Accelerated forward–backward algorithms for structured monotone inclusions

In this paper, we develop rapidly convergent forward–backward algorithms for computing zeroes of the sum of two maximally monotone operators. A modification of the classical forward–backward method is considered, by incorporating an inertial term (closed to the acceleration techniques introduced by...

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Bibliographic Details
Published in:Computational optimization and applications Vol. 88; no. 1; pp. 167 - 215
Main Authors: Maingé, Paul-Emile, Weng-Law, André
Format: Journal Article
Language:English
Published: New York Springer US 01.05.2024
Springer Nature B.V
Springer Verlag
Subjects:
Acceleration
Algorithms
Convergence
Convex and Discrete Geometry
Hilbert space
Inclusions
Management Science
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Preconditioning
Saddle points
Statistics
Optimal gradient method
Inertial-type algorithm
Nesterov-type algorithm
Restarting techniques
Global rate of convergence
90C25
65F22
90C06
Fast first-order method
global rate of convergence
restarting techniques
fast first-order method
inertial-type algorithm
optimal gradient method
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:In this paper, we develop rapidly convergent forward–backward algorithms for computing zeroes of the sum of two maximally monotone operators. A modification of the classical forward–backward method is considered, by incorporating an inertial term (closed to the acceleration techniques introduced by Nesterov), a constant relaxation factor and a correction term, along with a preconditioning process. In a Hilbert space setting, we prove the weak convergence to equilibria of the iterates ( x n ) , with worst-case rates of o ( n - 1 ) in terms of both the discrete velocity and the fixed point residual, instead of the rates of O ( n - 1 / 2 ) classically established for related algorithms. Our procedure can be also adapted to more general monotone inclusions. In particular, we propose a fast primal-dual algorithmic solution to some class of convex-concave saddle point problems. In addition, we provide a well-adapted framework for solving this class of problems by means of standard proximal-like algorithms dedicated to structured monotone inclusions. Numerical experiments are also performed so as to enlighten the efficiency of the proposed strategy.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-023-00547-3