Strong solvability in Orlicz spaces

Abstract

The main objective of the authors is to characterize strong solvability of optimization problems where convergence of the values to the optimum already implies norm-convergence of the approximations to the minimal solution. It turns out that strong solvability can be geometrically characterized by the local uniform convexity of the corresponding convex functional (local uniform convexity being appropriately defined). For bounded functionals we establish that in reflexive Banach spaces strong solvability is characterized by the Frechet-differentiability of the convex conjugate. These results are based in part on a paper of Asplund and Rockafellar on the duality of A-differentiability and B-convexity of conjugate pairs of convex functions, where B is the polar of A. Before we apply these results to Orlicz spaces, we turn to E-spaces introduced by Fan and Glicksberg. Using the properties of E-spaces we can show that for finite not purely atomic measures Frechet differentiability of an Orlicz space already implies its reflexivity. The main theorem gives — in 17 equivalent statements — a characterization of strong solvability, local uniform convexity, and Frechet differentiability of the dual space, in case LΦ is reflexive. It is remarkable that all these properties can also be equivalently expressed by the differentiability of Φ or the strict convexity of Ψ. In particular, LΦ is an E-space, if LΦ is reflexive and Φ is strictly convex. We discuss applications that refer to • Tychonov-regularization: local uniformly convex regularisations are sufficient to ensure convergence. As we have given a complete description of local uniform convexity in Orlicz spaces we can state such regularizing functionals explicitly. • Ritz method: it is well known that the Ritz procedure generates a minimizing sequence. Actual convergence of the minimal solutions on each subspace is achieved if the original problem is strongly solvable. • Greedy algorithms: the convergence proof makes use of the Kadec-Klee property of E-spaces.