Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

In view of the minimization of a nonsmooth nonconvex function f , we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that th...

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Vydané v:	Mathematical programming Ročník 137; číslo 1-2; s. 91 - 129
Hlavní autori:	Attouch, Hedy, Bolte, Jérôme, Svaiter, Benar Fux
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer-Verlag 01.02.2013 Springer Nature B.V
Predmet:	Algebra Algorithms Analysis Calculus of Variations and Optimal Control; Optimization Combinatorics Convergence Data smoothing Descent Full Length Paper Gauss-Seidel method Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Methods Minimization Numerical Analysis Optimization Splitting Studies Theoretical Tame optimization 65K15 o-minimal structures Nonconvex nonsmooth optimization Alternating minimization 34G25 Forward–backward splitting Descent methods 90C53 Semi-algebraic optimization Iterative thresholding 47J30 90C25 Proximal algorithms Sufficient decrease Block-coordinate methods Relative error 49M15 49M37 47J25 Kurdyka–Łojasiewicz inequality 47J35
ISSN:	0025-5610, 1436-4646
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Popis
Shrnutí:	In view of the minimization of a nonsmooth nonconvex function f , we prove an abstract convergence result for descent methods satisfying a sufficient-decrease 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.
Bibliografia:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-1 ObjectType-Feature-2 content type line 23
ISSN:	0025-5610 1436-4646
DOI:	10.1007/s10107-011-0484-9