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

2025-11-24

AI Summary

This paper addresses two canonical coupling constraints in distributed convex optimization: consensus constraints and global affine equality constraints. To overcome the limitations of existing methods—which rely on strong convexity or smoothness assumptions—we propose a novel linearized multiplier method that achieves non-ergodic $O(1/sqrt{k})$ sublinear convergence in both objective optimality residual and consensus violation, without requiring smoothness or strong convexity of the objective function. We establish, for the first time, a dual equivalence between consensus optimization and economic dispatch in power systems, ensuring global consensus of dual variables over the communication network. The theoretical analysis is grounded in convex optimization theory and the Slater condition, with no reliance on auxiliary regularization. Numerical experiments on the IEEE 118-bus system demonstrate that the proposed method outperforms state-of-the-art distributed algorithms in convergence speed for both objective and feasibility errors, as well as in dual consistency.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Solving distributed optimization with consensus and coupling constraints
• Developing non-ergodic algorithms for constrained convex optimization problems
• Addressing economic dispatch through dual consensus formulation

Innovation

• Linearized multiplier method for consensus optimization
• Non-ergodic convergence rates for distributed optimization
• Dual consensus formulation for economic dispatch problems