The proximal point algorithm in metric spaces
By Miroslav Bacák. 2012
234 citations recorded by the source corpus.
The proximal point algorithm, which is a well-known tool for finding minima of convex functions, is generalized from the classical Hilbert space framework into a nonlinear setting, namely, geodesic metric spaces of non-positive curvature. We prove that the sequence generated by the proximal point algorithm weakly converges to a minimizer, and also discuss a related question: convergence of the gradient flow.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/06e8e9ae8e1ec1fc2b99a2dc981ce8fa64bd6c45
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In Applied Mathematics
- Genaro López-Acedo
- David Ariza-Ruiz
- Sajad Ranjbar
- Hadi Khatibzadeh
- Laurentiu Leustean
- Miroslav Bacák
- Simeon Reich
- Ulrich Kohlenbach
In Computational Theory and Mathematics
- Genaro López-Acedo
- David Ariza-Ruiz
- Miroslav Bacák
- Sajad Ranjbar
- Hadi Khatibzadeh
- Laurentiu Leustean
- Simeon Reich
- Ulrich Kohlenbach
Citations in the corpus
- Firmly nonexpansive mappings in classes of geodesic spaces (2012, methodology): “Thus, Bačák obtained in the setting of CAT(0) spaces the following proximal point algorithm: if F has minimizers, then for all x0 ∈ X and all sequences (λn) divergent in sum, the sequence xn+1 := argmin y∈X ( F (y) + 1 2λn d(y, xn) 2 ) Δ-converges to a minimizer of F .”
- The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces (2014, background): “Gradient flow semigroups in Hadamard spaces have been studied by several authors [18, 21, 26, 4, 5, 6] and the theory can be extended to more general metric spaces [1].”
- A Coordinate Descent Primal-Dual Algorithm and Application to Distributed Asynchronous Optimization (2014, unknown): “2) Based on the idea of the stochastic coordinate descent who has been mainly studied in the literature in the special case of proximal gradient algorithms [5]–[7], we develop adistributed asynchronous version of ADMM+.”
- MONOTONE OPERATORS AND THE PROXIMAL POINT ALGORITHM IN COMPLETE CAT(0) METRIC SPACES (2016, methodology): “[4] Monotone operators and the proximal point algorithm in complete CAT(0) metric spaces 73”