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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Vydáno v:	Mathematical programming Ročník 182; číslo 1-2; s. 1 - 36
Hlavní autoři:	Bolte, Jérôme, Chen, Zheng, Pauwels, Edouard
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2020 Springer Nature B.V Springer Verlag
Témata:	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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Popis
Shrnutí:	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.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0025-5610 1436-4646
DOI:	10.1007/s10107-019-01382-3