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Fast alternating linearization methods for minimizing the sum of two convex functions

We present in this paper alternating linearization algorithms based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most iterations to obtain an -optimal solution, while our accelerated (i.e., fast) versions of th... Full description

Journal Title: Mathematical programming 2012-03-24, Vol.141 (1-2), p.349-382
Main Author: Goldfarb, Donald
Other Authors: Ma, Shiqian , Scheinberg, Katya
Format: Electronic Article Electronic Article
Language: English
Subjects:
Publisher: Berlin/Heidelberg: Springer Berlin Heidelberg
ID: ISSN: 0025-5610
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title: Fast alternating linearization methods for minimizing the sum of two convex functions
format: Article
creator:
  • Goldfarb, Donald
  • Ma, Shiqian
  • Scheinberg, Katya
subjects:
  • Algorithms
  • Calculus of Variations and Optimal Control
  • Optimization
  • Combinatorics
  • Computation
  • Convex analysis
  • Full Length Paper
  • Iterative methods
  • Lagrange multiplier
  • Linearization
  • Management science
  • Mathematical analysis
  • Mathematical and Computational Physics
  • Mathematical Methods in Physics
  • Mathematical models
  • Mathematics
  • Mathematics and Statistics
  • Mathematics of Computing
  • Methods
  • Numerical Analysis
  • Optimization
  • Principal components analysis
  • Splitting
  • Studies
  • Theoretical
ispartof: Mathematical programming, 2012-03-24, Vol.141 (1-2), p.349-382
description: We present in this paper alternating linearization algorithms based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most iterations to obtain an -optimal solution, while our accelerated (i.e., fast) versions of them require at most iterations, with little change in the computational effort required at each iteration. For both types of methods, we present one algorithm that requires both functions to be smooth with Lipschitz continuous gradients and one algorithm that needs only one of the functions to be so. Algorithms in this paper are Gauss-Seidel type methods, in contrast to the ones proposed by Goldfarb and Ma in (Fast multiple splitting algorithms for convex optimization, Columbia University, 2009 ) where the algorithms are Jacobi type methods. Numerical results are reported to support our theoretical conclusions and demonstrate the practical potential of our algorithms.
language: eng
source:
identifier: ISSN: 0025-5610
fulltext: no_fulltext
issn:
  • 0025-5610
  • 1436-4646
url: Link


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descriptionWe present in this paper alternating linearization algorithms based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most iterations to obtain an -optimal solution, while our accelerated (i.e., fast) versions of them require at most iterations, with little change in the computational effort required at each iteration. For both types of methods, we present one algorithm that requires both functions to be smooth with Lipschitz continuous gradients and one algorithm that needs only one of the functions to be so. Algorithms in this paper are Gauss-Seidel type methods, in contrast to the ones proposed by Goldfarb and Ma in (Fast multiple splitting algorithms for convex optimization, Columbia University, 2009 ) where the algorithms are Jacobi type methods. Numerical results are reported to support our theoretical conclusions and demonstrate the practical potential of our algorithms.
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subjectAlgorithms ; Calculus of Variations and Optimal Control; Optimization ; Combinatorics ; Computation ; Convex analysis ; Full Length Paper ; Iterative methods ; Lagrange multiplier ; Linearization ; Management science ; Mathematical analysis ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematical models ; Mathematics ; Mathematics and Statistics ; Mathematics of Computing ; Methods ; Numerical Analysis ; Optimization ; Principal components analysis ; Splitting ; Studies ; Theoretical
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descriptionWe present in this paper alternating linearization algorithms based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most iterations to obtain an -optimal solution, while our accelerated (i.e., fast) versions of them require at most iterations, with little change in the computational effort required at each iteration. For both types of methods, we present one algorithm that requires both functions to be smooth with Lipschitz continuous gradients and one algorithm that needs only one of the functions to be so. Algorithms in this paper are Gauss-Seidel type methods, in contrast to the ones proposed by Goldfarb and Ma in (Fast multiple splitting algorithms for convex optimization, Columbia University, 2009 ) where the algorithms are Jacobi type methods. Numerical results are reported to support our theoretical conclusions and demonstrate the practical potential of our algorithms.
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abstractWe present in this paper alternating linearization algorithms based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most iterations to obtain an -optimal solution, while our accelerated (i.e., fast) versions of them require at most iterations, with little change in the computational effort required at each iteration. For both types of methods, we present one algorithm that requires both functions to be smooth with Lipschitz continuous gradients and one algorithm that needs only one of the functions to be so. Algorithms in this paper are Gauss-Seidel type methods, in contrast to the ones proposed by Goldfarb and Ma in (Fast multiple splitting algorithms for convex optimization, Columbia University, 2009 ) where the algorithms are Jacobi type methods. Numerical results are reported to support our theoretical conclusions and demonstrate the practical potential of our algorithms.
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