The EPI-Distance Topology: Continuity and Stability Results with Applications to Convex Optimization Problems

Let Γ(X) denote the proper, lower semicontinuous, convex functions on a Banach space X, equipped with the completely metrizable topology of uniform convergence of distance functions on bounded sets. We show, subject to some standard constraint qualifications, that the operations of addition and restriction are continuous. These results are applied to convex well-posed optimization problems, as well as to convergence of approximated solutions in infinite dimensional convex programming, to linear functionals exposing convex sets, and to metric projections. Also, for any given function in Γ(X), we obtain results regarding the convergence of its inf-convolution with smoothing kernels.