A Cyclic Douglas-Rachford Iteration Scheme

In this paper we present two Douglas-Rachford inspired iteration schemes which can be applied directly to N-set convex feasibility problems in Hilbert space. Our main results are weak convergence of the methods to a point whose nearest point projections onto each of the N sets coincide. For affine subspaces, convergence is in norm. Initial results from numerical experiments, comparing our methods to the classical (product-space) Douglas-Rachford scheme, are promising.Comment: 22 pages, 7 figures, 4 table

The Cyclic Douglas-Rachford Method for Inconsistent Feasibility Problems

We analyse the behaviour of the newly introduced cyclic Douglas-Rachford algorithm for finding a point in the intersection of a finite number of closed convex sets. This work considers the case in which the target intersection set is possibly empty.Comment: 13 pages, 2 figures; references updated, figure 2 correcte

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

Global Behavior of the Douglas-Rachford Method for a Nonconvex Feasibility Problem

In recent times the Douglas-Rachford algorithm has been observed empirically to solve a variety of nonconvex feasibility problems including those of a combinatorial nature. For many of these problems current theory is not sufficient to explain this observed success and is mainly concerned with questions of local convergence. In this paper we analyze global behavior of the method for finding a point in the intersection of a half-space and a potentially non-convex set which is assumed to satisfy a well-quasi-ordering property or a property weaker than compactness. In particular, the special case in which the second set is finite is covered by our framework and provides a prototypical setting for combinatorial optimization problems