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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-Painlevé, 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).

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation
##### Keywords:
proper lower semicontinuous function; set convergence
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