Weak ψ-Sharp Minima in

Volume 2010, Article ID 154598,10pages doi:10.1155/2010/154598

Research Article

Weak ψ-Sharp Minima in

Vector Optimization Problems

S. Xu and S. J. Li

College of Mathematics and Statistics, Chongqing University, Chongqing 400030, China

Correspondence should be addressed to S. Xu,[email protected] Received 23 April 2010; Revised 15 July 2010; Accepted 13 August 2010 Academic Editor: N. J. Huang

Copyrightq2010 S. Xu and S. J. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present a sufficient and necessary condition for weakψ-sharp minima in infinite-dimensional spaces. Moreover, we develop the characterization of weak ψ-sharp minima by virtue of a nonlinear scalarization function.

1. Introduction

The notion of a weak sharp minimum in general mathematical program problems was first introduced by Ferris in1. It is an extension of sharp minimum in2. Weak sharp minima play important roles in the sensitivity analysis 3,4 and convergence analysis of a wide range of optimization algorithms5. Recently, the study of weak sharp solution set covers real-valued optimization problems5–8 and piecewise linear multiobjective optimization problems9–11.

Most recently, Bednarczuk12 defined weak sharp minima of order m for vector- valued mappings under an assumption that the order cone is closed, convex, and pointed and used the concept to prove upper H ¨olderness and H ¨older calmness of the solution set-valued mappings for a parametric vector optimization problem. In 13, Bednarczuk discussed the weak sharp solution set to vector optimization problems and presented some properties in terms of well-posedness of vector optimization problems. In14, Studniarski gave the definition of weakψ-sharp local Pareto minimum in vector optimization problems under the assumption that the order cone is convex and presented necessary and sufficient conditions under a variety of conditions. Though the notions in 12, 14 are different for vector optimization problems, they are equivalent for scalar optimization problems. They are a generalization of the weak sharp local minimum of orderm.

In this paper, motivated by the work in14,15, we present a sufficient and necessary condition of which a point is a weakψ-sharp minimum for a vector-valued mapping in the

infinite-dimensional spaces. In addition, we develop the characterization of weakψ-sharp minima in terms of a nonlinear scalarization function.

This paper is organized as follows. In Section 2, we recall the definitions of the local Pareto minimizer and weak ψ-sharp local minimizer for vector-valued optimization problems. InSection 3, we present a sufficient and necessary condition for weakψ-sharp local minimizer of vector-valued optimization problems. We also give an example to illustrate the optimality condition.

2. Preliminary Results

Throughout the paper,XandY are normed spaces.Bx, δdenotes the open ball with center x∈ Xand radiusδ > 0.Nxis the family of all neighborhoods ofx, and distx, Wis the distance from a pointxto a setW ⊂ X. The symbolsS^c, intSandbdsdenote, respectively, the complement, interior and boundary ofS.

LetD ⊂Y be a convex conecontaining 0. The cone defines an order structure onY, that is, a relation “≤” inY ×Y is defined by y1 ≤ y2 ⇔ y2−y1 ∈ D.Dis a proper cone if {0}/D /Y.

LetΩbe an open subset of X,S ⊂ Ω. Given a vector-valued map f : Ω → Y, the following abstract optimization is considered:

Min

fx:x∈S

. 2.1

In the sequel, we always assume thatDis a proper closed and convex cone.

Definition 2.1. One says that x0 is a local Pareto minimizer for 2.1, denoted by x0 ∈ LMinf, S, if there existsU∈ Nxfor which there is nox∈S∩Usuch that

fx−fx0∈−D\D. 2.2

If one can chooseU X, one will say thatx0 is a Pareto minimizer for 2.1, denoted by x0∈Minf, S.

Note that2.2may be replaced by the simple conditionfx−fx0∈−D\ {0}if we assume that the coneDis pointed.

Definition 2.2see14. Letψ : 0,∞ → 0,∞be a nondecreasing function with the propertyψt 0⇔t0such a family of functions is denoted byΨ. Letx0 ∈S. One says thatx0 is a weakψ-sharp local Pareto minimizer for2.1, denoted byx0 ∈ WSLψ, f, S, if there exist a constantα >0 andU∈ Nx0such that

fx D

∩B

fx0, αψdistx, W

∅, ∀x∈S∩U\W, 2.3

where

W :

x∈S:fx fx0

. 2.4

If one can chooseU X, one says x0 is a weakψ-sharp minimizer for2.1, denoted by x0∈WSψ, f, S. In particular, letψmt:t^mform1,2, . . . .Then, one says thatx0is a weak ψ-sharp local Pareto minimizer of ordermfor2.1ifx0 ∈WSLψm, f, S, and one says that x0is a weak sharp Pareto minimizer of ordermfor2.1ifx0∈WSψm, f, S.

