Stability Results for Ekeland's ε-Variational Principle and Cone Extremal Solutions

Given X a Banach space and f: X → ℝ ∪ {+∞} a proper lower semicontinuous function which is bounded from below, the Ekeland's ε-variational principle asserts the existence of a point x̄ in X, which we call ε-extremal with respect to f, which satisfies

f(u) > f(x̄) − ε‖u − x̄‖ for all u ∈ X, u ≠ x̄.

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 ε-ext f the set of such ε-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).