# Weak Sharp Minima in Set-Valued Optimization Problems

Authors: Ming-hao Jin; Jun-xiang Wang; Shu Xu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce the notion of a weakψ-sharp minimizer for set-valued optimization problems. We present some sufficient and necessary conditions that a pair point is a weak ψ-sharp minimizer through the outer limit of set-valued map and develop the characterization of the weak ψ-sharp minimizer in terms of a generalized nonlinear scalarization function. These results extend the corresponding ones in Studniarski (2007).


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## Body

## 1. Introduction

The notion of weak sharp minima in general mathematical program problems was first introduced by Ferris in [1]. It is a generalization of a sharp minimum in [2] to include the possibility of nonunique solution set. The study of weak sharp minima is motivated primarily by applications in convex and convex composite programming, where such minima commonly occur. Weak sharp minima plays an important role in the sensitivity analysis [3, 4] and convergence analysis of a wide range of optimization algorithms [5]. Recently, the study of weak sharp solution set covers real-valued optimization problems [5–8] and piecewise linear multiobjective optimization problems [9, 10].In [11], Bednarczuk defined weak sharp Pareto minima of order m for vector-valued mappings and used weak sharp Pareto minima to prove upper Hölderness and Hölder calmness of the solution set-valued mappings for parametric vector optimization problems. In [12], Studniarski gave the definition of weak ψ-sharp local Pareto minima in multiobjective optimization problems and presented necessary and sufficient conditions. In [13], Xu and Li established a sufficient and necessary condition for weak ψ-sharp local Pareto minima in vector optimization problems in infinite spaces, the approach is that they transformed weak ψ-sharp local Pareto minima of a vector-valued function to weak ψ-sharp local minima of a family of scalar functions. Most recently, Durea and Strugariu [14] introduced the definition of weak ψ-sharp local minima by an oriented distance function in set-valued optimization problems and established necessary optimality conditions in terms of Mordukhovich coderivative.In the paper, motivated by the work in [15, 16], we also introduce the notion of weak ψ-sharp minima, which is different from one in [14], and establish some sufficient and necessary conditions through the outer limit of set-valued map. In particular, we develop the characterization of the weak ψ-sharp minimizer in terms of the generalized nonlinear scalarization function.This paper is organized as follows. In Section2, we recall some basic definitions and give the notion of the weak ψ-sharp local minimizer for set-valued optimization problems. In Section 3, we present some sufficient and necessary conditions through the outer limit of the set-valued map. In Section 4, we establish a characterization of weak ψ-sharp local minima in terms of the generalized nonlinear scalarization function.


