Minimizing Irregular Convex Functions: Ulam Stability for Approximate Minima

The main concern of this article is to study Ulam stability of the set of ε-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 εapproximate minima is Hausdorff upper semicontinuous for the AttouchWets topology when the set C(X) of all the closed and nonempty convex subsets of X is equipped with the uniform Hausdorff topology. We prove that a proper lower semicontinuous convex function bounded below has Ulam-stable ε-approximate minima if and only if the boundary of any of its sublevel sets is bounded. 1. Statement of the problem In the sequel, (X, ‖ · ‖) will be a real normed linear space with closed unit ball BX and origin θX ; the closed unit ball and origin of the dual space X? will be denoted by BX? and θX? , respectively. We will often be forced to consider finite products of normed spaces, e.g., X×R, and in such spaces, the box norm will be understood. We denote the space of proper lower semicontinuous extended-real-valued convex mappings defined on X by Γ0(X). An important concept for our study is that of the Attouch-Wets topology τAW on C(X), the class of all the closed and convex subsets of X, which is nothing but the topology of the uniform convergence on bounded sets applied to the distance functionals to the sets from C(X); moreover this topology is metrizable [9]. This topology has been defined in different ways and we refer to section 2 for its historical presentation as it can be found for instance in Attouch’s book [3]. More precisely, we consider the Attouch-Wets topology on Γ0(X): given a sequence (fn)n∈N in Γ0(X), we say that it Attouch-Wets converges to f provided the sequence (epi fn)n∈N of the epigraphs of functions fn is convergent to epi f in the Attouch-Wets topology of the space C(X × R). Our study 2000 Mathematics Subject Classification. 49J53, 49K40.