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Necessary Conditions for Nonsmooth Optimization Problems with Operator Constraints in Metric Spaces

This paper concerns nonsmooth optimization problems involving operator constraints given by mappings on complete metric spaces with values in nonconvex subsets of Banach spaces. We derive general first-order necessary optimality conditions for such problems expressed via certain constructions of generalized derivatives for mappings on metric spaces and axiomatically defined subdifferentials for the distance function to nonconvex sets in Banach spaces. Our proofs are based on variational principles and perturbation/approximation techniques of modern variational analysis. The general necessary conditions obtained are specified in the case of optimization problems with operator constraints described by mappings taking values in approximately convex subsets of Banach spaces, which admit uniformly Gâteaux differentiable renorms (in particular, in any separable spaces).