Yet Another Convex Sets Subtraction with Application in Nondifferentiable Optimization

TL;DR

This paper proposes a new convex sets subtraction operation that forms a linear vector space, enabling novel analysis in nondifferentiable optimization, demonstrated through Lipschitz continuity of \\epsilon-subdifferentials.

Contribution

It introduces a new subtraction operation for convex sets, creating a linear vector space structure and applying it to analyze Lipschitz continuity in convex analysis.

Findings

Defined a convex sets subtraction as a collection of minimal convex sets

Established the vector space structure for convex set collections

Proved Lipschitz continuity of \\epsilon-subdifferentials using the new framework

Abstract

This paper introduces a new subtraction operation for convex sets, which defines their difference as a collection of inclusion-minimal convex sets with appropriate definitions of linear operations on them. With these operations the set of collections becomes a linear vector space with common zero and possibility to invert Minkowski summation. As the demonstration of usability of this concept the Lipschitz continuity of \(\epsilon\)-subdifferentials of convex analysis is proved in a novel way.