119891 (119909) (33)



119911 = 119909lowast

+ 120582120578 (119910 119909lowast

) (34)


119891 (119911) = 119891 (119909lowast

+ 120582120578 (119910 119909lowast

))

gt 119866minus1

(120582119866 (119891 (119910)) + (1 minus 120582)119866 (119891 (119909lowast

)))

gt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119909))) = 119891 (119909)

(35)




119894 119894 isin 119869 are 119866-



120598(119909) around 119909 such that



119891 (119909lowast

) lt 119891 (119909) (37)



119892119894(119909 + 120582120578 (119909

lowast

119909)) le 119866minus1

119894(120582119866119894(119892119894(119909lowast

)) + (1 minus 120582)119866119894(119892119894(119909)))

le 119866minus1

119894(120582119866119894(0) + (1 minus 120582)119866

119894(0)) = 0

(38)



119909 + 120582120578 (119909lowast

119909) isin 119863 (39)


119891 (119909 + 120582120578 (119909lowast

119909))

lt 119866minus1

(120582119866 (119891 (119909lowast


(40)


119891 (119909 + 120582120578 (119909lowast

119909))

lt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119909)))

= 119891 (119909) forall120582 isin (0 1)

(41)


119909 + 120582120578 (119909lowast

119909) isin 119870 cap 119873120598(119909) (42)





(43)


120578 (119909 119910)=

3

4minus 119910

if 119909 gt 1 0 le 119910 lt 1



1003816100381610038161003816 ge 1




minus5 minus1

2119910 if |119909| lt 1 minus1 le 119910 le 0

minus4119909 +3

2119910 if 119909 = 1 0 lt 119910 lt 1

3119909 +3






(44)


119894 119894 isin


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891 (119910 + 120582120578 (119909 119910)) le 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(9)



119891 (119910 + 120582120578 (119909 119910))

lt 119866minus1


(10)


119891 (119910 + 120582120578 (119909 119910))

le 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910))) = 119891 (119909)

(11)


119891 (119910 + 120572120578 (119909 119910)) gt 119891 (119909) (12)



119891 (119911120573) = 119891 (119910 + 120573120578 (119909 119910)) gt 119891 (119909) = 119891 (119910) (13)

From Condition C

119911120573= 119911120572+120573 minus 120572

1 minus 120572120578 (119909 119911

120572) (14)


119891 (119911120573)

= 119891(119911120572+120573 minus 120572

1 minus 120572120578 (119909 119911

120572))

lt 119866minus1

[120573 minus 120572

1 minus 120572119866 (119891 (119909)) + (1 minus

120573 minus 120572

1 minus 120572)119866 (119891 (119911

120572))]

lt 119866minus1

[120573 minus 120572

1 minus 120572119866 (119891 (119911

120572)) + (1 minus

120573 minus 120572

1 minus 120572)119866 (119891 (119911

120572))]

= 119891 (119911120572)

(15)


119911120572= 119911120573+ (1 minus

120572

120573) 120578 (119910 119911

120573) (16)


119891 (119911120572)

= 119891(119911120573+ (1 minus

120572

120573) 120578 (119910 119911

120573))

lt 119866minus1

[(1 minus120572

120573)119866 (119891 (119910)) +

120572

120573119866 (119891 (119911

120573))]

lt 119866minus1

[(1 minus120572

120573)119866 (119891 (119911

120573)) +

120572

120573119866 (119891 (119911

120573))]

= 119891 (119911120573)

(17)






119891 (119910 + 120572120578 (119909 119910)) lt 119866minus1

(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))

(18)



119891 (119910 + 120582120578 (119909 119910)) le 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(19)


119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910)

= 119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910 + 120572120578 (119909 119910) minus 120572120578 (119909 119910))

= 119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910 + 120572120578 (119909 119910)

+120578 (119910 119910 + 120572120578 (119909 119910)))

= 119910 minus120582

120572120578 (119910 119910 + 120572120578 (119909 119910))

= 119910 + 120582120578 (119909 119910)

(20)



119891 (119910 + 120582120578 (119909 119910))

= 119891 [119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910)]

le 119866minus1

[120582

120572119866 ( 119891 (119910 + 120572120578 (119909 119910))

+ (1 minus120582

120572)119866 (119891 (119910))]

lt 119866minus1

[120582

120572119866 (119866minus1

(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910))))

+(1 minus120582

120572)119866 (119891 (119910))]

= 119866minus1

[120582

120572(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))

+ (1 minus120582

120572)119866 (119891 (119910))]

= 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(21)

(ii) Let 120572 lt 120582 that is

0 lt1 minus 120582

1 minus 120572lt 1 (22)


