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

Chenyang Qiu, Zongli Lin

TL;DR

This work tackles distributed convex optimization on graphs with both consensus and global affine equality couplings. It introduces a linearized augmented Lagrangian method that yields non-ergodic $O(1/\sqrt{k})$ convergence for objective and feasibility without requiring smoothness or strong convexity, and extends the approach to economic dispatch through a dual consensus reformulation under convexity and Slater’s condition. Theoretical guarantees are complemented by numerical results on the IEEE 118-bus system, where the proposed method outperforms state-of-the-art baselines and achieves network-wide consensus of dual variables. By unifying consensus optimization and affine coupling via a distributed, primal–dual framework, the paper advances scalable, privacy-preserving optimization for large networks with practical relevance to power systems and beyond.

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.

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

TL;DR

This work tackles distributed convex optimization on graphs with both consensus and global affine equality couplings. It introduces a linearized augmented Lagrangian method that yields non-ergodic convergence for objective and feasibility without requiring smoothness or strong convexity, and extends the approach to economic dispatch through a dual consensus reformulation under convexity and Slater’s condition. Theoretical guarantees are complemented by numerical results on the IEEE 118-bus system, where the proposed method outperforms state-of-the-art baselines and achieves network-wide consensus of dual variables. By unifying consensus optimization and affine coupling via a distributed, primal–dual framework, the paper advances scalable, privacy-preserving optimization for large networks with practical relevance to power systems and beyond.

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.

Paper Structure

This paper contains 12 sections, 5 theorems, 46 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Consensus Optimization
  3. Problem Formulation
  4. Algorithm Development
  5. Convergence Analysis
  6. Extension to Economic Dispatch
  7. Problem Formulation
  8. Primal-Dual Relationship
  9. Algorithm Development
  10. Convergence Analysis
  11. Numerical Experiments
  12. Conclusion

Key Result

Lemma 1

Suppose Assumption ass: G_convex holds. If $\eta I - \rho W > 0$, then the sequences $y_k$ and $\lambda_k$ generated by the proposed algorithm satisfy the following inequality, where $\Omega = \eta I - \rho W$.

Figures (4)

  • Figure 1: The power mismatch under the five algorithms.
  • Figure 2: The absolute error of the total power generation and the total power demand under the five algorithms.
  • Figure 3: The power output of each generator under the proposed algorithm.
  • Figure 4: The consensus error of the variable $y_k$ under the proposed algorithm.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof