New Douglas-Rachford algorithmic structures and their convergence analyses

TL;DR

This paper introduces new algorithmic structures based on Douglas-Rachford operators, embedding them into String-Averaging and Block-Iterative frameworks, and analyzes their convergence for convex feasibility problems.

Contribution

It proposes novel DR-based algorithms within SAP and BIP frameworks, including a new multiple-set-DR operator, with convergence analysis using nonexpansive operator properties.

Findings

New DR algorithmic schemes including cyclic and averaged variants

Convergence established for all proposed schemes

Introduction of a multiple-set-DR operator

Abstract

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.