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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Vydáno v:	Computational optimization and applications Ročník 88; číslo 1; s. 167 - 215
Hlavní autoři:	Maingé, Paul-Emile, Weng-Law, André
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.05.2024 Springer Nature B.V Springer Verlag
Témata:	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
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Abstract	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.
AbstractList	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 (xn), with worst-case rates of o(1/n) in terms of both the discrete velocity and the fixed point residual, instead of the rates of O(1/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. 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. 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 (xn), 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.
Author	Weng-Law, André Maingé, Paul-Emile
Author_xml	– sequence: 1 givenname: Paul-Emile surname: Maingé fullname: Maingé, Paul-Emile email: Paul-Emile.Mainge@univ-antilles.fr organization: F.W.I., MEMIAD, Université des Antilles – sequence: 2 givenname: André surname: Weng-Law fullname: Weng-Law, André organization: F.W.I., MEMIAD, Université des Antilles
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Keywords	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
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Snippet	In this paper, we develop rapidly convergent forward–backward algorithms for computing zeroes of the sum of two maximally monotone operators. A modification of... In this paper, we develop rapidly convergent forward-backward algorithms for computing zeroes of the sum of two maximally monotone operators. A modification of...
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StartPage	167
SubjectTerms	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
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Title	Accelerated forward–backward algorithms for structured monotone inclusions
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