New Douglas-Rachford algorithmic structures and their convergence analyses

In this paper we study new algorithmic structures with Douglas- Rachford (DR) operators to solve convex feasibility problems. We propose to embed the basic two-set-DR algorithmic operator into the String-Averaging Projections (SAP) and into the Block-Iterative Pro- jection (BIP) algorithmic structures, thereby creating new DR algo- rithmic schemes that include the recently proposed cyclic Douglas- Rachford algorithm and the averaged DR algorithm as special cases. We further propose and investigate a new multiple-set-DR algorithmic operator. Convergence of all these algorithmic schemes is studied by using properties of strongly quasi-nonexpansive operators and firmly nonexpansive operators.Comment: SIAM Journal on Optimization, accepted for publicatio

Alternating Projections and Douglas-Rachford for Sparse Affine Feasibility

The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved nonconvex relaxations. In this work we consider elementary methods based on projections for solving a sparse feasibility problem without employing convex heuristics. In a recent paper Bauschke, Luke, Phan and Wang (2014) showed that, locally, the fundamental method of alternating projections must converge linearly to a solution to the sparse feasibility problem with an affine constraint. In this paper we apply different analytical tools that allow us to show global linear convergence of alternating projections under familiar constraint qualifications. These analytical tools can also be applied to other algorithms. This is demonstrated with the prominent Douglas-Rachford algorithm where we establish local linear convergence of this method applied to the sparse affine feasibility problem.Comment: 29 pages, 2 figures, 37 references. Much expanded version from last submission. Title changed to reflect new development

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

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