Remark 2.3. IfWis a closed set, condition2.3can be expressed as the following equivalent forms:

fx∈

fx0 B

0, αψdistx, W

−Dc

, ∀x∈S∩U\W, 2.5 d

fx−fx0,−D

≥αψdistx, W, ∀x∈S∩U\W. 2.6 Remark 2.4. In theDefinition 2.2, ifY R,D 0,∞, andψ ψm, then the relation2.6 becomes the following form:

fx−fx0≥αdistx, W^m, ∀x∈S∩U, 2.7 which is the well-known definition of a weak sharp minimizer of ordermfor2.1; see16.

3. Main Results

In this section, we first generalize the result of Theorem 1 in Studniarski 14 to infinite- dimensional spaces. Finally, we develop the characterization of weakψ-sharp minimizer by means of a nonlinear scalarization function.

LetD ⊂Y be a proper closed convex cone with intD /∅. The topological dual space ofY is denoted byY^∗. The polar cone toDisD^∗ {λ ∈Y^∗ :λ, y ≥0, ∀y ∈D}. It is well known that the coneD^∗contains aw^∗-compact convex setΛwith 0/∈Λsuch that

D^∗coneΛ {rλ:r≥0, λ∈Λ}. 3.1

The setΛis called a base for the dual coneD^∗. Recall that a pointλis an extremal point of a setΛif there exist no different pointsλ1, λ2∈Λandt∈0,1such thatλtλ1 1−tλ2. Theorem 3.1. Suppose thatf :X → Yis a vector-valued map. LetD⊂Y be a proper closed convex cone with intD /∅,x0 ∈S, andψ ∈Ψ.

iLetΛbe aw^∗-compact convex base ofD^∗ andQthe set of extremal points ofΛ. Suppose thatWdefined by2.4is a closed set. Then,x0 ∈WSLψ, f, Sif and only if there exist U∈ Nx, a constantα >0, a covering{Sλ:λ∈Q}ofS∩U, and

λ, fx

>

λ, fx0

αψdistx, W, ∀x∈Sλ∩U\W, ∀λ∈Q. 3.2

iiLetQ⊂D^∗\ {0}and assume thatD^∗cl cone coQ. Thenx0 ∈LMinf, Sif and only if there exists a covering{Sλ:λ∈Q}ofS∩Usuch that

λ, fx

>

λ, fx0

, ∀x∈Sλ∩U\W, ∀λ∈Q. 3.3

Proof. iPart “only if”: by assumption, there existβ >0 andU∈ Nx0such that fx−fx0 D

∩B

0, βψdistx, W

∅, ∀x∈S∩U\W. 3.4 Lete ∈ intD be a fixed point. Setβ0 inf_λ∈Λλ, e. SinceΛisw^∗-compact, the infimum is attained at a point ofQ. Namely,β0 min_λ∈Qλ, e. Clearly,λ, e>0 for anyλ∈Λ. Hence, β0>0.

For eachλ∈Q, we define

Sλ

x∈S∩U:

λ, fx

≥

λ, fx0 β

2eψdistx, Wβ0 . 3.5

We will show that

S∩U⊂

λ∈Q

Sλ. 3.6

Letx ∈ S∩U. Ifx ∈ W, thenfx fx0by2.4, hence,x ∈Sλfor allλ ∈ Q. Ifx /∈W, suppose thatx /∈Sλfor anyλ∈Q, then

λ, fx

<

λ, fx0 β

2eψdistx, Wβ0, ∀λ∈Q. 3.7

This relation, together with statementλ, e ≥β0yields

λ, fx0−fx β

2eψdistx, We

>0, ∀λ∈Q. 3.8

Obviously, for anyλ∈D^∗, the above relation becomes the following form:

λ, fx0−fx β

2eψdistx, We

≥0. 3.9

Consequently, by the bipolar theorem, one has

d:fx0−fx β

2eψdistx, We∈D. 3.10

Therefore,

fx−fx0 d β

2eψdistx, We, 3.11

andfx−fx0 d∈B0, βψdistx, W, which is a contradiction to3.4. We have thus proved thatSλcoversS∩U.

