A generalization of linearized alternating direction method of multipliers for solving two-block separable convex programming

• Published in: Journal of computational and applied mathematics Vol. 357; pp. 251 - 272
• Main Authors: Chang, Xiaokai, Liu, Sanyang, Zhao, Pengjun, Song, Dunjiang
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2019
• Subjects: Alternating direction method of multipliers; Indefinite proximal regularization; LASSO; Linearization; Proximal point algorithm; Separable convex programming; Separable convex programming; Alternating direction method of multipliers; 90C30; 90C25; LASSO; 65K10; Proximal point algorithm; Indefinite proximal regularization; Linearization
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2019.02.028

The linearized alternating direction method of multipliers (ADMM), with indefinite proximal regularization, has been proved to be efficient for solving separable convex optimization subject to linear constraints. In this paper, we present a generalization of linearized ADMM (G-LADMM) to solve two-block separable convex minimization model, which linearizes all the subproblems by choosing a proper positive-definite or indefinite proximal term and updates the Lagrangian multiplier twice in different ways. Furthermore, the proposed G-LADMM can be expressed as a proximal point algorithm (PPA), and all the subproblems are just to estimate the proximity operator of the function in the objective. We specify the domain of the proximal parameter and stepsizes to guarantee that G-LADMM is globally convergent. It turns out that our convergence domain of the proximal parameter and stepsizes is significantly larger than other convergence domains in the literature. The numerical experiments illustrate the improvements of the proposed G-LADMM to solve LASSO and image decomposition problems.