## 2. Preliminary Results

Throughout this paper, letX,  Y be real normed spaces. B(x,δ) denotes the open ball with center x∈X and radius δ>0, 𝒩(x) is the family of all neighborhoods of x, and dist(x,W) is the distance from the point x to the set W⊂X. The symbols Sc, clS, and intS denote, respectively, the complement, closure, and interior of S. Let D⊂Y be a convex cone (containing 0) with nonempty interior intD and let Y be partially ordered by D.LetF:X→2Y be a set-valued map. We denote the graph and domain of F, respectively, by
(2.1)GrF={(x,y)∈X×Y:y∈F(x)},DomF={x∈X:F(x)≠∅}.
If S is a subset of X, then F(S)=∪x∈SF(x) and the inverse set-valued map of F is F-1:Y→2X given by (y,x)∈GrF-1 if and only if (x,y)∈GrF.Definition 2.1.
Suppose thatD is a closed convex pointed ((-D)∩D={0}) cone. A point y0∈A⊂Y is called a strict efficient (resp., weak) point of A, denoted by y0∈StrDA (resp., y0∈WMinDA) if
(2.2)(A-y0)∩(-D∖{0})=∅(resp.(A-y0)∩(-intD)=∅).Given a set-valued mapF:X→2Y and a subset S of X, the following abstract optimization is considered:
(2.3)minF(x),s.t.x∈S.Definition 2.2.
Suppose thatD is a closed convex pointed cone. A point (x0,y0)∈GrF, with x0∈S, is said to be a local strict (resp., weak) minimizer of F over S, written as (x0,y0)∈LStrD(F,S)  (resp.,(x0,y0)∈LWMinD(F,S)), if there exists a neighborhood U of x0 in X such that
(2.4)y0∈StrD(F(U∩S))(resp.,y0∈WminD(F(U∩S))),thatis,∀x∈S∩U,(F(x)-y0)∩(-D∖{0})=∅(resp.∀x∈S∩U,(F(x)-y0)∩(-intD)=∅).We will say that(x0,y0) is a global strict (global weak) minimizers when U=X. The set of all global strict minimizers (resp., weak minimizers) is denoted by StrD(F,S)  (resp.,WMinD(F,S)).Definition 2.3.
Letψ:[0,+∞)→[0,+∞) be a nondecreasing function with the property ψ(t)=0⇔t=0 (such a family of functions is denoted by Ψ) and x0∈S. We say that a point pair (x0,y0)∈GrF∩(S×Y) is a weak ψ-sharp local Pareto minimizer for (2.3), denoted by (x0,y0)∈WSL(ψ,F,S), if there exists a constant α>0 and U∈𝒩(x0) such that
(2.5)(F(x)+D)∩B(y0,αψ(dist(x,W)))=∅,∀x∈S∩U∖W,
where
(2.6)W={x∈S:y0∈F(x)}=S∩F-1(y0).
If we choose U=X, we will say the point pair (x0,y0)∈GrF∩(S×Y) is a weak ψ-sharp minimizer for (2.3), denoted by (x0,y0)∈WS(ψ,F,S). In particular, let ψm(t)=tm for m=1,2,…. Then, we say (x0,y0)∈GrF∩(S×Y) is a weak ψ-sharp local minimizer of order m for (2.3) if (x0,y0)∈WSL(ψm,F,S).Obviously, condition (2.5) can be expressed in the following equivalent form:
(2.7)F(x)⊂(B(y0,αψ(dist(x,W)))-D)c,∀x∈S∩U∖W.Remark 2.4.
Clearly, if the mapF is a vector-valued function, the notion is equivalent to Definition 8.2.3 with ψ=ψm in [17] and the weak ψ-sharp local minimizer for vector optimizations in [12].Remark 2.5.
In [14], the definition of weak ψ-sharp local minimizer for set-valued optimization is given by the oriented distance function ▵. However, we establish the definition by the map F. When the map F is the real-valued function and the cone D=R+, our definition is equivalent to Definition  2.1 in [14].


