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Emil ERNST(Emil.Ernstunivcezanne.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 setvalued application which assigns to each function the set of its $\varepsilon$approximate minima is Hausdorff upper semicontinuous for the AttouchWets 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 Ulamstable $\varepsilon$approximate minima if and only if the boundary of any of its sublevel sets is bounded. Keywords: AttouchWets convergence, Hausdorff upper semicontinuity, Ulam stability, approximate minima Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: XLIM (UMRCNRS $6172$) and Universit\'e de Limoges, 123 Avenue A. Thomas, 87060 Limoges Cedex, France Download: [PDF] Entry Submitted: 02/18/2010 Modify/Update this entry  
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