NPGA: A Unified Algorithmic Framework for Decentralized Constraint-Coupled Optimization

TL;DR

This paper introduces NPGA, a new decentralized optimization algorithm with linear convergence, serving as a unifying framework that encompasses existing methods and enables the design of more efficient algorithms.

Contribution

The paper proposes NPGA, a novel primal-dual gradient algorithm with the weakest known convergence conditions, unifying and extending existing decentralized optimization algorithms.

Findings

NPGA achieves linear convergence under weak conditions.

Numerical experiments show NPGA outperforms existing algorithms.

NPGA serves as a flexible framework for designing new algorithms.

Abstract

This work focuses on a class of general decentralized constraint-coupled optimization problems. We propose a novel nested primal-dual gradient algorithm (NPGA), which can achieve linear convergence under the weakest known condition, and its theoretical convergence rate surpasses all known results. More importantly, NPGA serves not only as an algorithm but also as a unified algorithmic framework, encompassing various existing algorithms as special cases. By designing different network matrices, we can derive numerous versions of NPGA and analyze their convergences by leveraging the convergence results of NPGA conveniently, thereby enabling the design of more efficient algorithms. Finally, we conduct numerical experiments to compare the convergence rates of NPGA and existing algorithms, providing empirical evidence for the superior performance of NPGA.