Porosity of perturbed optimization problems in Banach spaces

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 infz∈Z{J(z)+‖x−z‖}, denoted by (x,J)-inf for x∈X. 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 z0∈Z such that J(z0)+‖x−z0‖=infz∈Z{J(z)+‖x−z‖} 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 x∈X∖Z0 such that the problem (x,J)-inf fails to be approximately compact, is a σ-porous set in X∖Z0, where Z0 denotes the set of all z∈Z such that z∈PZ(z). 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.