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Weak Sharp Minima in Vector Optimization Problems
Fixed Point Theory and Applications volume 2010, Article number: 154598 (2010)
Abstract
We present a sufficient and necessary condition for weak sharp minima in infinitedimensional spaces. Moreover, we develop the characterization of weak 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 in [1]. It is an extension of sharp minimum in [2]. Weak sharp minima play important roles 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 realvalued optimization problems [5–8] and piecewise linear multiobjective optimization problems [9–11].
Most recently, Bednarczuk [12] defined weak sharp minima of order for vectorvalued mappings under an assumption that the order cone is closed, convex, and pointed and used the concept to prove upper Hölderness and Hölder calmness of the solution setvalued 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 wellposedness of vector optimization problems. In [14], Studniarski gave the definition of weak 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 order .
In this paper, motivated by the work in [14, 15], we present a sufficient and necessary condition of which a point is a weak sharp minimum for a vectorvalued mapping in the infinitedimensional spaces. In addition, we develop the characterization of weak 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 sharp local minimizer for vectorvalued optimization problems. In Section 3, we present a sufficient and necessary condition for weak sharp local minimizer of vectorvalued optimization problems. We also give an example to illustrate the optimality condition.
2. Preliminary Results
Throughout the paper, and are normed spaces. denotes the open ball with center and radius . is the family of all neighborhoods of , and is the distance from a point to a set . The symbols , and denote, respectively, the complement, interior and boundary of .
Let be a convex cone (containing 0). The cone defines an order structure on , that is, a relation "" in is defined by . is a proper cone if .
Let be an open subset of , . Given a vectorvalued map , the following abstract optimization is considered:
In the sequel, we always assume that is a proper closed and convex cone.
Definition 2.1.
One says that is a local Pareto minimizer for (2.1), denoted by , if there exists for which there is no such that
If one can choose , one will say that is a Pareto minimizer for (2.1), denoted by .
Note that (2.2) may be replaced by the simple condition if we assume that the cone is pointed.
Definition 2.2 (see [14]).
Let be a nondecreasing function with the property (such a family of functions is denoted by ). Let . One says that is a weak sharp local Pareto minimizer for (2.1), denoted by , if there exist a constant and such that
where
If one can choose , one says is a weak sharp minimizer for (2.1), denoted by . In particular, let for Then, one says that is a weak sharp local Pareto minimizer of order for (2.1) if , and one says that is a weak sharp Pareto minimizer of order for (2.1) if .
Remark 2.3.
If is a closed set, condition (2.3) can be expressed as the following equivalent forms:
Remark 2.4.
In the Definition 2.2, if , , and , then the relation (2.6) becomes the following form:
which is the wellknown definition of a weak sharp minimizer of order for (2.1); see [16].
3. Main Results
In this section, we first generalize the result of Theorem in Studniarski [14] to infinitedimensional spaces. Finally, we develop the characterization of weak sharp minimizer by means of a nonlinear scalarization function.
Let be a proper closed convex cone with . The topological dual space of is denoted by . The polar cone to is . It is well known that the cone contains a compact convex set with such that
The set is called a base for the dual cone . Recall that a point is an extremal point of a set if there exist no different points and such that .
Theorem 3.1.
Suppose that is a vectorvalued map. Let be a proper closed convex cone with , , and .
(i)Let be a compact convex base of and the set of extremal points of . Suppose that defined by (2.4) is a closed set. Then, if and only if there exist , a constant , a covering of , and
(ii)Let and assume that . Then if and only if there exists a covering of such that
Proof.

(i)
Part "only if": by assumption, there exist and such that
(3.4)
Let be a fixed point. Set . Since is compact, the infimum is attained at a point of . Namely, . Clearly, for any . Hence, .
For each , we define
We will show that
Let . If , then by (2.4), hence, for all . If , suppose that for any , then
This relation, together with statement yields
Obviously, for any , the above relation becomes the following form:
Consequently, by the bipolar theorem, one has
Therefore,
and , which is a contradiction to (3.4). We have thus proved that covers .
Now, let and . From the procedure of the above proof, we see that . Hence, by (3.5), set , inequality (3.2) is true.
Part "if": we define . The supremum is attained at an extremal point because of the compactness of . So and for any . Hence, by assumption, we have
for and .
Now, suppose that for all , (3.4) is false, then there exist and such that
Let be a fixed point, and since is a cone, there is such that . Consequently,
Therefore,
There is from (3.15) such that
Since , there is such that . Moreover, and . Hence,
By choosing , we obtain a contradiction to (3.12).

(ii)
Part "only if": for each , we define,
(3.18)
Now, we will check that (3.6) holds true. Pick any . Suppose that for any , then
Hence, for any , . By applying the bipolar theorem, we have
Combing it with the assumption, we have
which is a contradiction to (3.19). So (3.6) holds and (3.3) is satisfied by the definition of .
Part "if": suppose that , then there exists such that
Indeed, can be replace by , because , , which is contradiction to (3.22). Hence, for , we have . In particular,
It follows from the assumption that
Therefore, by (3.3), we obtain
which contradicts relation (3.23).
Remark 3.2.
