Distributed Primal-Dual Method for Convex Optimization With Coupled Constraints

• Published in: IEEE transactions on signal processing Vol. 70; pp. 523 - 535
• Main Authors: Su, Yanxu, Wang, Qingling, Sun, Changyin
• Format: Journal Article
• Language: English
• Published: New York IEEE 2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Computational geometry; consensus; Constraints; Convergence; Convex analysis; Convex functions; Convexity; distributed constrained optimization; Iterative methods; Linear programming; non-ergodic convergence rate; Optimization; Peer-to-peer computing; Primal-dual algorithm; Signal processing algorithms; Sun
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2021.3123888

Distributed primal-dual methods have been widely used for solving large-scale constrained optimization problems. The majority of existing results focus on the problems with decoupled constraints. Some recent works have studied the problems subject to separable globally coupled constraints. This paper considers the distributed optimization problems with globally coupled constraints over networks without requiring the separability of the globally coupled constraints. This is made possible by the local estimates of the constraint violations. For solving such a problem, we propose a primal-dual algorithm in the augmented Lagrangian framework, combining the average consensus technique. We first establish a non-ergodic convergence rate of <inline-formula><tex-math notation="LaTeX">\mathcal {O}(1/k)</tex-math></inline-formula> in terms of the objective residual for solving a distributed constrained convex optimization problem, where <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula> is the iteration counter. Specifically, the global objective function is the aggregate of the local convex and possibly non-smooth costs, and the coupled constraint is the sum of the local linear equality constraints. The numerical results illustrate the performance of the proposed method.