Minimizing Irregular Convex Functions: Ulam Stability for Approximate Minima

The main concern of this article is to study Ulam stability of the set of epsilon-approximate minima of a proper lower semicontinuous convex function bounded below on a real normed space X, when the objective function is subjected to small perturbations (in the sense of Attouch & Wets). More precisely, we characterize the class all proper lower semicontinuous convex functions bounded below such that the set-valued application which assigns to each function the set of its epsilon-approximate minima is Hausdorff upper semi-continuous for the Attouch-Wets topology when the set C(X) of all the closed and nonempty convex subsets of X is equipped with the Hausdorff topology. We prove that a proper lower semicontinuous convex function bounded below has Ulam-stable epsilon-approximate minima if and only if the boundary of any of its sublevel sets is bounded.