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Weak Sharp Minima in Set-Valued Optimization Problems
We introduce the notion of a weak ψ-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).
The notion of weak sharp minima in general mathematical program problems was first introduced by Ferris in . It is a generalization of a sharp minimum in  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 . Recently, the study of weak sharp solution set covers real-valued optimization problems [5–8] and piecewise linear multiobjective optimization problems [9, 10].
In , Bednarczuk defined weak sharp Pareto minima of order 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 , Studniarski gave the definition of weak -sharp local Pareto minima in multiobjective optimization problems and presented necessary and sufficient conditions. In , Xu and Li established a sufficient and necessary condition for weak -sharp local Pareto minima in vector optimization problems in infinite spaces, the approach is that they transformed weak -sharp local Pareto minima of a vector-valued function to weak -sharp local minima of a family of scalar functions. Most recently, Durea and Strugariu  introduced the definition of weak -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 -sharp minima, which is different from one in , and establish some sufficient and necessary conditions through the outer limit of set-valued map. In particular, we develop the characterization of the weak -sharp minimizer in terms of the generalized nonlinear scalarization function.
This paper is organized as follows. In Section 2, we recall some basic definitions and give the notion of the weak -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 -sharp local minima in terms of the generalized nonlinear scalarization function.
2. Preliminary Results
Throughout this paper, let , be real normed spaces. denotes the open ball with center and radius , is the family of all neighborhoods of , and is the distance from the point to the set . The symbols , , and denote, respectively, the complement, closure, and interior of . Let be a convex cone (containing 0) with nonempty interior and let be partially ordered by .
Let be a set-valued map. We denote the graph and domain of , respectively, by If is a subset of , then and the inverse set-valued map of is given by if and only if .
Definition 2.1. Suppose that is a closed convex pointed cone. A point is called a strict efficient (resp., weak) point of , denoted by (resp., ) if
Given a set-valued map and a subset of , the following abstract optimization is considered:
Definition 2.2. Suppose that is a closed convex pointed cone. A point , with , is said to be a local strict (resp., weak) minimizer of over , written as , if there exists a neighborhood of in such that
We will say that is a global strict (global weak) minimizers when . The set of all global strict minimizers (resp., weak minimizers) is denoted by .
Definition 2.3. Let be a nondecreasing function with the property (such a family of functions is denoted by ) and . We say that a point pair is a weak -sharp local Pareto minimizer for (2.3), denoted by , if there exists a constant and such that where If we choose , we will say the point pair is a weak -sharp minimizer for (2.3), denoted by . In particular, let for . Then, we say is a weak -sharp local minimizer of order for (2.3) if .
Obviously, condition (2.5) can be expressed in the following equivalent form:
Remark 2.5. In , the definition of weak -sharp local minimizer for set-valued optimization is given by the oriented distance function . However, we establish the definition by the map . When the map is the real-valued function and the cone , our definition is equivalent to Definition 2.1 in .
3. Optimality Conditions for Weak -Sharp Minimizer for Set-Valued Optimization
In this section, we present sufficient and necessary conditions that a point pair is a weak -sharp local minimizer in set-valued optimization problems.
Theorem 3.1. Let , , and . Assume that defined in (2.6) is a closed set. Then, if and only if
Proof. Part “only if”: suppose that (3.1) is false, then there exist sequences , , such that and
Hence, for any , there is such that
By assumption, there exist and such that (2.5) holds. In particular, for , there exists such that for each , we have that which is contradiction to (2.5).
Part “if”: suppose that the relation (2.5) is false, then for any and , there exist and such that In particular, choosing , there exist and and such that that is, Hence, for sufficiently large , we have which contradicts (3.1).
From Theorem 3.1, we easily obtain the following result.
Corollary 3.2. Let , , and . Assume that defined in (2.6) is a closed set. If , then,
Theorem 3.3. Let , , and . Let , . Assume that defined in (2.6) is a closed set and . Then, the following statements are equivalent:(i),(ii).
Proof. By assumption and Theorem 3.1, there exist sequences , , such that and
Let , where
Consider the first component of the vector . Let . Then, there is an infinite set such that . We have (it can be taken ), since . Now, let us consider the second component of sequence . Let . Hence, there exists an infinite set such that . We still have (it can be taken ). So, we have . Continuing the process, we obtain a vector and an infinite set such that .
Since , taking the limit on both sides of the equation, we have Therefore, . Namely,
If , by Theorem 3.1, the result is true. So, we suppose that some components of are . Reordering to , let with for and for , with . Hence, from relation (3.13), we see that there exist such that Since, for sufficiently large and for , . Let Clearly, one has Namely, By Theorem 3.1, we derive the result.
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  to discuss the weak -sharp minimizer in set-valued optimization problems.
Let be a proper closed convex cone and . Let be a fixed point.
Definition 4.1 (see ). The generalized nonlinear scalarization function is defined by A nonempty set is said to be -proper if .
Next, we present several properties about the generalized nonlinear scalarization function .
Lemma 4.2 (see ). is -proper if and only if .
Lemma 4.3 (see ). Let and be a nonempty subset of . Then,
Lemma 4.4 (see ). Let and . If , then .
Given a set-valued map and . Define by
Now, we consider weak -sharp local minimizer for a set-valued map through the weak sharp local minimizer of the scalarization function .
Theorem 4.5. Let and . Suppose that defined in (2.6) is a closed set and . Then,
Proof. Part “only if”: assume that , there exist and such that
Since is an open set,
Note that, when is a closed set,
By Lemma 4.3, we have
On the other hand, since , in terms of Lemma 4.4, we get
This relation, together with (4.9), yields
that is, .
Part “if”: by assumption, there exist and such that Since , by applying Lemma 4.4, we get . Thus, we have Once more using Lemma 4.3, one has Furthermore, which implies that Since , there exists a number such that . Moreover, Hence, from (4.18), we obtain Combining it with relation (4.17), we deduce that By the definition of weak -sharp local minimum, we have .
Corollary 4.6. Let , , and . Then,
This research was partially supported by Heilongjiang Department of Education Science and Technology Research Project (Grant no.: 12521457) and the National Natural Science Foundation of China (no. 11071267).
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