Decentralized proximal splitting algorithms for composite constrained convex optimization

• Published in: Journal of the Franklin Institute Vol. 359; no. 14; pp. 7482 - 7509
• Main Authors: Zheng, Lifeng, Ran, Liang, Li, Huaqing, Feng, Liping, Wang, Zheng, Lü, Qingguo, Xia, Dawen
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.09.2022
• ISSN: 0016-0032; 1879-2693
• DOI: 10.1016/j.jfranklin.2022.07.053

•Consider a class of decentralized convex optimization problems with local feasible sets, equality and inequality constraints.•Integrate the inequality constraints into local cost function by means of suitable barrier function terms, and avoid the unapproximable property of proximal functions with respect to inequality sets.•Propose a synchronous full-decentralized primal-dual proximal splitting algorithm and its randomized version that removes the global clock coordinator, both of which enjoy private uncoordinated step-sizes. This paper concentrates on a class of decentralized convex optimization problems subject to local feasible sets, equality and inequality constraints, where the global objective function consists of a sum of locally smooth convex functions and non-smooth regularization terms. To address this problem, a synchronous full-decentralized primal-dual proximal splitting algorithm (Syn-FdPdPs) is presented, which avoids the unapproximable property of the proximal operator with respect to inequality constraints via logarithmic barrier functions. Following the proposed decentralized protocol, each agent carries out local information exchange without any global coordination and weight balancing strategies introduced in most consensus algorithms. In addition, a randomized version of the proposed algorithm (Rand-FdPdPs) is conducted through subsets of activated agents, which further removes the global clock coordinator. Theoretically, with the help of asymmetric forward-backward-adjoint (AFBA) splitting technique, the convergence results of the proposed algorithms are provided under the same local step-size conditions. Finally, the effectiveness and practicability of the proposed algorithms are demonstrated by numerical simulations on the least-square and least absolute deviation problems.