Linear Convergence of First- and Zeroth-Order Primal-Dual Algorithms for Distributed Nonconvex Optimization

• 成果类型: Article
• 署名作者: Yi, Xinlei; Zhang, Shengjun; Yang, Tao; Chai, Tianyou; Johansson, Karl H.
• 署名单位: Royal Institute of Technology; University of North Texas System; University of North Texas Denton; Northeastern University - China
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2021.3108501
• 发表日期: 2022
• 页码: 4194-4201
• 关键词: convergence; cost function; Convex functions; COSTS; Technological innovation; Lyapunov methods; Laplace equations; Distributed nonconvex optimization; first-order algorithm; linear convergence; primal-dual algorithm; zeroth-order algorithm

摘要

This article considers the distributed nonconvex optimization problem of minimizing a global cost function formed by a sum of local cost functions by using local information exchange. We first consider a distributed first-order primal-dual algorithm. We show that it converges sublinearly to a stationary point if each local cost function is smooth and linearly to a global optimum under an additional condition that the global cost function satisfies the Polyak-Lojasiewicz condition. This condition is weaker than strong convexity, which is a standard condition for proving linear convergence of distributed optimization algorithms, and the global minimizer is not necessarily unique. Motivated by the situations where the gradients are unavailable, we then propose a distributed zeroth-order algorithm, derived from the considered first-order algorithm by using a deterministic gradient estimator, and show that it has the same convergence properties as the considered first-order algorithm under the same conditions. The theoretical results are illustrated by numerical simulations.