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Passivity-Based Local Design Conditions for Global Optimality in Distributed Convex Optimization

Pol Jane-Soneira, Charles Muller, Felix Strehle, Sören Hohmann

TL;DR

This work introduces a passivity-based framework for distributed convex optimization that enables local, heterogeneous agent designs while guaranteeing global optimality. By decomposing the network into agent dynamics and edge controllers linked through a generalized incidence structure, the method achieves consensus and convergence without requiring global initialization or multi-variable exchanges. It provides explicit local design conditions that ensure the global optimum satisfies standard optimality or KKT conditions, and it addresses both undirected and directed communication graphs, including constrained problems via projected dynamics. Numerical results illustrate plug-and-play interoperability and robustness to group reconfigurations, highlighting practical scalability in large, dynamic networks. The approach offers a flexible, scalable path to globally optimal distributed optimization in diverse applications.

Abstract

In recent times, various distributed optimization algorithms have been proposed for whose specific agent dynamics global optimality and convergence is proven. However, there exist no general conditions for the design of such algorithms. In this paper, we leverage passivity theory to fi rst establish a distributed optimization framework with local design requirements for the agent dynamics in both unconstrained and constrained problems with undirected communication topologies. Under the roof of these requirements, the agents may use heterogeneous optimization algorithms without compromising global optimality and convergence. Subsequently, we propose some exemplary agent systems that comply with the established requirements. Compared to existing approaches, our algorithms do not require any global initialization nor communication of multiple variables. Consequently, the agents may leave or rejoin the networked optimization without compromising convergence to the correct global optimizer. Furthermore, we show that for unconstrained optimization, an extension to directed communication topologies is possible. Simulation results illustrate the plug-and-play capabilities and interoperability of the proposed agent dynamics.

Passivity-Based Local Design Conditions for Global Optimality in Distributed Convex Optimization

TL;DR

This work introduces a passivity-based framework for distributed convex optimization that enables local, heterogeneous agent designs while guaranteeing global optimality. By decomposing the network into agent dynamics and edge controllers linked through a generalized incidence structure, the method achieves consensus and convergence without requiring global initialization or multi-variable exchanges. It provides explicit local design conditions that ensure the global optimum satisfies standard optimality or KKT conditions, and it addresses both undirected and directed communication graphs, including constrained problems via projected dynamics. Numerical results illustrate plug-and-play interoperability and robustness to group reconfigurations, highlighting practical scalability in large, dynamic networks. The approach offers a flexible, scalable path to globally optimal distributed optimization in diverse applications.

Abstract

In recent times, various distributed optimization algorithms have been proposed for whose specific agent dynamics global optimality and convergence is proven. However, there exist no general conditions for the design of such algorithms. In this paper, we leverage passivity theory to fi rst establish a distributed optimization framework with local design requirements for the agent dynamics in both unconstrained and constrained problems with undirected communication topologies. Under the roof of these requirements, the agents may use heterogeneous optimization algorithms without compromising global optimality and convergence. Subsequently, we propose some exemplary agent systems that comply with the established requirements. Compared to existing approaches, our algorithms do not require any global initialization nor communication of multiple variables. Consequently, the agents may leave or rejoin the networked optimization without compromising convergence to the correct global optimizer. Furthermore, we show that for unconstrained optimization, an extension to directed communication topologies is possible. Simulation results illustrate the plug-and-play capabilities and interoperability of the proposed agent dynamics.

Paper Structure

This paper contains 27 sections, 22 theorems, 84 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Convex Analysis
  4. Passivity Theory
  5. Problem Formulation and Main Idea
  6. Local Design Requirements for Global Optimality and Convergence
  7. Global Optimality Conditions
  8. Local Design Requirements for Global Optimality
  9. Unconstrained Optimization
  10. Constrained Optimization
  11. Communication Topology
  12. Generalized Undirected Topology
  13. Directed Graphs and Non-Symmetric Communication Structure
  14. Local Design Requirements for Convergence with Undirected Communication Topologies
  15. Agent and Controller Dynamics
  16. ...and 12 more sections

Key Result

Proposition 6

Assume the optimization problem eq:optimization_problem is convex and there are no constraints. Then, ${\mathrm{y}}^*$ is an optimizer if and only if holds. If eq:optimization_problem is strictly convex, ${\mathrm{y}}^*$ is unique.

Figures (7)

  • Figure 1: Block diagram of the proposed algorithm structure, composed of agent systems $\Sigma_i$ and controller systems $\Pi_k$.
  • Figure 2: Exemplary network composed of 4 agents (nodes) and communication links in blue (edges).
  • Figure 3: Different technical realization of the controller systems.
  • Figure 4: Network composed of 3 agents and 2 controllers with a generalized symmetric communication structure.
  • Figure 5: Network composed of 3 agents and 2 controllers with a directed, non-symmetric communication structure.
  • ...and 2 more figures

Theorems & Definitions (55)

  • Definition 1
  • Definition 2: arcak2016networks
  • Definition 3: arcak2016networks,simpson2018equilibrium
  • Definition 4
  • Definition 5: simpson2018equilibrium
  • Example 1
  • Proposition 6: boyd2004convex
  • Proposition 7: boyd2004convex
  • Theorem 8
  • proof
  • ...and 45 more