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

Application of projection algorithms to differential equations: boundary value problems

The Douglas-Rachford method has been employed successfully to solve many kinds of non-convex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary valued problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well-suited to parallelization. We explore the stability of the method by applying it to several examples of BVPs, including cases where the traditional Newton's method fails

The Douglas–Rachford algorithm for convex and nonconvex feasibility problems

The Douglas–Rachford algorithm is an optimization method that can be used for solving feasibility problems. To apply the method, it is necessary that the problem at hand is prescribed in terms of constraint sets having efficiently computable nearest points. Although the convergence of the algorithm is guaranteed in the convex setting, the scheme has demonstrated to be a successful heuristic for solving combinatorial problems of different type. In this self-contained tutorial, we develop the convergence theory of projection algorithms within the framework of fixed point iterations, explain how to devise useful feasibility problem formulations, and demonstrate the application of the Douglas–Rachford method to said formulations. The paradigm is then illustrated on two concrete problems: a generalization of the “eight queens puzzle” known as the “(m, n)-queens problem”, and the problem of constructing a probability distribution with prescribed moments.FJAA and RC were partially supported by Ministerio de Economía, Industria y Competitividad (MINECO) and European Regional Development Fund (ERDF), grant MTM2014-59179-C2-1-P. FJAA was supported by the Ramón y Cajal program by MINECO and ERDF (RYC-2013-13327) and by the Ministerio de Ciencia, Innovación y Universidades and ERDF, grant PGC2018-097960-B-C22. RC was supported by MINECO and European Social Fund (BES-2015-073360) under the program “Ayudas para contratos predoctorales para la formación de doctores 2015”