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Distributed Constraint-Coupled Optimization: Harnessing ADMM-consensus for robustness

Mohamed Abdelmouamin Messilem, Guido Carnevale, Ruggero Carli

TL;DR

The paper addresses distributed optimization over networks for minimizing the sum of local costs under coupling constraints, proposing a primal–dual scheme derived from an alternative Lagrangian and proven to converge linearly via time-scale separation. It introduces consensus-based proxies to maintain distributed operation and establishes a rigorous convergence framework that blends optimization and consensus analyses. A robust asynchronous implementation is developed to tolerate packet losses, with theoretical guarantees. Numerical results in three-phase microgrid ancillary services demonstrate practical effectiveness and scalability under realistic communication conditions.

Abstract

In this paper, we consider a network of agents that jointly aim to minimise the sum of local functions subject to coupling constraints involving all local variables. To solve this problem, we propose a novel solution based on a primal-dual architecture. The algorithm is derived starting from an alternative definition of the Lagrangian function, and its convergence to the optimal solution is proved using recent advanced results in the theory of time-scale separation in nonlinear systems. The rate of convergence is shown to be linear under standard assumptions on the local cost functions. Interestingly, the algorithm is amenable to a direct implementation to deal with asynchronous communication scenarios that may be corrupted by other non-idealities such as packet loss. We numerically test the validity of our approach on a real-world application related to the provision of ancillary services in three-phase low-voltage microgrids.

Distributed Constraint-Coupled Optimization: Harnessing ADMM-consensus for robustness

TL;DR

The paper addresses distributed optimization over networks for minimizing the sum of local costs under coupling constraints, proposing a primal–dual scheme derived from an alternative Lagrangian and proven to converge linearly via time-scale separation. It introduces consensus-based proxies to maintain distributed operation and establishes a rigorous convergence framework that blends optimization and consensus analyses. A robust asynchronous implementation is developed to tolerate packet losses, with theoretical guarantees. Numerical results in three-phase microgrid ancillary services demonstrate practical effectiveness and scalability under realistic communication conditions.

Abstract

In this paper, we consider a network of agents that jointly aim to minimise the sum of local functions subject to coupling constraints involving all local variables. To solve this problem, we propose a novel solution based on a primal-dual architecture. The algorithm is derived starting from an alternative definition of the Lagrangian function, and its convergence to the optimal solution is proved using recent advanced results in the theory of time-scale separation in nonlinear systems. The rate of convergence is shown to be linear under standard assumptions on the local cost functions. Interestingly, the algorithm is amenable to a direct implementation to deal with asynchronous communication scenarios that may be corrupted by other non-idealities such as packet loss. We numerically test the validity of our approach on a real-world application related to the provision of ancillary services in three-phase low-voltage microgrids.

Paper Structure

This paper contains 12 sections, 3 theorems, 37 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Distributed Algorithm Design
  4. Distributed Algorithm Analysis
  5. Algorithm Reformulation
  6. Optimization-Oriented Part Analysis
  7. Consensus-Oriented Part Analysis
  8. Proof of Theorem \ref{['th:convegence']}
  9. Robust Distributed Algorithm
  10. Numerical Simulations
  11. Conclusions
  12. Proof of Lemma \ref{['lemma:centralized_method']}

Key Result

Theorem 1

Consider Algorithm algo:algo and let Assumptions ass:cost and ass:full_row hold. Then, for all $\kappa, \rho > 0$ and $\beta \in (0,1)$, there exist $\bar{\gamma} > 0$ such that, for all $\gamma \in (0,\bar{\gamma})$, $(\boldsymbol{x}_{i}^0,\lambda_{i}^0,\boldsymbol{z}_{i}^{0}) \in \mathbb{R}^{n_{i} for all $t \in \mathbb{N}$ and some $a_1, a_2 >0$. $\blacksquare$

Figures (1)

  • Figure 1: Numerical simulation results, comparing Synchronous and asynchronous approach.

Theorems & Definitions (4)

  • Remark 1
  • Theorem 1
  • Lemma 1
  • Lemma 2