The multiproximal linearization method for convex composite problems

Composite minimization involves a collection of smooth functions which are aggregated in a nonsmooth manner. In the convex setting, we design an algorithm by linearizing each smooth component in accordance with its main curvature. The resulting method, called the Multiprox method, consists in solvin...

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Veröffentlicht in:	Mathematical programming Jg. 182; H. 1-2; S. 1 - 36
Hauptverfasser:	Bolte, Jérôme, Chen, Zheng, Pauwels, Edouard
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2020 Springer Nature B.V Springer Verlag
Schlagworte:	Algorithms Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Complexity Constraints Curvature Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Methods Numerical Analysis Operators (mathematics) Optimization Optimization and Control Theoretical 90C30 Convex optimization 90C25 90C47 Composite optimization First order methods Proximal Gauss–Newton’s method Prox-linear method Complexity
ISSN:	0025-5610, 1436-4646
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Abstract	Composite minimization involves a collection of smooth functions which are aggregated in a nonsmooth manner. In the convex setting, we design an algorithm by linearizing each smooth component in accordance with its main curvature. The resulting method, called the Multiprox method, consists in solving successively simple problems (e.g., constrained quadratic problems) which can also feature some proximal operators. To study the complexity and the convergence of this method, we are led to prove a new type of qualification condition and to understand the impact of multipliers on the complexity bounds. We obtain explicit complexity results of the form O ( 1 k ) involving new types of constant terms. A distinctive feature of our approach is to be able to cope with oracles involving moving constraints. Our method is flexible enough to include the moving balls method, the proximal Gauss–Newton’s method, or the forward–backward splitting, for which we recover known complexity results or establish new ones. We show through several numerical experiments how the use of multiple proximal terms can be decisive for problems with complex geometries.
AbstractList	Composite minimization involves a collection of smooth functions which are aggregated in a nonsmooth manner. In the convex setting, we design an algorithm by linearizing each smooth component in accordance with its main curvature. The resulting method, called the Multiprox method, consists in solving successively simple problems (e.g., constrained quadratic problems) which can also feature some proximal operators. To study the complexity and the convergence of this method, we are led to prove a new type of qualification condition and to understand the impact of multipliers on the complexity bounds. We obtain explicit complexity results of the form O(1k) involving new types of constant terms. A distinctive feature of our approach is to be able to cope with oracles involving moving constraints. Our method is flexible enough to include the moving balls method, the proximal Gauss–Newton’s method, or the forward–backward splitting, for which we recover known complexity results or establish new ones. We show through several numerical experiments how the use of multiple proximal terms can be decisive for problems with complex geometries. Composite minimization involves a collection of smooth functions which are aggregated in a nonsmooth manner. In the convex setting, we design an algorithm by linearizing each smooth component in accordance with its main curvature. The resulting method, called the Multiprox method, consists in solving successively simple problems (e.g., constrained quadratic problems) which can also feature some proximal operators. To study the complexity and the convergence of this method, we are led to prove a new type of qualification condition and to understand the impact of multipliers on the complexity bounds. We obtain explicit complexity results of the form O ( 1 k ) involving new types of constant terms. A distinctive feature of our approach is to be able to cope with oracles involving moving constraints. Our method is flexible enough to include the moving balls method, the proximal Gauss–Newton’s method, or the forward–backward splitting, for which we recover known complexity results or establish new ones. We show through several numerical experiments how the use of multiple proximal terms can be decisive for problems with complex geometries. Composite minimization involves a collection of smooth functions which are aggregated in a nonsmooth manner. In the convex setting, we design an algorithm by linearizing each smooth component in accordance with its main curvature. The resulting method, called the Multiprox method, consists in solving successively simple problems (e.g. constrained quadratic problems) which can also feature some proximal operators. To study the complexity and the convergence of this method we are led to prove a new type of qualification condition and to understand the impact of multipliers on the complexity bounds. We obtain explicit complexity results of the form $O(\frac{1}{k})$ involving new types of constant terms. A distinctive feature of our approach is to be able to cope with oracles involving moving constraints. Our method is flexible enough to include the moving balls method, the proximal Gauss-Newton's method, or the forward-backward splitting, for which we recover known complexity results or establish new ones. We show through several numerical experiments how the use of multiple proximal terms can be decisive for problems with complex geometries.
Author	Bolte, Jérôme Chen, Zheng Pauwels, Edouard
Author_xml	– sequence: 1 givenname: Jérôme surname: Bolte fullname: Bolte, Jérôme organization: Toulouse School of Economics, Université Toulouse Capitole – sequence: 2 givenname: Zheng orcidid: 0000-0002-3573-1546 surname: Chen fullname: Chen, Zheng email: z_chen@zju.edu.cn organization: School of Aeronautics and Astronautics, Zhejiang University – sequence: 3 givenname: Edouard surname: Pauwels fullname: Pauwels, Edouard organization: IRIT, Université Paul Sabatier
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Copyright	Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019 Mathematical Programming is a copyright of Springer, (2019). All Rights Reserved. Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019. Distributed under a Creative Commons Attribution 4.0 International License
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Keywords	90C30 Convex optimization 90C25 90C47 Composite optimization First order methods Proximal Gauss–Newton’s method Prox-linear method Complexity
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Snippet	Composite minimization involves a collection of smooth functions which are aggregated in a nonsmooth manner. In the convex setting, we design an algorithm by...
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SubjectTerms	Algorithms Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Complexity Constraints Curvature Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Methods Numerical Analysis Operators (mathematics) Optimization Optimization and Control Theoretical
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Title	The multiproximal linearization method for convex composite problems
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