## 3. Optimality Conditions for Weakψ-Sharp Minimizer for Set-Valued Optimization

In this section, we present sufficient and necessary conditions that a point pair is a weakψ-sharp local minimizer in set-valued optimization problems.Theorem 3.1.
LetF:X→2Y, x0∈S, and ψ∈Ψ. Assume that W defined in (2.6) is a closed set. Then, (x0,y0)∈WSL(ψ,F,S) if and only if
(3.1)0∉limsupx→x0,x∈SF(x)-y0+Dψ(dist(x,W)).Proof.
Part “only if”: suppose that (3.1) is false, then there exist sequences xk∈S∖W, yk∈F(xk), dk∈D such that xk→x0 and
(3.2)limk→∞yk+dk-y0ψ(dist(xk,W))=0.
Hence, for any ϵ>0, there is k0=k0(ϵ) such that
(3.3)∥yk-y0+dk∥<ϵψ(dist(xk,W)),∀k≥k0.
Namely, yk+dk∈B(y0,ϵψ(dist(xk,W))).
By assumption, there existα>0 and U=B(x0,δ) such that (2.5) holds. In particular, for ϵ=min{α,δ}, there exists k0=k0(ϵ) such that for each k≥k0, we have that xk∈S∩B(x0,δ)∖W(3.4)yk+dk∈B(y0,ϵψ(dist(xk,W)))⊂B(y0,αψ(dist(xk,W))),
which is contradiction to (2.5).
Part “if”: suppose that the relation (2.5) is false, then for any δ>0 and α>0, there exist x∈S∩B(x0,δ)∖W and y∈F(x) such that
(3.5)(y+D)∩B(y0,αψ(dist(x,W)))≠∅.
In particular, choosing α=δ=1/k, there exist xk∈S∩B(x0,1/k)∖W and yk∈F(xk) and dk∈D such that
(3.6)(yk+dk)∈B(y0,1kψ(dist(xk,W))),
that is,
(3.7)∥yk+dk-y0∥ψ(dist(xk,W))<1k.
Hence, for sufficiently large k, we have
(3.8)∥yk+dk-y0∥ψ(dist(xk,W))→0,
which contradicts (3.1).From Theorem3.1, we easily obtain the following result.Corollary 3.2.
LetF:X→2Y, x0∈S, and ψ∈Ψ. Assume that W defined in (2.6) is a closed set. If (x0,y0)∈WSL(ψ,F,S), then,
(3.9)(limsupx→x0,x∈SF(x)-y0ψ(dist(x,W)))∩(-D)=∅.Theorem 3.3.
LetY=Rp, D=R+p=[0,+∞)p, and R-+p=[0,+∞]p. Let F:X→2Y, x0∈S. Assume that W defined in (2.6) is a closed set and ψ∈Ψ. Then, the following statements are equivalent:(i)
(x0,y0)∉WSL(ψ,F,S),(ii)
(limsupx→x0,x∈S((F(x)-y0)/ψ(dist(x,W))))∩(-R-+p)≠∅.Proof.
(
i
⇒
ii
) By assumption and Theorem 3.1, there exist sequences xk∈S∖W,  yk∈F(xk), dk∈R+p such that xk→x0 and
(3.10)limk→∞yk+dk-y0ψ(dist(xk,W))=0.
Let bk=ak+ck, where
(3.11)ak=yk-y0ψ(dist(xk,W)),ck=dkψ(dist(xk,W))∈R+p.
Consider the first component of the vectorck=(ck1,ck2,…,ckp). Let c1=limsupk→∞ck1. Then, there is an infinite set K1⊂𝒩 such that limK1∋k→∞ck1=c1. We have c1≥0 (it can be taken +∞), since ck1≥0. Now, let us consider the second component of sequence (ck)k∈K1. Let c2=limsupK1∋k→∞ck2. Hence, there exists an infinite set K2⊂K1 such that limK2∋k→∞ck2=c2. We still have c2≥0 (it can be taken +∞). So, we have limK2∋k→∞(ck1,ck2)=(c1,c2). Continuing the process, we obtain a vector c=(c1,c2,…,cp)∈R-+p and an infinite set Kp⊂𝒩 such that c=limKp∋k→∞ck.
Sincebk=ak+ck, taking the limit on both sides of the equation, we have
(3.12)0=limKp∋k→∞bk=limKp∋k→∞ak+limKp∋k→∞ck.
Therefore, d-:=limKp∋k→∞ak=-c≤0. Namely,
(3.13)(limsupx→x0,x∈SF(x)-y0ψ(dist(x,W)))∩(-R-+p)≠∅.
(
i
⇐
ii
) If -d-∈R+p, by Theorem 3.1, the result is true. So, we suppose that some components of d- are -∞. Reordering to d-, let d-=(d-1,d-2,…,d-n,d-n+1,…,d-p) with d-i=-∞ for i=1,2,…,n and d-i∈(-∞,0] for i>n, with n≥1. Hence, from relation (3.13), we see that there exist xk∈S∖W,xk→x0,yk∈F(xk) such that
(3.14)limk→∞yki-y0iψ(dist(xk,W))=-∞,i=1,2,…,n.
Since, for sufficiently large k and for i=1,2,…,n,  dki∶=-(yki-y0i)>0. Let
(3.15)dk=(dk1,…,dkn,-ψ(dist(xk,W))d-n+1,…,-ψ(dist(xk,W))d-p)∈R+p.
Clearly, one has
(3.16)limk→∞yk-y0+dkψ(dist(xk,W))=0.
Namely,
(3.17)0∈limsupx→x0,x∈SF(x)-y0+R+pψ(dist(x,W)).
By Theorem 3.1, we derive the result.


