Minimization of entropy functionals revisited

Abstract

Integral functionals based on convex normal integrands are minimized subject to finitely many moment constraints. The integrands are assumed to be strictly convex but not autonomous or differentiable. The effective domain of the value function is described by a modification of the concept of convex core. The minimization is viewed as a primal problem and studied together with a dual one in the framework of convex duality. Main results assume a dual constraint qualification but dispense with the primal constraint qualification. Minimizers and generalized minimizers are explicitly described whenever the primal value is finite. Existence of a generalized dual solution is established whenever the dual value is finite. A generalized Pythagorean identity is presented using Bregman distance and a correction term. Results are applied to minimization of Bregman distances.