119910 + 120572120578 (119909 119910) + (1 minus1 minus 120582

1 minus 120572) 120578 (119909 119910 + 120572120578 (119909 119910))

= 119910 + [120572 + (1 minus1 minus 120582

1 minus 120572) (1 minus 120572)] 120578 (119909 119910)

= 119910 + 120582120578 (119909 119910)

(23)


119891 (119910 + 120582120578 (119909 119910))

= 119891 [119910 + 120572120578 (119909 119910) + (1 minus1 minus 120582

1 minus 120572) 120578 (119909 119910 + 120572120578 (119909 119910))]

le 119866minus1


1 minus 120572)119866 (119891 (119909))

+1 minus 120582

1 minus 120572119866 (119891 (119910 + 120572120578 (119909 119910))) ]

lt 119866minus1


1 minus 120572)119866 (119891 (119909))

+1 minus 120582

1 minus 120572119866 (119866minus1

(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))) ]

= 119866minus1


1 minus 120572)119866 (119891 (119909))

+1 minus 120582

1 minus 120572(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))]

= 119866minus1

( 120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(24)




Example 14 Let

119891 (119909) = ln(32|119909| +

7

4)

120578 (119909 119910) =


(25)



119891 (119910 + 120582120578 (119909 119910))

= 119891(4 minus7

2120582) = ln(3

2

10038161003816100381610038161003816100381610038164 minus

7

2120582

1003816100381610038161003816100381610038161003816+7

4)

= ln(314minus21

4120582)

= 119866minus1

(3

2(120582 |119909| + (1 minus 120582)

10038161003816100381610038161199101003816100381610038161003816) +

7

4)

= 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(26)



1-preinvex func-



2119866minus1

1is convex on







the same 120578 on 119870



119866 (120582119910 + (1 minus 120582) 119911) ge 120582119866 (119910) + (1 minus 120582)119866 (119911) (27)


119891(119883) that is

119866minus1


119866(120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906))

ge 120582119866 (119866minus1

(119909)) + (1 minus 120582)119866 (119866minus1

(119906))

= 120582119909 + (1 minus 120582) 119906

(28)


119866minus1

119866(120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906))

ge 119866minus1

(120582119866 (119866minus1

(119909)) + (1 minus 120582)119866 (119866minus1

(119906)))

= 119866minus1

(120582119909 + (1 minus 120582) 119906)

(29)

Thus

120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906) ge 119866minus1

(120582119909 + (1 minus 120582) 119906)

(30)


2(119905) = 119905 then

119892(119905) = 1198662119866minus1












min119891 (119909) (P)

119892119894(119909) le 0 119894 isin 119869 = 1 119898 (31)



119863 = 119909 isin 119870 119892119894(119909) le 0 119894 isin 119869 (32)



119891 (119909lowast

) gt 119891 (119909) (33)



119911 = 119909lowast

+ 120582120578 (119910 119909lowast

) (34)


119891 (119911) = 119891 (119909lowast

+ 120582120578 (119910 119909lowast

))

gt 119866minus1

(120582119866 (119891 (119910)) + (1 minus 120582)119866 (119891 (119909lowast

)))

gt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119909))) = 119891 (119909)

(35)




119894 119894 isin 119869 are 119866-



120598(119909) around 119909 such that



119891 (119909lowast

) lt 119891 (119909) (37)



119892119894(119909 + 120582120578 (119909

lowast

119909)) le 119866minus1

119894(120582119866119894(119892119894(119909lowast

)) + (1 minus 120582)119866119894(119892119894(119909)))

le 119866minus1

119894(120582119866119894(0) + (1 minus 120582)119866

119894(0)) = 0

(38)



119909 + 120582120578 (119909lowast

119909) isin 119863 (39)


119891 (119909 + 120582120578 (119909lowast

119909))

lt 119866minus1

(120582119866 (119891 (119909lowast


(40)


119891 (119909 + 120582120578 (119909lowast

119909))

lt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119909)))

= 119891 (119909) forall120582 isin (0 1)

(41)


119909 + 120582120578 (119909lowast

119909) isin 119870 cap 119873120598(119909) (42)





(43)


120578 (119909 119910)=

3

4minus 119910

if 119909 gt 1 0 le 119910 lt 1



1003816100381610038161003816 ge 1




minus5 minus1

2119910 if |119909| lt 1 minus1 le 119910 le 0

minus4119909 +3

2119910 if 119909 = 1 0 lt 119910 lt 1

3119909 +3






(44)


119894 119894 isin


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891 (119910 + 120582120578 (119909 119910))

