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Minimizing irregular convex functions: Ulam stability for approximate minima

Emil ERNST(Emil.Ernst***at***univ-cezanne.fr)
Michel THERA(michel.thera***at***unilim.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: 02/18/2010
Entry Accepted: 02/18/2010
Entry Last Modified: 02/18/2010

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