Convex optimization and the epi-distance topology

Abstract:

Let $\Gamma (X)$ denote the proper, lower semicontinuous, convex functions on a Banach space $X$, equipped with the completely metrizable topology $\tau$ of uniform convergence of distance functions on bounded sets. A function $f$ in $\Gamma (X)$ is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of $f \in \Gamma (X)$ is the minimal condition that guarantees strong convergence of approximate minima of $\tau$-approximating functions to the minimum of $f$. Moreover, we show that most functions in $\langle \Gamma (X),{\tau _{aw}}\rangle$ are well-posed, and that this fails if $\Gamma (X)$ 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.