Convex optimization and the epi-distance topology

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