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Bolte, J., Sabach, S., Teboulle, M. and Vaisbourd, T. (2018) First Order Methods beyond Convexity and Lipschitz Gradient Continuity with Applications to Quadratic Inverse Problems. SIAM Journal on Optimization, 28, 2131-2151.
https://doi.org/10.1137/17M1138558

has been cited by the following article:

• TITLE: Convergence of Bregman Alternating Direction Method of Multipliers for Nonseparable Nonconvex Objective with Linear Constraints

  AUTHORS: Xiaotong Zeng, Junping Yao, Haoming Xia

  KEYWORDS: Nonseparable Nonconvex Optimization, Bregman ADMM, Kurdyka-Lojasiewicz Inequality

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.12 No.2, February 29, 2024

  ABSTRACT: In this paper, our focus lies on addressing a two-block linearly constrained nonseparable nonconvex optimization problem with coupling terms. The most classical algorithm, the alternating direction method of multipliers (ADMM), is employed to solve such problems typically, which still requires the assumption of the gradient Lipschitz continuity condition on the objective function to ensure overall convergence from the current knowledge. However, many practical applications do not adhere to the conditions of smoothness. In this study, we justify the convergence of variant Bregman ADMM for the problem with coupling terms to circumvent the issue of the global Lipschitz continuity of the gradient. We demonstrate that the iterative sequence generated by our approach converges to a critical point of the issue when the corresponding function fulfills the Kurdyka-Lojasiewicz inequality and certain assumptions apply. In addition, we illustrate the convergence rate of the algorithm.