Optimality Conditions for Nondifferentiable Multiobjective Semi-Infinite Programming Problems

D Barilla,A Puglisi,G Caristi

doi:10.1155/2016/5367190

Abstract

We have considered a multiobjective semi-infinite programming problem with a feasible set defined by inequality constraints. First we studied a Fritz-John type necessary condition. Then, we introduced two constraint qualifications and derive the weak and strong Karush-Kuhn-Tucker (KKT in brief) types necessary conditions for an efficient solution of the considered problem. Finally an extension of a Caristi-Ferrara-Stefanescu result for the (Φ,ρ)-invexity is proved, and some sufficient conditions are presented under this weak assumption. All results are given in terms of Clark subdifferential.

Highlights

The Clarke subdifferential is a natural generalization of the classical derivative, since it is known that when function φ is continuously differentiable at x, ∂cφ(x) = {∇φ(x)}
In the following theorem we summarize some important properties of the Clarke directional derivative and the Clarke subdifferential from [1] which are widely used in what follows
We have considered the following multiobjective semi-infinite programming problem: (MOSIP) inf (f1 (x), f2 (x), . . . , fp (x))

Summary

Preliminaries and Introduction

We briefly overview some notions of convex analysis and nonsmooth analysis widely used in the formulations and proofs of the main results of the paper. Given a nonempty set A ⊆ Rn, we denote with A, ri(A), conv(A), and cone(A) the closure of A, the relative interior of A, convex hull, and convex cone (containing the origin) generated by A, respectively. The Clarke subdifferential is a natural generalization of the classical derivative, since it is known that when function φ is continuously differentiable at x, ∂cφ(x) = {∇φ(x)}. For differentiable MOSIP where T is finite, necessary conditions of KKT type have been established under various constraint qualifications in [4]. Glover et al in [7] considered a nondifferentiable convex MOSIP and presented optimality theorems for it. For a nonsmooth MOSIP, the “basic constraint qualification” has been studied by Chuong and Kim in [8], who have given optimality and duality conditions of Karush-KuhnTucker (KKT, briefly) type for the problem which involves the notion of Mordukhovich subdifferential.

Necessary Conditions

Sufficient Conditions