OPTIMALITY AND SUBOPTIMALITY CONDITIONS IN NONSMOOTH VARIATIONAL PROBLEMS

Boris Mordukhovich Department of Mathematics Wayne State University

This talk is devoted to some recent developments in nonsmooth variational analysis and their applications to necessary optimality and suboptimality conditions for problems of constrained optimization in infinitedimensional spaces. We mostly focus on a geometric approach to generalized differentiation and nonsmooth optimization based on the so-called extremal principle, which can be viewed as a variational counterpart of the classical separation principle in the case of nonconvex sets. First we discuss refined tools of generalized differentiation for nonconvex sets, set-valued mappings, and extended-real-valued functions that allow us to develop comprehensive calculus rules of generalized differentiation and apply them to the study of constrained optimization problems. Then we present necessary conditions for optimal and suboptimal (almost optimal) solutions to nonsmooth optimization problems with equality, inequality, and geometric constraints in infinite-dimensional spaces. A major advantage of suboptimality conditions is that they do not assume the existence of optimal solutions (which is not always the case in infinite-dimensional optimization, particularly in optimal control) while select feasible solutions that are almost optimal and almost satisfy necessary conditions for optimality.