On finite-dimensional set-inclusive constraint systems: local analysis and related optimality conditions

Abstract: In the present paper, some aspects of the finite-dimensional theory of set-inclusive generalized equations are studied. Set-inclusive generalized equations are problems arising in several contexts of optimization and variational analysis, involving multi-valued mappings and cones. The aim of this paper is to propose an approach to the local analysis of the solution set to such kind of generalized equations. In particular, a study of the contingent cone to the solution set is carry out by means of first-order approximations of set-valued mappings, which are expressed by prederivatives. Such an approach emphasizes the role of the metric increase property for set-valued mappings, as a condition triggering crucial error bound estimates for the tangential description of solution sets. Some of the results obtained through this kind of analysis are then exploited for formulating necessary optimality conditions, which are valid for problems with constraints formalized by set-inclusive generalized equations.