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.
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XLIM (UMR-CNRS 172$) and Universit\'e de Limoges, 123 Avenue A. Thomas, 87060 Limoges Cedex, France
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