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. We show for prototypical feasibility problems that circumcentered-reflection method (CRM), 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 spherical surrogates for the Lyapunov function. We study the more general class of operators that minimize such surrogates. In particular, we introduce a new 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. This makes possible the first primal/dual implementation of a method that seeks the center of spiraling iterates, which we describe, and provide a computed example (basis pursuit).
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