Yet Another Convex Sets Subtraction with Application in Nondifferentiable Optimization

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.