Weak -Sharp Minima in Vector Optimization Problems
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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 Copyright q 2010 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 weak ψ-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 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 real-valued optimization problems 5–8 and piecewise linear multiobjective optimization problems 9–11. Most recently, Bednarczuk 12 defined weak sharp minima of order m for vectorvalued mappings under an assumption that the order cone is closed, convex, and pointed and used the concept to prove upper Holderness and Holder 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. In 14, Studniarski gave the definition of weak ψ-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 m. In this paper, motivated by the work in 14, 15, we present a sufficient and necessary condition of which a point is a weak ψ-sharp minimum for a vector-valued mapping in the
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Fixed Point Theory and Applications
infinite-dimensional spaces. In addition, we develop the characterization of weak ψ-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. In Section 3, we present a sufficient and necessary condition for weak ψ-sharp local minimizer of vector-valued optimization problems. We also give an example to illustrate the optimality condition.
2. Preliminary Results Throughout the paper, X and Y are normed spaces. Bx, δ denotes the open ball with c
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