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Proximal alternating linearized minimization for nonconvex and nonsmooth problems

We introduce a proximal alternating linearized minimization (PALM) algorithm for solving a broad class of nonconvex and nonsmooth minimization problems. Building on the powerful Kurdyka–Łojasiewicz property, we derive a self-contained convergence analysis framework and establish that each bounded se... Full description

Journal Title: Mathematical programming 2013, Vol.146 (1-2), p.459-494
Main Author: Bolte, Jérôme
Other Authors: Sabach, Shoham , Teboulle, Marc
Format: Electronic Article Electronic Article
Language: English
Subjects:
Publisher: Berlin/Heidelberg: Springer Berlin Heidelberg
ID: ISSN: 0025-5610
Link: https://hal.inria.fr/hal-00916090
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recordid: cdi_hal_primary_oai_HAL_hal_00916090v1
title: Proximal alternating linearized minimization for nonconvex and nonsmooth problems
format: Article
creator:
  • Bolte, Jérôme
  • Sabach, Shoham
  • Teboulle, Marc
subjects:
  • Algorithms
  • Analysis
  • Byproducts
  • Calculus of Variations and Optimal Control
  • Optimization
  • Combinatorics
  • Construction
  • Control
  • Data smoothing
  • Full Length Paper
  • Functions (mathematics)
  • Management science
  • Mathematical analysis
  • Mathematical and Computational Physics
  • Mathematical Methods in Physics
  • Mathematical programming
  • Mathematics
  • Mathematics and Statistics
  • Mathematics of Computing
  • Minimization
  • Numerical Analysis
  • Optimization
  • Optimization and Control
  • Palm
  • Studies
  • Theoretical
ispartof: Mathematical programming, 2013, Vol.146 (1-2), p.459-494
description: We introduce a proximal alternating linearized minimization (PALM) algorithm for solving a broad class of nonconvex and nonsmooth minimization problems. Building on the powerful Kurdyka–Łojasiewicz property, we derive a self-contained convergence analysis framework and establish that each bounded sequence generated by PALM globally converges to a critical point. Our approach allows to analyze various classes of nonconvex-nonsmooth problems and related nonconvex proximal forward–backward algorithms with semi-algebraic problem’s data, the later property being shared by many functions arising in a wide variety of fundamental applications. A by-product of our framework also shows that our results are new even in the convex setting. As an illustration of the results, we derive a new and simple globally convergent algorithm for solving the sparse nonnegative matrix factorization problem.
language: eng
source:
identifier: ISSN: 0025-5610
fulltext: no_fulltext
issn:
  • 0025-5610
  • 1436-4646
url: Link


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descriptionWe introduce a proximal alternating linearized minimization (PALM) algorithm for solving a broad class of nonconvex and nonsmooth minimization problems. Building on the powerful Kurdyka–Łojasiewicz property, we derive a self-contained convergence analysis framework and establish that each bounded sequence generated by PALM globally converges to a critical point. Our approach allows to analyze various classes of nonconvex-nonsmooth problems and related nonconvex proximal forward–backward algorithms with semi-algebraic problem’s data, the later property being shared by many functions arising in a wide variety of fundamental applications. A by-product of our framework also shows that our results are new even in the convex setting. As an illustration of the results, we derive a new and simple globally convergent algorithm for solving the sparse nonnegative matrix factorization problem.
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subjectAlgorithms ; Analysis ; Byproducts ; Calculus of Variations and Optimal Control; Optimization ; Combinatorics ; Construction ; Control ; Data smoothing ; Full Length Paper ; Functions (mathematics) ; Management science ; Mathematical analysis ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematical programming ; Mathematics ; Mathematics and Statistics ; Mathematics of Computing ; Minimization ; Numerical Analysis ; Optimization ; Optimization and Control ; Palm ; Studies ; Theoretical
ispartofMathematical programming, 2013, Vol.146 (1-2), p.459-494
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abstractWe introduce a proximal alternating linearized minimization (PALM) algorithm for solving a broad class of nonconvex and nonsmooth minimization problems. Building on the powerful Kurdyka–Łojasiewicz property, we derive a self-contained convergence analysis framework and establish that each bounded sequence generated by PALM globally converges to a critical point. Our approach allows to analyze various classes of nonconvex-nonsmooth problems and related nonconvex proximal forward–backward algorithms with semi-algebraic problem’s data, the later property being shared by many functions arising in a wide variety of fundamental applications. A by-product of our framework also shows that our results are new even in the convex setting. As an illustration of the results, we derive a new and simple globally convergent algorithm for solving the sparse nonnegative matrix factorization problem.
copBerlin/Heidelberg
pubSpringer Berlin Heidelberg
doi10.1007/s10107-013-0701-9
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