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Distributed Consensus Optimization with Consensus ALADIN

Xu Du, Jingzhe Wang

TL;DR

This work addresses distributed consensus optimization (DC) by introducing Consensus ALADIN, which converts the original ALADIN coupled QP into a consensus QP with a global variable $z$ to coordinate across agents. Building on this, the authors develop two efficient variants: BFGS Consensus ALADIN, which uses a BFGS Hessian approximation to reduce communication and computation, and Reduced Consensus ALADIN, which further lowers cost by substituting a scaled identity for the Hessian and exploiting a closed-form $z$ update. The paper establishes global convergence for convex DC and local linear convergence for non-convex DC, and demonstrates effectiveness on a non-convex sensor allocation problem, where RCA outperforms Consensus ADMM and BFGS CAALADIN achieves the best performance. The results suggest that Consensus ALADIN offers scalable, communication-efficient, and theoretically grounded solutions for large-scale distributed optimization tasks in applications like sensor networks and federated settings.

Abstract

TThe paper proposes the Consensus Augmented Lagrange Alternating Direction Inexact Newton (Consensus ALADIN) algorithm, a novel approach for solving distributed consensus optimization problems (DC). Consensus ALADIN allows each agent to independently solve its own nonlinear programming problem while coordinating with other agents by solving a consensus quadratic programming (QP) problem. Building on this, we propose Broyden-Fletcher-Goldfarb-Shanno (BFGS) Consensus ALADIN, a communication-and-computation-efficient Consensus ALADIN.BFGS Consensus ALADIN improves communication efficiency through BFGS approximation techniques and enhances computational efficiency by deriving a closed form for the consensus QP problem. Additionally, by replacing the BFGS approximation with a scaled identity matrix, we develop Reduced Consensus ALADIN, a more computationally efficient variant. We establish the convergence theory for Consensus ALADIN and demonstrate its effectiveness through application to a non-convex sensor allocation problem.

Distributed Consensus Optimization with Consensus ALADIN

TL;DR

This work addresses distributed consensus optimization (DC) by introducing Consensus ALADIN, which converts the original ALADIN coupled QP into a consensus QP with a global variable to coordinate across agents. Building on this, the authors develop two efficient variants: BFGS Consensus ALADIN, which uses a BFGS Hessian approximation to reduce communication and computation, and Reduced Consensus ALADIN, which further lowers cost by substituting a scaled identity for the Hessian and exploiting a closed-form update. The paper establishes global convergence for convex DC and local linear convergence for non-convex DC, and demonstrates effectiveness on a non-convex sensor allocation problem, where RCA outperforms Consensus ADMM and BFGS CAALADIN achieves the best performance. The results suggest that Consensus ALADIN offers scalable, communication-efficient, and theoretically grounded solutions for large-scale distributed optimization tasks in applications like sensor networks and federated settings.

Abstract

TThe paper proposes the Consensus Augmented Lagrange Alternating Direction Inexact Newton (Consensus ALADIN) algorithm, a novel approach for solving distributed consensus optimization problems (DC). Consensus ALADIN allows each agent to independently solve its own nonlinear programming problem while coordinating with other agents by solving a consensus quadratic programming (QP) problem. Building on this, we propose Broyden-Fletcher-Goldfarb-Shanno (BFGS) Consensus ALADIN, a communication-and-computation-efficient Consensus ALADIN.BFGS Consensus ALADIN improves communication efficiency through BFGS approximation techniques and enhances computational efficiency by deriving a closed form for the consensus QP problem. Additionally, by replacing the BFGS approximation with a scaled identity matrix, we develop Reduced Consensus ALADIN, a more computationally efficient variant. We establish the convergence theory for Consensus ALADIN and demonstrate its effectiveness through application to a non-convex sensor allocation problem.

Paper Structure

This paper contains 17 sections, 3 theorems, 34 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Preliminaries of ALADIN
  4. Limitations of ALADIN for Solving DC
  5. Consensus Augmented Lagrange Alternating Direction Inexact Newton Method
  6. Consensus ALADIN: From Coupled QP to Consensus QP
  7. Algorithms
  8. BFGS Consensus ALADIN
  9. Reduced Consensus ALADIN
  10. Convergence Analysis
  11. Global Convergence Analysis
  12. Local Convergence Analysis
  13. Numerical Example
  14. Conclusion
  15. Comparison Between Reduced Consensus ALADIN and Consensus ADMM
  16. ...and 2 more sections

Key Result

Theorem 1

Let the local objective $f_i$s of Problem eq: DC be closed, proper, strictly convex. Let the strong duality of Problem eq: DC hold. Let matrices $B_i\succ 0$s be constant. Let $x_i^*=z^*$ denote the primal solution and $\lambda^*$ denotes the dual solution of Problem eq: DC, then is satisfied with the iterations of Algorithm alg:BFGS ALADIN2. Here, $\Pi$ is a class $\mathcal{K}$ function nonlinea

Figures (1)

  • Figure 1: Numerical comparison on a non-convex case \ref{['eq: nonconvex problem']}.

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Theorem 3