A convergent relaxation of the Douglas–Rachford algorithm

Abstract

This paper proposes an algorithm for solving structured optimization problems, which covers both the backward-backward and the Douglas-Rachford algorithms as special cases, and analyzes its convergence. The set of fixed points of the corresponding operator is characterized in several cases. Convergence criteria of the algorithm in terms of general fixed point iterations are established. When applied to nonconvex feasibility including potentially inconsistent problems, we prove local linear convergence results under mild assumptions on regularity of individual sets and of the collection of sets. In this special case, we refine known linear convergence criteria for the Douglas-Rachford (DR) algorithm. As a consequence, for feasibility problem with one of the sets being affine, we establish criteria for linear and sublinear convergence of convex combinations of the alternating projection and the DR methods. These results seem to be new. We also demonstrate the seemingly improved numerical performance of this algorithm compared to the RAAR algorithm for both consistent and inconsistent sparse feasibility problems.

Some New Characterizations of Intrinsic Transversality in Hilbert Spaces

“…A number of dual characterizations of transversality, especially in the Euclidean space setting, have been established [25][26][27]35,36,38,40] and applied, for example, [40,44,49,52]. The situation is very much different for subtransversality.…”

“…conditions for linear convergence of computational algorithms [3,14,16,35,40,41,43,49,52]. We refer the reader to the papers [25-27, 32-36, 38] by Kruger and his collaborators for a variety of their sufficient and/or necessary conditions in both primal and dual spaces.…”

Convex combination of alternating projection and Douglas–Rachford operators for phase retrieval

“…Inexact versions of RAAR were also proposed and analyzed in [36]. The DRAP algorithm [52] is another relaxation of DR:…”

“…1 Proposition 2 extends the applicable scope of this type of convergence results 2 to cover also phase retrieval problems with amplitude constraint. Second, applying the analysis scheme developed by Luke et al [42], we establish another convergence criterion for the DRAP algorithm (Theorem 2) by integrating the physical properties of the phase retrieval problem [36] into the earlier known results for DRAP [52]. Recall that the analysis of the latter article involves only abstract mathematical notions in the general setting of set feasibility.…”

“…As a result, this algorithm can not be naively applied to practical problems which intrinsically involve noise and model approximation. This fact has motivated a number of its efficient relaxation schemes such as the usage of the Krasnoselski-Mann relaxation, the Fienup's hybrid input-output (HIO) algorithm [15], the relaxed averaged alternating reflections (RAAR) algorithm [35,36], and the DRAP algorithm [52].…”