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