Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems

We consider projection algorithms for solving (nonconvex) feasibility problems in Euclidean spaces. Of special interest are the Method of Alternating Projections (MAP) and the Douglas-Rachford or Averaged Alternating Reflection Algorithm (AAR). In the case of convex feasibility, firm nonexpansiveness of projection mappings is a global property that yields global convergence of MAP and for consistent problems AAR. Based on (\epsilon, \delta)-regularity of sets developed by Bauschke, Luke, Phan and Wang in 2012, a relaxed local version of firm nonexpansiveness with respect to the intersection is introduced for consistent feasibility problems. Together with a coercivity condition that relates to the regularity of the intersection, this yields local linear convergence of MAP for a wide class of nonconvex problems,Comment: 22 pages, no figures, 30 reference

Characterizations of Super-regularity and its Variants

Convergence of projection-based methods for nonconvex set feasibility problems has been established for sets with ever weaker regularity assumptions. What has not kept pace with these developments is analogous results for convergence of optimization problems with correspondingly weak assumptions on the value functions. Indeed, one of the earliest classes of nonconvex sets for which convergence results were obtainable, the class of so-called super-regular sets introduced by Lewis, Luke and Malick (2009), has no functional counterpart. In this work, we amend this gap in the theory by establishing the equivalence between a property slightly stronger than super-regularity, which we call Clarke super-regularity, and subsmootheness of sets as introduced by Aussel, Daniilidis and Thibault (2004). The bridge to functions shows that approximately convex functions studied by Ngai, Luc and Th\'era (2000) are those which have Clarke super-regular epigraphs. Further classes of regularity of functions based on the corresponding regularity of their epigraph are also discussed.Comment: 15 pages, 2 figure

Optimal Convergence Rates for Generalized Alternating Projections

Generalized alternating projections is an algorithm that alternates relaxed projections onto a finite number of sets to find a point in their intersection. We consider the special case of two linear subspaces, for which the algorithm reduces to a matrix teration. For convergent matrix iterations, the asymptotic rate is linear and decided by the magnitude of the subdominant eigenvalue. In this paper, we show how to select the three algorithm parameters to optimize this magnitude, and hence the asymptotic convergence rate. The obtained rate depends on the Friedrichs angle between the subspaces and is considerably better than known rates for other methods such as alternating projections and Douglas-Rachford splitting. We also present an adaptive scheme that, online, estimates the Friedrichs angle and updates the algorithm parameters based on this estimate. A numerical example is provided that supports our theoretical claims and shows very good performance for the adaptive method.Comment: 20 pages, extended version of article submitted to CD

Linear Convergence of the Douglas-Rachford Method for Two Closed Sets

In this paper, we investigate the Douglas-Rachford method for two closed (possibly nonconvex) sets in Euclidean spaces. We show that under certain regularity conditions, the Douglas-Rachford method converges locally with R-linear rate. In convex settings, we prove that the linear convergence is global. Our study recovers recent results on the same topic