An Adaptive Lagrangian-Based Scheme for Nonconvex Composite Optimization

Nadav Hallak () and Marc Teboulle ()
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Nadav Hallak: Faculty of Industrial Engineering and Management, The Technion, Haifa 3200003, Israel
Marc Teboulle: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv 69978, Israel

Mathematics of Operations Research, 2023, vol. 48, issue 4, 2337-2352

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.

Keywords: Primary: 90C30; 49M37; 65K10; functional composite optimization; augmented Lagrangian-based methods; nonconvex and nonsmooth minimization; proximal multiplier method; alternating minimization (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:48:y:2023:i:4:p:2337-2352

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