Now, letx∈Sλ∩U\Wandλ ∈Q. From the procedure of the above proof, we see thatS∩U\W⊂ ∪_λ∈QSλ. Hence, by3.5, setαββ0/4e, inequality3.2is true.

Part “if”: we defineβ1sup_λ∈Λλ, e. The supremum is attained at an extremal point because of thew^∗-compactness ofΛ. Soβ1max_λ∈Qλ, e>0 andβ⁻¹₁ λ, e ≤1 for anyλ∈Q.

Hence, by assumption, we have λ, fx>

λ, fx0

αψdistx, W≥

λ, fx0

β₁⁻¹αψdistx, Wλ, e, 3.12 forx∈Sλ∩U\Wandλ∈Q.

Now, suppose that for allβ > 0,3.4is false, then there existx ∈ S∩U\W and d∈Dsuch that

f x

−fx0 d∈B

0, βψdistx, W

. 3.13

Lete∈intDbe a fixed point, and sinceDis a cone, there isk >0 such thatB0,1⊂ke−D.

Consequently,

B

0, βψdistx, W

⊂kβψdistx, We−D. 3.14

Therefore,

f x

−fx0 d∈kβψdistx, We−D. 3.15

There isd∈Dfrom3.15such that f

x

−fx0 kβψdistx, We− dd

. 3.16

Sincex ∈S∩U\W ⊂

λ∈QSλ\W, there isλ ∈Qsuch thatx ∈Sλ. Moreover,Λ ⊂D^∗ anddd∈D. Hence,

λ, f x

−

λ, fx0

kβψ dist

x, W λ, e

−

λ, dd

≤kβψ dist

x, W λ, e

. 3.17 By choosingββ⁻¹₁ αk⁻¹, we obtain a contradiction to3.12.

iiPart “only if”: for eachλ∈Q, we define,

Sλ

x∈S∩U:

λ, fx

≥

λ, fx0

. 3.18

Now, we will check that3.6holds true. Pick anyx ∈ S∩U. Suppose thatx /∈Sλ for any λ∈Q, then

λ, fx−fx0

<0, ∀λ∈Q. 3.19

Hence, for anyλ∈cl cone coQD^∗,λ, fx−fx0 ≤0. By applying the bipolar theorem, we have

fx−fx0∈ −D, 3.20

Combing it with the assumption, we have

fx−fx0∈−D∩D, 3.21

which is a contradiction to3.19. So3.6holds and3.3is satisfied by the definition ofSλ. Part “if”: suppose thatx0/∈LMinf, S, then there existsx∈S∩Usuch that

fx−fx0∈ −D\D. 3.22

Indeed,x∈S∩Ucan be replace byx∈S∩U\W, becausex∈W,fx−fx0 0, which is contradiction to3.22. Hence, forx∈S∩U\W, we haveλ, fx−fx0 ≤0, ∀λ∈D^∗. In particular,

λ, fx−fx0 ≤0, ∀λ∈Q. 3.23

It follows from the assumption that

∪λ∈QSλ∩U

\W⊃S∩U\W. 3.24

Therefore, by3.3, we obtain

λ, fx−fx0

>0, ∀λ∈Q, ∀x∈Sλ∩U\W, 3.25

which contradicts relation3.23.

Remark 3.2. By takingUXin parti resp.,iiofTheorem 3.1, we obtain a necessary and sufficient condition forx0 to be in WSψ, f, S resp., Minf, S. In particular, if we choose Y R^pandDR^pandQ{λ1, λ2, . . . , λp}, then, we obtain Theorem 1 in14.

Finally, we apply the nonlinear scalarization function to discuss the weak ψ-sharp minimizer in vector optimization problems.

LetD ⊂ Y be a closed and convex cone with nonempty interior intD. Given a fixed pointe∈intDandy∈Y, the nonlinear scalarization functionξ:Y → Ris defined by

ξ y

inf

t:te∈yD

. 3.26

This function plays an important role in the context of nonconvex vector optimization problems and has excellent properties such as continuousness, convexity, and strict monotonicity onY. More results about the function can be found in17.

In what follows, we present several properties about the nonlinear scalarization function.

