- Minimizing irregular convex functions: Ulam stability for approximate minima Emil ERNST(Emil.Ernstuniv-cezanne.fr) Michel THERA(michel.theraunilim.fr) Abstract: The main concern of this article is to study Ulam stability of the set of $\varepsilon$-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 $\varepsilon$-approximate minima is Hausdorff upper semicontinuous for the Attouch-Wets topology when the set $\mathcal{C}(X)$ 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 $\varepsilon$-approximate minima if and only if the boundary of any of its sublevel sets is bounded. Keywords: Attouch-Wets convergence, Hausdorff upper semi-continuity, Ulam stability, approximate minima Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: XLIM (UMR-CNRS $6172$) and Universit\'e de Limoges, 123 Avenue A. Thomas, 87060 Limoges Cedex, France Download: [PDF]Entry Submitted: 02/18/2010Entry Accepted: 02/18/2010Entry Last Modified: 02/18/2010Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society.