#### Title

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

#### Document Type

Technical Report

#### Abstract

This paper concerns nonsmooth optimization problems involving operator constraints given by mappings on complete metric spaces with values in nonconvcx 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 arc 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 dDScribcd by mappings taking values in approximately convex subsets of Banach spaces, which admit uniformly Gateaux differentiable renorms (in particular, in any separable spaces).

#### Number in Series

2008.07

#### Disciplines

Applied Mathematics | Mathematics

#### AMS Subject Classification

49J53, 49J52, 49K27, 90C48

#### Recommended Citation

Mordukhovich, Boris S. and Mou, Libin, "Necessary Conditions for Nonsmooth Optimization Problems with Operator Constraints in Metric Spaces" (2008). *Mathematics Research Reports*. 58.

http://digitalcommons.wayne.edu/math_reports/58

## Comments

Dedicated to Stephen Simons in honor of his 70th birthday.