Feasibility problems via paramonotone operators in a convex setting

• Published in: Optimization Vol. 73; no. 10; pp. 3055 - 3086
• Main Authors: Camacho, J., Cánovas, M.J., Martínez-Legaz, J.E., Parra, J.
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 02-10-2024; Taylor & Francis LLC
• Subjects: Banach spaces; convex inequalities; Convexity; displacement mapping; Distance function; distance to feasibility; Euclidean geometry; Euclidean space; Feasibility; Hilbert space; Intersections; Linear systems; Operators (mathematics); paramonotone operators; Perturbation; Translations
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2024.2313690

This paper is focussed 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 focussed 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.