Geodesic
Existence of minimizers and necessary conditions in set-valued optimization with equilibrium constraints
Applications of Mathematics, Tome 52 (2007) no. 6, pp. 453-472.

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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.
DOI : 10.1007/s10492-007-0027-0
Classification : 49J52, 49J53, 90C29, 90C30, 90C33
Keywords: variational analysis; nonsmooth and set-valued optimization; equilibrium constraints; existence of optimal solutions; necessary optimality conditions; generalized differentiation 
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Bao, Truong Q.; Mordukhovich, Boris S. Existence of minimizers and necessary conditions in set-valued optimization with equilibrium constraints. Applications of Mathematics, Tome 52 (2007) no. 6, pp. 453-472. doi : 10.1007/s10492-007-0027-0. http://geodesic.mathdoc.fr/articles/10.1007/s10492-007-0027-0/

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