Porosity of perturbed optimization problems in Banach spaces

Abstract: Let X be a Banach space and Z a nonempty closed subset of X. Let be a lower semicontinuous function bounded from below. This paper is concerned with the perturbed optimization problem , denoted by -inf for . In the case when X is compactly fully 2-convex, it is proved in the present paper that the set of all points x in X for which there does not exist such that is a σ-porous set in X. Furthermore, if X is assumed additionally to be compactly locally uniformly convex, we verify that the set of all points such that the problem -inf fails to be approximately compact, is a σ-porous set in , where denotes the set of all such that . Moreover, a counterexample to which some results of Ni [R.X. Ni, Generic solutions for some perturbed optimization problem in nonreflexive Banach space, J. Math. Anal. Appl. 302 (2005) 417–424] fail is provided. [Copyright &y& Elsevier]