Extended Set Difference : Inverse Operation of Minkowski Summation

TL;DR

This paper introduces the extended set difference for compact convex sets, generalizing previous differences, with a constructive optimization-based approach, broadening set arithmetic capabilities.

Contribution

It defines a new extended set difference that guarantees existence for all pairs of convex sets and provides a computational method via linear optimization.

Findings

Guarantees existence for all pairs of convex sets

Provides bounds on solution variety

Offers a linear optimization-based computation method

Abstract

This paper introduces the extended set difference, a generalization of the Hukuhara and generalized Hukuhara differences, defined for compact convex sets in $\mathbb{R}^d$. The proposed difference guarantees existence for any pair of such sets, offering a broader framework for set arithmetic. The difference may not be necessarily unique, but we offer a bound on the variety of solutions. The definition of the extended set difference is formulated through an optimization problem, which provides a constructive approach to its computation. The paper explores the properties of this new difference, including its stability under orthogonal transformations and its robustness to perturbations of the input sets. We propose a method to compute this difference through a formulated linear optimization problem.