Conic cancellation laws and some applications

TL;DR

This paper extends Radström's cancellation law to conic variants in normed vector spaces, providing new tools for set optimization, subdifferential calculus, and stability analysis.

Contribution

It introduces conic versions of Radström's cancellation law applicable to set optimization and explores their implications for subdifferential calculus and optimality conditions.

Findings

Developed conic Radström cancellation laws for finite and infinite dimensional spaces.

Applied these laws to subdifferential calculus for set-valued maps.

Provided necessary optimality conditions and stability results for set optimization.

Abstract

We discuss, on finite and infinite dimensional normed vector spaces, some versions of Radstr\"{o}m cancellation law (or lemma) that are suited for applications to set optimization problems. In this sense, we call our results "conic" variants of the celebrated result of Radstr\"{o}m, since they involve the presence of an ordering cone on the underlying space. Several adaptations to this context of some topological properties of sets are studied and some applications to subdifferential calculus associated to set-valued maps and to necessary optimality conditions for constrained set optimization problems are given. Finally, a stability problem is considered.