One Optimization Problem with Convex Set-Valued Mapping and Duality

Abstract

This study focuses on the formulation and analysis of problems that are dual to those defined by convex set-valued mappings. Various important classes of optimization problems—such as the classical problems of mathematical and linear programming, as well as extremal problems arising in economic dynamics models—can be reduced to problems of this type. The dual problem proposed in this work is constructed on the basis of the duality theorem connecting the operations of addition and infimal convolution of convex functions, a result that has been previously applied to compact-valued mappings. It appears that, under the so-called nondegeneracy condition, this construction serves as a fundamental approach for deriving duality theorems and establishing both necessary and sufficient optimality conditions. Furthermore, alternative conditions that partially replace the nondegeneracy assumption may also prove valuable for addressing other issues within convex analysis.