Nonconvex Notions of Regularity and Convergence of Fundamental Algorithms for Feasibility Problems

• Published in: SIAM journal on optimization Vol. 23; no. 4; pp. 2397 - 2419
• Main Authors: Hesse, Robert, Luke, D. Russell
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
• Subjects: Algorithms; Approximation; Euclidean space; Feasibility; Projectors
• ISSN: 1052-6234; 1095-7189; 1095-7189
• DOI: 10.1137/120902653

We consider projection algorithms for solving (nonconvex) feasibility problems in Euclidean spaces. Of special interest are the method of alternating projections (AP) and the Douglas--Rachford algorithm (DR). In the case of convex feasibility, firm nonexpansiveness of projection mappings is a global property that yields global convergence of AP and for consistent problems DR. A notion of local subfirm nonexpansiveness with respect to the intersection is introduced for consistent feasibility problems. This, together with a coercivity condition that relates to the regularity of the collection of sets at points in the intersection, yields local linear convergence of AP for a wide class of nonconvex problems and even local linear convergence of nonconvex instances of the DR algorithm. [PUBLICATION ABSTRACT]