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. 1