Convex Optimization and the Epi-Distance Topology

Let Γ (X) denote the proper, lower semicontinuous, convex functions on a Banach space X, equipped with the completely metrizable topology τ of uniform convergence of distance functions on bounded sets. A function f in Γ (X) is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of f ∈ Γ (X) is the minimal condition that guarantees strong convergence of approximate minima of τ-approximating functions to the minimum of f. Moreover, we show that most functions in Γ (X),τ aw are well-posed, and that this fails if Γ (X) is topologized by the weaker topology of Mosco convergence, whenever X is infinite dimensional. Applications to metric projections are also given, including a fundamental characterization of approximative compactness.