Linearized generalized ADMM-based algorithm for multi-block linearly constrained separable convex programming in real-world applications

We study a multi-block separable convex optimization problem with the linear constraints, where the objective function is the sum of m individual convex functions without overlapping variables. The linearized version of the generalized alternating direction method of multipliers (L-GADMM) is particularly efficient for the two-block separable convex programming problem and its convergence was proved when two blocks of variables are alternatively updated. However, the convergence analysis and practical applications of the extended L-GADMM (m >= 3) are in their early stages. In this paper, we extend this algorithm to the general case where the objective function consists of the sum of m-block convex functions. Theoretically, we prove global convergence of the new method and establish the worst-case convergence rate in the ergodic and nonergodic senses for the proposed algorithm. The efficiency of the new method is further demonstrated through numerical results on the calibration of the correlation matrices.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper studies a multi-block separable convex optimization problem where the objective function is the sum of individual convex functions without overlapping variables. The linearized version of the generalized alternating direction method of multipliers (L-GADMM) has been proven to be efficient for two-block separable convex programming problems, and its convergence has been analyzed. However, the analysis and applications of the extended L-GADMM (m >= 3) are still in their early stages. In this paper, the algorithm is extended to the general case, and global convergence and convergence rates are theoretically established. The efficiency of the method is demonstrated through numerical results.