Feasibility problems via paramonotone operators in a convex setting

TL;DR

This paper investigates properties of paramonotone operators in Banach and Hilbert spaces, applying these to convex feasibility problems, including perturbation analysis for inconsistent systems and set intersection issues.

Contribution

It introduces new theoretical insights into paramonotone operators and applies these to solve convex feasibility problems with perturbation estimates.

Findings

Operators that are paramonotone and bimonotone are constant on their domains.

Provides bounds for the minimal perturbations needed to reach feasibility in convex systems.

Exact solutions for linear systems in the context of perturbation estimates.

Abstract

This paper is focused on some properties of paramonotone operators on Banach spaces and their application to certain feasibility problems for convex sets in a Hilbert space and convex systems in the Euclidean space. In particular, it shows that operators that are simultaneously paramonotone and bimonotone are constant on their domains, and this fact is applied to tackle two particular situations. The first one, closely related to simultaneous projections, deals with a finite amount of convex sets with an empty intersection and tackles the problem of finding the smallest perturbations (in the sense of translations) of these sets to reach a nonempty intersection. The second is focused on the distance to feasibility; specifically, given an inconsistent convex inequality system, our goal is to compute/estimate the smallest right-hand side perturbations that reach feasibility. We advance that this work derives lower and upper estimates of such a distance, which become the exact value when confined to linear systems.