Lemma 3.3see17. For any fixede∈intD,y∈Y, andr∈R. One has iξy< r ⇔re∈yintD,

iiξy> r ⇔re/∈yD.

iiiξy r ⇔re∈ybdD.

Given a vector-valued mapf :X → Y, definef:X → Y by

fx fx−fx0. 3.27

Next, we consider weakψ-sharp local minimizer for a vector-valued mapfthrough a weak sharp local minimizer of a scalar functionξ◦f:X → R.

Theorem 3.4. Letx0∈S⊂X. Suppose thatWdefined by2.4is a closed set. Then, x0∈WSL

ψ, f, S

⇐⇒x0∈WSL

ψ, ξ◦f, S

. 3.28

Proof. Part “only if”: let us assume thatx0 ∈ WSLψ, f, S. Thus, there existα >0 andU ∈ Nx0such that

fx−fx0 D

∩B

0, αψdistx, W

∅, ∀x∈S∩U\W. 3.29

Note that, whenWis a closed set, α

4eψdistx, We∈B

0, αψdistx, W

∀x∈S∩U\W. 3.30

Therefore,

α

4eψdistx, We /∈fx−fx0 D ∀x∈S∩U\W. 3.31

By usingLemma 3.3ii, one has

ξ

fx−fx0

> α

4eψdistx, W ∀x∈S∩U\W. 3.32

According toLemma 3.3iii, one has

ξ

fx0−fx0

0. 3.33

This relation, together with3.32yields

ξ

fx−fx0

> ξ

fx0−fx0 α

4eψdistx, W, ∀x∈S∩U\W. 3.34

Namely,

ξ◦f x>

ξ◦f

x0 α

4eψdistx, W, ∀x∈S∩U\W, 3.35

that is,x0∈WSLψ, ξ◦f, S.

Part “if”: by assumption, there existβ >0 andU∈ Nx0such that

ξ fx

> ξ fx 0

βψdistx, W, ∀x∈S∩U\W. 3.36

In terms ofLemma 3.3iii, we have

ξ fx 0

ξ

fx0−fx0

0. 3.37

Hence,

ξ

fx−fx0

> βψdistx, W, ∀x∈S∩U\W. 3.38

Once more usingLemma 3.3ii, one has

βψdistx, We /∈fx−fx0 D, ∀x∈S∩U\W, 3.39

which implies that

βψdistx, We−D

∩

fx−fx0 D

∅, ∀x∈S∩U\W. 3.40

Sincee∈intD, there exists some number >0 such thatB0, ⊂e−D. Moreover,

B0, λ⊂λe−D, ∀λ >0. 3.41

Hence, it follows from the relation that B

0, βψdistx, W

⊂βψdistx, We−D, ∀x∈S∩U\W. 3.42

Combing it with relation3.40, we deduce that B

0, βψdistx, W

∩

fx−fx0 D

∅, ∀x∈S∩U\W. 3.43

Letαβ, by the definition of weakψ-sharp local minimizer, we havex0∈WSLψ, f, S.

It is possible to illustrateTheorem 3.4by means of adapting a simple example given in 14.

Example 3.5. Letnp2,S Ω R², andDR²and letf f1, f2:R² → R²be defined by

f1

x¹, x²

:max 0,min

x¹, x²

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

x¹, ifx²≥x¹>0, x², ifx¹> x²>0, 0, ifx¹≤0 orx²≤0,

f2

x¹, x²

:max 0,min

−x¹, x²

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−x¹, ifx²≥ −x¹ >0, x², if −x¹> x²>0, 0, ifx¹≥0 orx²≤0,

3.44

We chooseUR². UsingDefinition 2.2, we derive thatx0 0,0∈WSψ1, f, S.

Lete 1,1. From Corollary 1.46 in17, we haveξ◦fx max_1≤i≤2fix. Observe that

W

x:fx 0,0

x:x²≤0

∪

x:x¹0

. 3.45

It is easy to verify thatfix distx, Wfor allx∈S\W. Using relation2.7, we show that x0 0,0∈WSψ1, ξ◦f, S. Hence, condition 3.28withψψ1holds forα∈0,1.

Acknowledgments

This paper was partially supported by the National Natural Science Foundation of China Grant no. 10871216and Chongqing University Postgraduates Science and Innovation Fund Project no. 201005B1A0010338. The authors would like to thank the anonymous referees for their valuable comments and suggestions, which helped to improve the paper, and are grateful to Professor M. Studniarski for providing the paper14.

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