The Cyclic Douglas-Rachford Algorithm with r-sets-Douglas-Rachford Operators

The Douglas-Rachford (DR) algorithm is an iterative procedure that uses sequential reflections onto convex sets and which has become popular for convex feasibility problems. In this paper we propose a structural generalization that allows to use $r$-sets-DR operators in a cyclic fashion. We prove convergence and present numerical illustrations of the potential advantage of such operators with $r>2$ over the classical $2$-sets-DR operators in a cyclic algorithm.Comment: Accepted for publication in Optimization Methods and Software (OMS) July 17, 201

Computable Centering Methods for Spiraling Algorithms and their Duals, with Motivations from the theory of Lyapunov Functions

Splitting methods like Douglas--Rachford (DR), ADMM, and FISTA solve problems whose objectives are sums of functions that may be evaluated separately, and all frequently show signs of spiraling. Circumcentering reflection methods (CRMs) have been shown to obviate spiraling for DR for certain feasibility problems. Under conditions thought to typify local convergence for splitting methods, we first show that Lyapunov functions generically exist. We then show for prototypical feasibility problems that CRMs, subgradient projections, and Newton--Raphson are all describable as gradient-based methods for minimizing Lyapunov functions constructed for DR operators, with the former returning the minimizers of quadratic surrogates for the Lyapunov function. Motivated thereby, we introduce a centering method that shares these properties but with the added advantages that it: 1) does not rely on subproblems (e.g. reflections) and so may be applied for any operator whose iterates spiral; 2) provably has the aforementioned Lyapunov properties with few structural assumptions and so is generically suitable for primal/dual implementation; and 3) maps spaces of reduced dimension into themselves whenever the original operator does. We then introduce a general approach to primal/dual implementation of a centering method and provide a computed example (basis pursuit), the first such application of centering. The new centering operator we introduce works well, while a similar primal/dual adaptation of CRM fails to solve the problem, for reasons we explain

The Douglas–Rachford algorithm for a hyperplane and a doubleton

The Douglas–Rachford algorithm is a popular algorithm for solving both convex and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent case, the general nonconvex inconsistent case is far from being fully understood. In this paper, we focus on the most simple nonconvex inconsistent case: when one set is a hyperplane and the other a doubleton (i.e., a two-point set). We present a characterization of cycling in this case which—somewhat surprisingly—depends on whether the ratio of the distance of the points to the hyperplane is rational or not. Furthermore, we provide closed-form expressions as well as several concrete examples which illustrate the dynamical richness of this algorithm