Weak Sharp Minima for Convex Infinite Optimization Problems in Normed Linear Spaces

Abstract

In the present paper, we study the characterization issue of the weak sharp minima properties for convex infinite optimization problems in normed linear spaces. We develop a new approach to establish several complete geometric characterizations for the global/bounded/local weak sharp minima property, which extend/improve the corresponding ones in this direction by removing/relaxing the key topological assumptions made on the index set. As by-products, some complete characterizations of the global/bounded/local weak sharp minima are obtained for a subset of the level set of a given convex function (not necessarily the level set itself) in terms of the normal cones and the subdifferentials of the involved convex subset and convex function. These characterization results are of independent interest in extending/improving the existing ones on characterizing the weak sharp minima for convex optimization problems.