Implicit Tracking-Based Distributed Constraint-Coupled Optimization

TL;DR

This paper introduces a novel implicit tracking approach for distributed optimization with coupled constraints, enabling distributed handling of violations and achieving convergence without strict convexity assumptions.

Contribution

The paper proposes the IDEA and Proj-IDEA algorithms, the first distributed methods with constant step-size for this class of problems, and analyzes their convergence properties.

Findings

IDEA achieves exponential convergence under strong convexity.

Proj-IDEA can handle local constraints without strict convexity.

Numerical experiments validate theoretical convergence results.

Abstract

A class of distributed optimization problem with a globally coupled equality constraint and local constrained sets is studied in this paper. For its special case where local constrained sets are absent, an augmented primal-dual gradient dynamics is proposed and analyzed, but it cannot be implemented distributedly since the violation of the coupled constraint needs to be used. Benefiting from the brand-new comprehending of a classical distributed unconstrained optimization algorithm, the novel implicit tracking approach is proposed to track the violation distributedly, which leads to the birth of the \underline{i}mplicit tracking-based \underline{d}istribut\underline{e}d \underline{a}ugmented primal-dual gradient dynamics (IDEA). A projected variant of IDEA, i.e., Proj-IDEA, is further designed to deal with the general case where local constrained sets exist. With the aid of the Lyapunov stability theory, the convergences of IDEA and Pro-IDEA over undigraphs and digraphs are analyzed respectively. As far as we know, Proj-IDEA is the first constant step-size distributed algorithm which can solve the studied problem without the need of the strict convexity of local cost functions. Besides, if local cost functions are strongly convex and smooth, IDEA can achieve exponential convergence with a weaker condition about the coupled constraint. Finally, numerical experiments are taken to corroborate our theoretical results.