Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods
Résumé
In view of the minimization of a nonsmooth nonconvex function f , we prove an abstract convergence result for descent methods satisfying a sufficientdecrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka-Łojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward-backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss-Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.
Mots clés
Semi-algebraic optimization
Tame optimization
Kurdyka-Łojasiewicz inequality
Descent methods
Relative error
Sufficient decrease
Forward-backward splitting
Alternating minimization
Proximal algorithms
Iterative thresholding
Block-coordinate methods
o-minimal structures
Nonconvex nonsmooth optimization
Domaines
Optimisation et contrôle [math.OC]
Origine : Fichiers produits par l'(les) auteur(s)
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