Entropic Proximal Mappings with Applications to Nonlinear Programming

We introduce a family of new transforms based on imitating the proximal mapping of Moreau and the associated Moreau-Yosida proximal approximation of a function. The transforms are constructed in terms of the φ-divergence functional (a generalization of the relative entropy) and of Bregman's measure of distance. An analogue of Moreau's theorem associated with these entropy-like distances is proved. We show that the resulting Entropic Proximal Maps share properties similar to the proximal mapping and provide a fairly general framework for constructing approximation and smoothing schemes for optimization problems. Applications of the results to the construction of generalized augmented Lagrangians for nonlinear programs and the minimax problem are presented.