= 119891 [119910 +120582

120572120578 (119910 + 120572120578 (119909 119910) 119910)]

le 119866minus1

[120582

120572119866 ( 119891 (119910 + 120572120578 (119909 119910))

+ (1 minus120582

120572)119866 (119891 (119910))]

lt 119866minus1

[120582

120572119866 (119866minus1

(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910))))

+(1 minus120582

120572)119866 (119891 (119910))]

= 119866minus1

[120582

120572(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))

+ (1 minus120582

120572)119866 (119891 (119910))]

= 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(21)

(ii) Let 120572 lt 120582 that is

0 lt1 minus 120582

1 minus 120572lt 1 (22)


119910 + 120572120578 (119909 119910) + (1 minus1 minus 120582

1 minus 120572) 120578 (119909 119910 + 120572120578 (119909 119910))

= 119910 + [120572 + (1 minus1 minus 120582

1 minus 120572) (1 minus 120572)] 120578 (119909 119910)

= 119910 + 120582120578 (119909 119910)

(23)


119891 (119910 + 120582120578 (119909 119910))

= 119891 [119910 + 120572120578 (119909 119910) + (1 minus1 minus 120582

1 minus 120572) 120578 (119909 119910 + 120572120578 (119909 119910))]

le 119866minus1


1 minus 120572)119866 (119891 (119909))

+1 minus 120582

1 minus 120572119866 (119891 (119910 + 120572120578 (119909 119910))) ]

lt 119866minus1


1 minus 120572)119866 (119891 (119909))

+1 minus 120582

1 minus 120572119866 (119866minus1

(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))) ]

= 119866minus1


1 minus 120572)119866 (119891 (119909))

+1 minus 120582

1 minus 120572(120572119866 (119891 (119909)) + (1 minus 120572)119866 (119891 (119910)))]

= 119866minus1

( 120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(24)




Example 14 Let

119891 (119909) = ln(32|119909| +

7

4)

120578 (119909 119910) =


(25)



119891 (119910 + 120582120578 (119909 119910))

= 119891(4 minus7

2120582) = ln(3

2

10038161003816100381610038161003816100381610038164 minus

7

2120582

1003816100381610038161003816100381610038161003816+7

4)

= ln(314minus21

4120582)

= 119866minus1

(3

2(120582 |119909| + (1 minus 120582)

10038161003816100381610038161199101003816100381610038161003816) +

7

4)

= 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119910)))

(26)



1-preinvex func-



2119866minus1

1is convex on







the same 120578 on 119870



119866 (120582119910 + (1 minus 120582) 119911) ge 120582119866 (119910) + (1 minus 120582)119866 (119911) (27)


119891(119883) that is

119866minus1


119866(120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906))

ge 120582119866 (119866minus1

(119909)) + (1 minus 120582)119866 (119866minus1

(119906))

= 120582119909 + (1 minus 120582) 119906

(28)


119866minus1

119866(120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906))

ge 119866minus1

(120582119866 (119866minus1

(119909)) + (1 minus 120582)119866 (119866minus1

(119906)))

= 119866minus1

(120582119909 + (1 minus 120582) 119906)

(29)

Thus

120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906) ge 119866minus1

(120582119909 + (1 minus 120582) 119906)

(30)


2(119905) = 119905 then

119892(119905) = 1198662119866minus1












min119891 (119909) (P)

119892119894(119909) le 0 119894 isin 119869 = 1 119898 (31)



119863 = 119909 isin 119870 119892119894(119909) le 0 119894 isin 119869 (32)



119891 (119909lowast

) gt 119891 (119909) (33)



119911 = 119909lowast

+ 120582120578 (119910 119909lowast

) (34)


119891 (119911) = 119891 (119909lowast

+ 120582120578 (119910 119909lowast

))

gt 119866minus1

(120582119866 (119891 (119910)) + (1 minus 120582)119866 (119891 (119909lowast

)))

gt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119909))) = 119891 (119909)

(35)




119894 119894 isin 119869 are 119866-



120598(119909) around 119909 such that



119891 (119909lowast

) lt 119891 (119909) (37)



119892119894(119909 + 120582120578 (119909

lowast

119909)) le 119866minus1

119894(120582119866119894(119892119894(119909lowast

)) + (1 minus 120582)119866119894(119892119894(119909)))

le 119866minus1

119894(120582119866119894(0) + (1 minus 120582)119866

119894(0)) = 0

(38)



119909 + 120582120578 (119909lowast

119909) isin 119863 (39)


119891 (119909 + 120582120578 (119909lowast

119909))

lt 119866minus1

(120582119866 (119891 (119909lowast


(40)


