Connectedness of Cone-Efficient Solution Set for Cone-Quasiconvex Multiobjective Programming in Locally Convex Spaces

Abstract

This paper deals with the connectedness of the cone-efficient solution set for vector optimization in locally convex Hausdorff topological vector spaces.The connectedness of the cone-efficient solution set is proved for multiobjective programming defined by a continuous cone-quasiconvex mapping on a compact convex set of alternatives. The generalized saddle theorem plays a key role in the proof.

On Connectedness of Efficient Solution Sets for a Star Cone-Quasiconvex Vector Optimization Problem

“…Corollary 3.1 improves the main theorem in [24] where K is assumed to have a weakly compact base and a nonempty interior. For example the natural ordering cone in l ∞ has nonempty interior and K +i = ∅, but without weakly compact base.…”

“…2. Theorem 3.1 improves Theorem 3.7 in Zhou and Hu [24] where the cone K is assumed to have nonempty interior and a weakly compact base. In fact each of both conditions is very restrictive.…”

“…In particular, the connectedness of efficient (minimal) solution set is proved for a strictly cone-quasiconvex objective mapping by Fu and Zhou [5] under the assumption that efficient solution set is closed, and for a cone-quasiconvex objective mapping by Gong [7] and Song [18,19] under the assumption that the image of the feasible set is cone-convex. Very recently, Zhou and Hu [24] proved that the efficient (minimal) solution set for a continuous, star cone-quasiconvex objective mapping is connected under the assumptions that the ordering cone has a nonempty interior and a weakly compact base. We know that each of the above conditions imposed on the cone is too strong.…”

A Geometric Programming Approach to Convex Multi-Objective Programming Problems

“…The basic advantages of the convex optimization problem for solving a problem very reliably and efficiently using interior-point methods or other special methods have been shown by Boyd et al [6]. The connectedness properties of quasi-convex problems using cone-efficient set of the solution have been shown by Zhou [16]. An and Liu [1] have proven different necessary and sufficient conditions for getting weakly Pareto solutions and weakly efficient solutions of convex multi-objective programming problems.…”

Pareto Optimal Solution Analysis of Convex Multi-Objective Programming Problem

“…The relations of effective solutions between the large-scale multiobjective programming and its subproblems have been studied and the existence of effective solution has been investigated [10]. X. Zhou proved the connectedness of cone-efficient solution set for cone-quasiconvex multiobjective programming in topological vector spaces [11]. Y. Li and Q. Zhang obtained some Kuhn-Tucker type optimality conditions for a class of non-smooth multiobjective semi-infinite programming involving generalized convexity and some non-smooth non-convex functions.…”