Variational analysis in normed Spaces with applications to constrained optimization

TL;DR

This paper develops a generalized differential theory for variational analysis in incomplete normed spaces, enabling new calculus rules and optimality conditions for constrained optimization problems, including nonconvex cases.

Contribution

It introduces a novel variational analysis framework in incomplete normed spaces, avoiding traditional completeness assumptions, and applies it to derive new optimality conditions.

Findings

New calculus rules for generalized derivatives in normed spaces

Refined necessary optimality conditions for nonconvex constrained problems

Applicability to semi-infinite and semidefinite programming

Abstract

This paper is devoted to developing and applications of a generalized differential theory of variational analysis that allows us to work in incomplete normed spaces, without employing conventional variational techniques based on completeness and limiting procedures. The main attention is paid to generalized derivatives and subdifferentials of the Dini-Hadamard type with the usage of mild qualification conditions revolved around metric subregularity. In this way we develop calculus rules of generalized differentiation in normed spaces without imposing restrictive normal compactness assumptions and the like and then apply them to general problems of constrained optimization. Most of the obtained results are new even in finite dimensions. Finally, we derive refined necessary optimality conditions for nonconvex problems of semi-infinite and semidefinite programming.