119891 (119909 + 120582120578 (119909lowast

119909))

lt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119909)))

= 119891 (119909) forall120582 isin (0 1)

(41)


119909 + 120582120578 (119909lowast

119909) isin 119870 cap 119873120598(119909) (42)





(43)


120578 (119909 119910)=

3

4minus 119910

if 119909 gt 1 0 le 119910 lt 1



1003816100381610038161003816 ge 1




minus5 minus1

2119910 if |119909| lt 1 minus1 le 119910 le 0

minus4119909 +3

2119910 if 119909 = 1 0 lt 119910 lt 1

3119909 +3






(44)


119894 119894 isin


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











the same 120578 on 119870



119866 (120582119910 + (1 minus 120582) 119911) ge 120582119866 (119910) + (1 minus 120582)119866 (119911) (27)


119891(119883) that is

119866minus1


119866(120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906))

ge 120582119866 (119866minus1

(119909)) + (1 minus 120582)119866 (119866minus1

(119906))

= 120582119909 + (1 minus 120582) 119906

(28)


119866minus1

119866(120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906))

ge 119866minus1

(120582119866 (119866minus1

(119909)) + (1 minus 120582)119866 (119866minus1

(119906)))

= 119866minus1

(120582119909 + (1 minus 120582) 119906)

(29)

Thus

120582119866minus1

(119909) + (1 minus 120582)119866minus1

(119906) ge 119866minus1

(120582119909 + (1 minus 120582) 119906)

(30)


2(119905) = 119905 then

119892(119905) = 1198662119866minus1












min119891 (119909) (P)

119892119894(119909) le 0 119894 isin 119869 = 1 119898 (31)



119863 = 119909 isin 119870 119892119894(119909) le 0 119894 isin 119869 (32)



119891 (119909lowast

) gt 119891 (119909) (33)



119911 = 119909lowast

+ 120582120578 (119910 119909lowast

) (34)


119891 (119911) = 119891 (119909lowast

+ 120582120578 (119910 119909lowast

))

gt 119866minus1

(120582119866 (119891 (119910)) + (1 minus 120582)119866 (119891 (119909lowast

)))

gt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119909))) = 119891 (119909)

(35)




119894 119894 isin 119869 are 119866-



120598(119909) around 119909 such that



119891 (119909lowast

) lt 119891 (119909) (37)



119892119894(119909 + 120582120578 (119909

lowast

119909)) le 119866minus1

119894(120582119866119894(119892119894(119909lowast

)) + (1 minus 120582)119866119894(119892119894(119909)))

le 119866minus1

119894(120582119866119894(0) + (1 minus 120582)119866

119894(0)) = 0

(38)



119909 + 120582120578 (119909lowast

119909) isin 119863 (39)


119891 (119909 + 120582120578 (119909lowast

119909))

lt 119866minus1

(120582119866 (119891 (119909lowast


(40)


119891 (119909 + 120582120578 (119909lowast

119909))

lt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119909)))

= 119891 (119909) forall120582 isin (0 1)

(41)


119909 + 120582120578 (119909lowast

119909) isin 119870 cap 119873120598(119909) (42)





(43)


120578 (119909 119910)=

3

4minus 119910

if 119909 gt 1 0 le 119910 lt 1



1003816100381610038161003816 ge 1




minus5 minus1

2119910 if |119909| lt 1 minus1 le 119910 le 0

minus4119909 +3

2119910 if 119909 = 1 0 lt 119910 lt 1

3119909 +3






(44)


119894 119894 isin


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909 + 120582120578 (119909lowast

119909) isin 119863 (39)


119891 (119909 + 120582120578 (119909lowast

119909))

lt 119866minus1

(120582119866 (119891 (119909lowast


(40)


119891 (119909 + 120582120578 (119909lowast

119909))

lt 119866minus1

(120582119866 (119891 (119909)) + (1 minus 120582)119866 (119891 (119909)))

= 119891 (119909) forall120582 isin (0 1)

(41)


119909 + 120582120578 (119909lowast

119909) isin 119870 cap 119873120598(119909) (42)





(43)


120578 (119909 119910)=

3

4minus 119910

if 119909 gt 1 0 le 119910 lt 1



1003816100381610038161003816 ge 1




minus5 minus1

2119910 if |119909| lt 1 minus1 le 119910 le 0

minus4119909 +3

2119910 if 119909 = 1 0 lt 119910 lt 1

3119909 +3






(44)


119894 119894 isin


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Semistrict -Preinvexity and Optimality in Nonlinear Programming · 2019. 7....

Documents

Transcript of Research Article Semistrict -Preinvexity and Optimality in Nonlinear Programming · 2019. 7....