On Local Convergence of the Method of Alternating Projections

Abstract

Abstract. The method of alternating projections is a classical tool to solve feasibility problems. Here we prove local convergence of alternating projections between subanalytic sets A, B under a mild regularity hypothesis on one of the sets. We show that the speed of convergence is O(k −ρ ) for some ρ ∈ (0, ∞).

Transversality and Alternating Projections for Nonconvex Sets

“…It does not cover classically well-understood cases like two distinct intersecting lines in R 3 , a gap discussed in some recent work [1,4,5,17,34].…”

“…We first discuss its provenance. A new and general framework for analyzing the convergence of nonconvex alternating projections (using properties of short sequences of iterates) was submitted for publication and disseminated in December 2013-see [32]. That manuscript did not show that transversality implies linear convergence (Theorem 2.1 above).…”

“…Some relationships may be found in [14], and extensive discussions appear in [34] and [24,25]. Our focus here is rather a clear and complete presentation of the new and subtle geometric argument opened up by the idea of intrinsic transversality, proving linear convergence.…”

“…We remark that the technique we present here applies to much broader classes of setsthose "definable in an o-minimal structure" [11]. Under a further regularity condition, [34,Corollary 9] proves convergence of the full alternating projection sequence in the semi-algebraic (or, more broadly, subanalytic) setting.…”

A convergent relaxation of the Douglas–Rachford algorithm

“…As a result, the question about convergence of a solving method can often be reduced to checking whether certain regularity properties of the problem data are satisfied. There have been a considerable number of papers studying these two ingredients of convergence analysis in order to establish sharper convergence criteria in various circumstances, especially those applicable to algorithms for solving nonconvex problems [5,12,13,19,26,27,[31][32][33]38,42,45]. This paper suggests an algorithm called T λ , which covers both the backwardbackward and the DR algorithms as special cases of choosing the parameter λ ∈ [0, 1], and analyzes its convergence.…”