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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... Full description

Journal Title: Mathematical Programming Series A, 2011-08-20, Vol.137 (1-2), p.91-129
Main Author: Attouch, Hedy
Other Authors: Bolte, Jérôme , Svaiter, Benar Fux
Format: Electronic Article Electronic Article
Language: English
Subjects:
o
Publisher: Berlin/Heidelberg: Springer-Verlag
ID: ISSN: 0025-5610
Link: https://hal.archives-ouvertes.fr/hal-00790042
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title: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods
format: Article
creator:
  • Attouch, Hedy
  • Bolte, Jérôme
  • Svaiter, Benar Fux
subjects:
  • Algebra
  • algebraic optimization
  • Algorithms
  • Alternating minimization
  • Analysis
  • backward splitting
  • Block
  • Calculus of Variations and Optimal Control
  • Optimization
  • Combinatorics
  • Control
  • Convergence
  • coordinate methods
  • Data smoothing
  • Descent
  • Descent methods
  • Forward
  • Full Length Paper
  • Gauss-Seidel method
  • Iterative thresholding
  • Kurdyka
  • Mathematical analysis
  • Mathematical and Computational Physics
  • Mathematical Methods in Physics
  • Mathematical programming
  • Mathematics
  • Mathematics and Statistics
  • Mathematics of Computing
  • Methods
  • minimal structures
  • Minimization
  • Nonconvex nonsmooth optimization
  • Numerical Analysis
  • o
  • Optimization
  • Optimization and Control
  • Proximal algorithms
  • Relative error
  • sadco
  • Semi
  • Splitting
  • Studies
  • Sufficient decrease
  • Tame optimization
  • Theoretical
  • Łojasiewicz inequality
ispartof: Mathematical Programming, Series A, 2011-08-20, Vol.137 (1-2), p.91-129
description: 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.
language: eng
source:
identifier: ISSN: 0025-5610
fulltext: no_fulltext
issn:
  • 0025-5610
  • 1436-4646
url: Link


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titleConvergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods
creatorAttouch, Hedy ; Bolte, Jérôme ; Svaiter, Benar Fux
creatorcontribAttouch, Hedy ; Bolte, Jérôme ; Svaiter, Benar Fux
descriptionIn 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.
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subjectAlgebra ; algebraic optimization ; Algorithms ; Alternating minimization ; Analysis ; backward splitting ; Block ; Calculus of Variations and Optimal Control; Optimization ; Combinatorics ; Control ; Convergence ; coordinate methods ; Data smoothing ; Descent ; Descent methods ; Forward ; Full Length Paper ; Gauss-Seidel method ; Iterative thresholding ; Kurdyka ; Mathematical analysis ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematical programming ; Mathematics ; Mathematics and Statistics ; Mathematics of Computing ; Methods ; minimal structures ; Minimization ; Nonconvex nonsmooth optimization ; Numerical Analysis ; o ; Optimization ; Optimization and Control ; Proximal algorithms ; Relative error ; sadco ; Semi ; Splitting ; Studies ; Sufficient decrease ; Tame optimization ; Theoretical ; Łojasiewicz inequality
ispartofMathematical Programming, Series A, 2011-08-20, Vol.137 (1-2), p.91-129
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descriptionIn 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.
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7Calculus of Variations and Optimal Control; Optimization
8Combinatorics
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10Convergence
11coordinate methods
12Data smoothing
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17Gauss-Seidel method
18Iterative thresholding
19Kurdyka
20Mathematical analysis
21Mathematical and Computational Physics
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25Mathematics and Statistics
26Mathematics of Computing
27Methods
28minimal structures
29Minimization
30Nonconvex nonsmooth optimization
31Numerical Analysis
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34Optimization and Control
35Proximal algorithms
36Relative error
37sadco
38Semi
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40Studies
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titleConvergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods
authorAttouch, Hedy ; Bolte, Jérôme ; Svaiter, Benar Fux
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41Sufficient decrease
42Tame optimization
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atitleConvergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods
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stitleMath. Program
date2011-08-20
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volume137
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abstractIn 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.
copBerlin/Heidelberg
pubSpringer-Verlag
doi10.1007/s10107-011-0484-9
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