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Asymptotic Convergence Analysis of the Forward-Backward Splitting Algorithm

Author:

Ciyou ZhuAuthors Info & Claims

Mathematics of Operations Research, Volume 20, Issue 2

Pages 449 - 464

Published: 01 May 1995 Publication History

Abstract

The asymptotic convergence of the forward-backward splitting algorithm for solving equations of type 0 ∈ Tz is analyzed, where T is a multivalued maximal monotone operator in the n-dimensional Euclidean space. When the problem has a nonempty solution set, and T is split in the form T = F + h with F being maximal monotone and h being co-coercive with modulus greater than ½, convergence rates are shown, under mild conditions, to be linear, superlinear or sublinear depending on how rapidly F^-1 and h^-1 grow in the neighborhoods of certain specific points.

As a special case, when both F and h are polyhedral functions, we get R-linear convergence and 2-step Q-linear convergence without any further assumptions on the strict monotonicity on T or on the uniqueness of the solution. As another special case when h = 0, the splitting algorithm reduces to the proximal point algorithm, and we get new results, which complement R. T. Rockafellar's and F. J. Luque's earlier results on the proximal point algorithm.

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Cited By

Vuong PStrodiot J(2018)The Glowinski---Le Tallec splitting method revisited in the framework of equilibrium problems in Hilbert spacesJournal of Global Optimization10.1007/s10898-017-0575-070:2(477-495)Online publication date: 1-Feb-2018
https://dl.acm.org/doi/10.1007/s10898-017-0575-0
Haubruge SNguyen VStrodiot J(1998)Convergence Analysis and Applications of the Glowinski---Le Tallec Splitting Method for Finding a Zero of the Sum of Two Maximal Monotone OperatorsJournal of Optimization Theory and Applications10.5555/3182671.318290197:3(645-673)Online publication date: 1-Jun-1998
https://dl.acm.org/doi/10.5555/3182671.3182901

Index Terms

Asymptotic Convergence Analysis of the Forward-Backward Splitting Algorithm
1. Mathematics of computing
  1. Discrete mathematics
  2. Mathematical software
2. Theory of computation
  1. Randomness, geometry and discrete structures

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cover image Mathematics of Operations Research

Mathematics of Operations Research Volume 20, Issue 2

May 1995

256 pages

ISSN:0364-765X

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INFORMS

Linthicum, MD, United States

Publication History

Published: 01 May 1995

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Cited By

Vuong PStrodiot J(2018)The Glowinski---Le Tallec splitting method revisited in the framework of equilibrium problems in Hilbert spacesJournal of Global Optimization10.1007/s10898-017-0575-070:2(477-495)Online publication date: 1-Feb-2018
https://dl.acm.org/doi/10.1007/s10898-017-0575-0
Haubruge SNguyen VStrodiot J(1998)Convergence Analysis and Applications of the Glowinski---Le Tallec Splitting Method for Finding a Zero of the Sum of Two Maximal Monotone OperatorsJournal of Optimization Theory and Applications10.5555/3182671.318290197:3(645-673)Online publication date: 1-Jun-1998
https://dl.acm.org/doi/10.5555/3182671.3182901

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