Non-Ergodic Convergence Algorithms for Distributed Consensus and Coupling-Constrained Optimization

TL;DR

This paper introduces a non-ergodic, linearized method of multipliers for distributed convex optimization with consensus and affine coupling constraints, achieving sublinear convergence rates without requiring smoothness or strong convexity.

Contribution

The paper develops a novel non-ergodic algorithm with proven convergence rates for distributed convex optimization problems, including the economic dispatch problem, extending the applicability of such methods.

Findings

Achieves O(1/√k) convergence rates for objective and consensus violation.

Demonstrates faster reduction of errors in numerical experiments on IEEE 118-bus system.

Ensures dual variables reach network-wide consensus.

Abstract

We study distributed convex optimization with two ubiquitous forms of coupling: consensus constraints and global affine equalities. We first design a linearized method of multipliers for the consensus optimization problem. Without smoothness or strong convexity, we establish non-ergodic sublinear rates of order O(1/\sqrt{k}) for both the objective optimality and the consensus violation. Leveraging duality, we then show that the economic dispatch problem admits a dual consensus formulation, and that applying the same algorithm to the dual economic dispatch yields non-ergodic O(1/\sqrt{k}) decay for the error of the summation of the cost over the network and the equality-constraint residual under convexity and Slater's condition. Numerical results on the IEEE 118-bus system demonstrate faster reduction of both objective error and feasibility error relative to the state-of-the-art baselines, while the dual variables reach network-wide consensus.