An Adaptive Lagrangian-Based Scheme for Nonconvex Composite Optimization

Nadav Hallak, Marc Teboulle*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

This paper develops a novel adaptive, augmented, Lagrangian-based method to address the comprehensive class of nonsmooth, nonconvex models with a nonlinear, functional composite structure in the objective. The proposed method uses an adaptive mechanism for the update of the feasibility penalizing elements, essentially turning our multiplier type method into a simple alternating minimization procedure based on the augmented Lagrangian function from some iteration onward. This allows us to avoid the restrictive and, until now, mandatory surjectivity-type assumptions on the model. We establish the iteration complexity of the proposed scheme to reach an ε-critical point. Moreover, we prove that the limit point of every bounded sequence generated by a procedure that employs the method with strictly decreasing levels of precision is a critical point of the problem. Our approach provides novel results even in the simpler composite linear model, in which the surjectivity of the linear operator is a baseline assumption.

Original languageEnglish
Pages (from-to)2337-2352
Number of pages16
JournalMathematics of Operations Research
Volume48
Issue number4
DOIs
StatePublished - 2023

Funding

FundersFunder number
Israel Science Foundation2619-20, 637/21, 1844-16

    Keywords

    • alternating minimization
    • augmented Lagrangian-based methods
    • functional composite optimization
    • nonconvex and nonsmooth minimization
    • proximal multiplier method

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