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Stability results for Ekeland’s \(\varepsilon\)-variational principle and cone extremal solutions. (English) Zbl 0779.49015
Summary: Given \(X\) a Banach space and \(f:X\to\mathbb{R}\cup\{+\infty\}\) a proper lower semicontinuous function which is bounded from below, the Ekeland’s \(\varepsilon\)-variational principle asserts the existence of a point \(\overline x\) in \(X\), which we call \(\varepsilon\)-extremal with respect to \(f\), which satisfies \(f(u)>f(\overline x)-\varepsilon\| u-\overline x\|\) for all \(u\in X\), \(u\neq\overline x\). By using set convergence notions (Kuratowski-PainlevĂ©, Mosco, bounded Hausdorff) and their epigraphical versions we study the (semi) continuity properties of the mapping which to \(f\) associates \(\varepsilon-\text{ext} f\) the set of such \(\varepsilon\)-extremal points. The key for the geometrical understanding of such properties is to consider the equivalent Phelps extremization principle which, given a closed set \(D\) in \(X\) and a partial ordering with respect to a pointed cone, associates the set of elements of \(D\) maximal with respect to this order. Direct or potential applications are given in various fields (multicriteria optimization, numerical algorithmic, calculus of variations).

49J45 Methods involving semicontinuity and convergence; relaxation
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