A Unified Distributed Method for Constrained Networked Optimization via Saddle-Point Dynamics

This article develops a unified distributed method for solving two classes of constrained networked optimization problems, i.e., optimal consensus problem and resource allocation problem with nonidentical set constraints. We first transform these two constrained networked optimization problems into a unified saddle-point problem framework with set constraints. Subsequently, two projection-based primal-dual algorithms via optimistic gradient descent ascent method and extra-gradient method are developed for solving constrained saddle-point problems. It is shown that the developed algorithms achieve exact convergence to a saddle point with an ergodic convergence rate O(1/κ) for general convex-concave functions. Based on the proposed primal-dual algorithms via saddle-point dynamics, we develop unified distributed algorithm design and convergence analysis for these two networked optimization problems. Finally, two numerical examples are presented to demonstrate the theoretical results.