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For many problems, some of which are reviewed in the paper, popularalgorithms like Douglas--Rachford (DR), ADMM, and FISTA produce approximatingsequences that show signs of spiraling toward the solution. We present ameta-algorithm that exploits such dynamics to potentially enhance performance.The strategy of this meta-algorithm is to iteratively build and minimizesurrogates for the Lyapunov function that captures those dynamics.  As a first motivating application, we show that for prototypical feasibilityproblems the circumcentered-reflection method (CRM), subgradient projections,and Newton--Raphson are all describable as gradient-based methods forminimizing Lyapunov functions constructed for DR operators, with the formerreturning the minimizers of spherical surrogates for the Lyapunov function.  As a second motivating application, we introduce a new method that sharesthese properties but with the added advantages that it: 1) does not rely onsubproblems (e.g. reflections) and so may be applied for any operator whoseiterates have the spiraling property; 2) provably has the aforementionedLyapunov properties with few structural assumptions and so is genericallysuitable for primal/dual implementation; and 3) maps spaces of reduceddimension into themselves whenever the original operator does. This makespossible the first primal/dual implementation of a method that seeks the centerof spiraling iterates. We describe this method, and provide a computed example(basis pursuit).

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