Dynamical Primal-Dual Nesterov Accelerated Method and Its Application to Network Optimization

• Published in: IEEE transactions on automatic control Vol. 68; no. 3; pp. 1760 - 1767
• Main Authors: Zeng, Xianlin, Lei, Jinlong, Chen, Jie
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.03.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Computational geometry; Continuous-time algorithm; Convergence; Convex functions; Convexity; Cost function; Damping; Distributed algorithms; Gradient methods; Nesterov’s accelerated method; network optimization; Optimization; Parameters; primal-dual method; Trajectory
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2022.3152720

This article develops a continuous-time primal-dual accelerated method with an increasing damping coefficient for a class of convex optimization problems with affine equality constraints. This article analyzes critical values for parameters in the proposed method and prove that the rate of convergence in terms of the duality gap function is <inline-formula><tex-math notation="LaTeX">O(\frac{1}{t^2})</tex-math></inline-formula> by choosing suitable parameters. As far as we know, this is the first continuous-time primal-dual accelerated method that can obtain the optimal rate. Then, this article applies the proposed method to two network optimization problems, a distributed optimization problem with consensus constraints and a distributed extended monotropic optimization problem, and obtains two variant distributed algorithms. Finally, numerical simulations are given to demonstrate the efficacy of the proposed method.