Porosity of Generalized Mutually Maximization Problem

Abstract

Let C be a closed bounded convex subset of a Banach space X with 0 being an interior point of C and pC(.) be the Minkowski functional with respect to C. A generalized mutually maximization problem is said to be well posed if it has a unique solution (x, z) and every maximizing sequence converges strongly to (x, z). Under the assumption that the modulus of convexity with respect to pC(.) is strictly positive, we show that the collection of all subsets in the admissible family such that the generalized mutually maximization problem fail to be well-posed is porous in the admissible family. These extend and sharpen some recent results due to De Blasi, Myjak and Papini, Li, Li and Xu, and Ni, etc.