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

Author

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  • Eyal Cohen

    (Tel-Aviv University)

  • Nadav Hallak

    (Faculty of Industrial Engineering and Management, The Technion)

  • Marc Teboulle

    (Tel-Aviv University)

Abstract

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.

Suggested Citation

  • Eyal Cohen & Nadav Hallak & Marc Teboulle, 2022. "A Dynamic Alternating Direction of Multipliers for Nonconvex Minimization with Nonlinear Functional Equality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 324-353, June.
  • Handle: RePEc:spr:joptap:v:193:y:2022:i:1:d:10.1007_s10957-021-01929-5
    DOI: 10.1007/s10957-021-01929-5
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    References listed on IDEAS

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    1. Jérôme Bolte & Shoham Sabach & Marc Teboulle, 2018. "Nonconvex Lagrangian-Based Optimization: Monitoring Schemes and Global Convergence," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1210-1232, November.
    2. Radu Ioan Bot & Dang-Khoa Nguyen, 2020. "The Proximal Alternating Direction Method of Multipliers in the Nonconvex Setting: Convergence Analysis and Rates," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 682-712, May.
    3. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
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    Cited by:

    1. Christian Kanzow & Patrick Mehlitz, 2022. "Convergence Properties of Monotone and Nonmonotone Proximal Gradient Methods Revisited," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 624-646, November.

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