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A Dynamic Alternating Direction of Multipliers for Nonconvex Minimization with Nonlinear Functional Equality Constraints

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2022年第1-3期193卷 324-353页

作者： Cohen, Eyal Hallak, Nadav Teboulle, Marc Tel Aviv Univ Sch Math Sci Tel Aviv Israel Technion Fac Ind Engn & Management Haifa Israel

This paper studies the minimization of a broad class of nonsmooth nonconvex objective functions subject to nonlinear functional equality constraints, where the gradients of the differentiable parts in the objective and the constraints are only locally Lipschitz continuous. We propose a specific proximal linearized alternating direction method of multipliers in which the proximal parameter is generated dynamically, and we design an explicit and tractable backtracking procedure to generate it. We prove subsequent convergence of the method to a critical point of the problem, and global convergence when the problem's data are semialgebraic. These results are obtained with no dependency on the explicit manner in which the proximal parameter is generated. As a byproduct of our analysis, we also obtain global convergence guarantees for the proximal gradient method with a dynamic proximal parameter under local Lipschitz continuity of the gradient of the smooth part of the nonlinear sum composite minimization model.

关键词： augmented lagrangian-based methods Nonconvex and nonsmooth minimization Proximal gradient method Kurdyka-Lojasiewicz property Global convergence

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An Adaptive lagrangian-based Scheme for Nonconvex Composite Optimization

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MATHEMATICS OF OPERATIONS RESEARCH 2023年第4期48卷 2337-2352页

作者： Hallak, Nadav Teboulle, Marc Technion Fac Ind Engn & Management IL-3200003 Haifa Israel Tel Aviv Univ Sch Math Sci IL-69978 Ramat Aviv Israel

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 epsilon-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.

关键词： functional composite optimization augmented lagrangian-based methods nonconvex and nonsmooth minimization proximal multiplier method alternating minimization

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