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

TL;DR

This paper introduces a unified distributed saddle-point approach with projection-based primal-dual algorithms for solving constrained networked optimization problems, achieving exact convergence with an ergodic rate of O(1/k).

Contribution

It develops a unified saddle-point framework and primal-dual algorithms for constrained networked optimization, providing convergence guarantees and a general solution approach.

Findings

Algorithms achieve exact convergence to saddle points.

Convergence rate of O(1/k) for convex-concave functions.

Numerical examples validate theoretical results.

Abstract

This paper develops a unified distributed method for solving two classes of constrained networked optimization problems, i.e., optimal consensus problem and resource allocation problem with non-identical 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 (OGDA) method and Extra-gradient (EG) 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 $O(1/k)$ 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.