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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Bibliographic Details
Published in:Mathematical programming Vol. 182; no. 1-2; pp. 1 - 36
Main Authors: Bolte, Jérôme, Chen, Zheng, Pauwels, Edouard
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2020
Springer Nature B.V
Springer Verlag
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-019-01382-3