Existence of minimizers and necessary conditions for set-valued optimization with equilibrium constraints.
Mathematics and Computer Science
In this paper we study set-valued optimization problems with equilibrium constraints (SOPECs) described by parametric generalized equations in the form $0 \in G(x) + Q(x)$, where both $G$ and $Q$ are set-valued mappings between infinite-dimensional spaces. Such models particularly arise from certain optimization-related problems governed by set-valued variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish general results on the existence of optimal solutions under appropriate assumptions of the Palais-Smale type and then derive necessary conditions for optimality in the models under consideration by using advanced tools of variational analysis and generalized differentiation.
Bao, Truong Q.; Mordukhovich, Boris S. Existence of minimizers and necessary conditions in set-valued optimization with equilibrium constraints. Appl. Math. 52 (2007), no. 6, 453–472.