By taking in part (i) (resp., (ii)) of Theorem 3.1, we obtain a necessary and sufficient condition for to be in (resp., ). In particular, if we choose and and , then, we obtain Theorem in [14].
Finally, we apply the nonlinear scalarization function to discuss the weak sharp minimizer in vector optimization problems.
Let be a closed and convex cone with nonempty interior . Given a fixed point and , the nonlinear scalarization function is defined by
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 on . More results about the function can be found in [17].
In what follows, we present several properties about the nonlinear scalarization function.
Lemma 3.3 (see [17]).
For any fixed , , and . One has
(i),
(ii).
(iii).
Given a vectorvalued map , define by
Next, we consider weak sharp local minimizer for a vectorvalued map through a weak sharp local minimizer of a scalar function .
Theorem 3.4.
Let . Suppose that defined by (2.4) is a closed set. Then,
Proof.
Part "only if": let us assume that . Thus, there exist and such that
Note that, when is a closed set,
Therefore,
By using Lemma 3.3(ii), one has
According to Lemma 3.3(iii), one has
This relation, together with (3.32) yields
Namely,
that is, .
Part "if": by assumption, there exist and such that
In terms of Lemma 3.3(iii), we have
Hence,
Once more using Lemma 3.3(ii), one has
which implies that
Since , there exists some number such that . Moreover,
Hence, it follows from the relation that
Combing it with relation (3.40), we deduce that
Let , by the definition of weak sharp local minimizer, we have .
It is possible to illustrate Theorem 3.4 by means of adapting a simple example given in [14].
Example 3.5.
Let , and let be defined by
We choose . Using Definition 2.2, we derive that .
Let . From Corollary in [17], we have . Observe that
It is easy to verify that for all . Using relation (2.7), we show that . Hence, condition (3.28) with holds for .
References
Ferris MC: Weak sharp minima and penalty functions in mathematical programming. Computer Sciences Department, University of Wisconsin, Madison, Wis, USA; June 1988.
Polyak BT: Sharp Minima, Institue of Control Sciences Lecture Notes. USSR, Moscow, Russia; 1979. Presented at the IIASA Workshop on Generalized Lagrangians and Their Applications, IIASA, Laxenburg, Austria, 1979
Henrion R, Outrata J: A subdifferential condition for calmness of multifunctions. Journal of Mathematical Analysis and Applications 2001,258(1):110–130. 10.1006/jmaa.2000.7363
Lewis AS, Pang JS: Error bounds for convex inequality systems. In Proceedings of the 5th Symposium on Generalized Convexity, 1996, LuminyMarseille, France Edited by: Crouzeix JP.
Burke JV, Ferris MC: Weak sharp minima in mathematical programming. SIAM Journal on Control and Optimization 1993,31(5):1340–1359. 10.1137/0331063
Burke JV, Deng S: Weak sharp minima revisited. I. Basic theory. Control and Cybernetics 2002,31(3):439–469.
Burke JV, Deng S: Weak sharp minima revisited. II. Application to linear regularity and error bounds. Mathematical Programming B 2005, 104: 235–261. 10.1007/s1010700506152
Burke JV, Deng S: Weak sharp minima revisited. III. Error bounds for differentiable convex inclusions. Mathematical Programming B 2009, 116: 37–56. 10.1007/s1010700701308
Deng S, Yang XQ: Weak sharp minima in multicriteria linear programming. SIAM Journal on Optimization 2004,15(2):456–460.
Zheng XY, Yang XQ: Weak sharp minima for piecewise linear multiobjective optimization in normed spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,68(12):3771–3779. 10.1016/j.na.2007.04.018
Zheng XY, Yang XM, Teo KL: Sharp minima for multiobjective optimization in Banach spaces. SetValued Analysis 2006,14(4):327–345. 10.1007/s1122800600237
Bednarczuk EM: Weak sharp efficiency and growth condition for vectorvalued functions with applications. Optimization 2004,53(5–6):455–474. 10.1080/02331930412331330478
Bednarczuk E: On weak sharp minima in vector optimization with applications to parametric problems. Control and Cybernetics 2007,36(3):563–570.
Studniarski M: Weak sharp minima in multiobjective optimization. Control and Cybernetics 2007,36(4):925–937.
FloresBazán F, Jiménez B: Strict efficiency in setvalued optimization. SIAM Journal on Control and Optimization 2009,48(2):881–908. 10.1137/07070139X
Studniarski M, Ward DE: Weak sharp minima: characterizations and sufficient conditions. SIAM Journal on Control and Optimization 1999,38(1):219–236. 10.1137/S0363012996301269
Chen GY, Huang X, Yang X: Vector Optimization, SetValued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems. Volume 541. Springer, Berlin, Germany; 2005:x+306.
Acknowledgments
This paper was partially supported by the National Natural Science Foundation of China (Grant no. 10871216) and 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 paper [14].
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Xu, S., Li, S. Weak Sharp Minima in Vector Optimization Problems. Fixed Point Theory Appl 2010, 154598 (2010). https://doi.org/10.1155/2010/154598
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DOI: https://doi.org/10.1155/2010/154598
Keywords
 Convex Cone
 Vector Optimization Problem
 Multiobjective Optimization Problem
 Closed Convex Cone
 Weak Sharp Minimum