## 4. Scalarization

Scalarization is one of the most important procedures in vector optimization. In this section, we apply a generalized nonlinear scalarization function introduced by Hernández and Rodríguez-Marín in [18] to discuss the weak ψ-sharp minimizer in set-valued optimization problems.LetD be a proper closed convex cone and intD≠∅. Let e∈intD be a fixed point.Definition 4.1 (see [18]).
The generalized nonlinear scalarization functionG:2Y→R∪{-∞} is defined by
(4.1)G(A)=inf{t∈R:te∈A+D}.
A nonempty set A⊂Y is said to be D-proper if A+D≠Y.Next, we present several properties about the generalized nonlinear scalarization functionG.Lemma 4.2 (see [18]).
A is D-proper if and only if G(A)>-∞.Lemma 4.3 (see [16]).
Letr∈R and A be a nonempty subset of Y. Then,
(4.2)G(A)>r⟺re∉cl(A+D).Lemma 4.4 (see [18]).
LetA⊂Y and y0∈A. If y0∈StrDA, then G(A-y0)=0.Given a set-valued mapF:X→2Y and (x0,y0)∈GrF. Define F-:X→2Y by
(4.3)F-(x)=F(x)-y0.Now, we consider weakψ-sharp local minimizer for a set-valued map F through the weak sharp local minimizer of the scalarization function G∘F-:X→R∪{-∞}.Theorem 4.5.
Letx0∈S and (x0,y0)∈GrF. Suppose that W defined in (2.6) is a closed set and y0∈StrDF(x0). Then,
(4.4)(x0,y0)∈WSL(ψ,F,S)⟺x0∈WSL(ψ,G∘F-,S).Proof.
Part “only if”: assume that(x0,y0)∈WSL(ψ,F,S), there exist α>0 and U∈𝒩(x0) such that
(4.5)(F(x)-y0+D)∩B(0,αψ(dist(x,W)))=∅,∀x∈S∩U∖W.
Since B(0,αψ(dist(x,W)) is an open set,
(4.6)B(0,αψ(dist(x,W)))⊂(cl(F(x)-y0+D))c.
Note that, when W is a closed set,
(4.7)α4∥e∥ψ(dist(x,W))e∈B(0,αψ(dist(x,W))).
Hence,
(4.8)α4∥e∥ψ(dist(x,W))e∉cl(F(x)-y0+D).
By Lemma 4.3, we have
(4.9)G(F(x)-y0)>α4∥e∥ψ(dist(x,W)).
On the other hand, since y0∈StrF(x0), in terms of Lemma 4.4, we get
(4.10)G(F(x0)-y0)=0.
This relation, together with (4.9), yields
(4.11)G(F(x)-y0)>G(F(x0)-y0)+α4∥e∥ψ(dist(x,W)),∀x∈S∩U∖W.
Namely,
(4.12)(G∘F-)(x)>(G∘F-)(x0)+α4∥e∥ψ(dist(x,W)),∀x∈S∩U∖W,
that is, x0∈WSL(ψ,G∘F-,S).
Part “if”: by assumption, there existβ>0 and U∈𝒩(x0) such that
(4.13)G(F-(x))>G(F-(x0))+βψ(dist(x,W)),∀x∈S∩U∖W.
Since y0∈StrDF(x0), by applying Lemma 4.4, we get G(F(x0)-y0)=0. Thus, we have
(4.14)G(F(x)-y0)>βψ(dist(x,W)),∀x∈S∩U∖W.
Once more using Lemma 4.3, one has
(4.15)βψ(dist(x,W))e∉cl(F(x)-y0+D).
Furthermore,
(4.16)βψ(dist(x,W))e∉F(x)-y0+D,
which implies that
(4.17)(βψ(dist(x,W))e-D)∩(F(x)-y0+D)=∅,∀x∈S∩U∖W.
Since e∈intD, there exists a number ϵ>0 such that B(0,ϵ)⊂e-D. Moreover,
(4.18)B(0,λϵ)⊂λe-D,∀λ>0.
Hence, from (4.18), we obtain
(4.19)B(0,ϵβψ(dist(x,W)))⊂βψ(dist(x,W))e-D.
Combining it with relation (4.17), we deduce that
(4.20)B(0,ϵβψ(dist(x,W)))∩(F(x)-y0+D)=∅,∀x∈S∩U∖W.
By the definition of weak ψ-sharp local minimum, we have (x0,y0)∈WSL(ψ,F,S).In Theorem4.5, if the map F is a vector-valued function and the function G becomes the nonlinear scalarization function g, we easily obtain the following result, which is Theorem  3.4 in [13].Corollary 4.6.
Letx0∈S, f:X→Y, and f-(x)=f(x)-f(x0). Then,
(4.21)x0∈WSL(ψ,f,S)⟺x0∈WSL(ψ,g∘f-,S).



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