Abstract
AbstractFor many problems, some of which are reviewed in the paper, popular algorithms like Douglas–Rachford (DR), ADMM, and FISTA produce approximating sequences that show signs of spiraling toward the solution. We present a meta-algorithm that exploits such dynamics to potentially enhance performance. The strategy of this meta-algorithm is to iteratively build and minimize surrogates for the Lyapunov function that captures those dynamics. As a first motivating application, we show that for prototypical feasibility problems the circumcentered-reflection method, 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. As a second motivating application, 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 have the spiraling property; (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. We describe this method, and provide a computed example (basis pursuit).
Funder
Hong Kong Research Grants Council
Australian Mathematical Society
Curtin University
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Control and Optimization
Cited by
2 articles.
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