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A Heuristic Alternating Direction Method of Multipliers Framework for Distributed and Centralized Tree-Constrained Optimization: Applications to Hop-Constrained Spanning Tree Multicommodity Flow Design

Yacine Mokhtari

Abstract

This paper presents centralized and distributed Alternating Direction Method of Multipliers (ADMM) frameworks for solving large-scale nonconvex optimization problems with binary decision variables subject to spanning tree or rooted arborescence constraints. We address the combinatorial complexity by introducing a continuous relaxation of the binary variables and enforcing agreement through an augmented Lagrangian formulation. The algorithms alternate between solving a convex continuous subproblem and projecting onto the tree-feasible set, reducing to a Minimum Spanning Tree or Minimum Weight Rooted Arborescence problem, both solvable in polynomial time. The distributed algorithm enables agents to cooperate via local communication, enhancing scalability and robustness. We apply the framework to multicommodity flow design with hop-constrained spanning trees. Numerical experiments demonstrate that our methods yield high-quality feasible solutions in many cases, achieving near-optimal performance.

A Heuristic Alternating Direction Method of Multipliers Framework for Distributed and Centralized Tree-Constrained Optimization: Applications to Hop-Constrained Spanning Tree Multicommodity Flow Design

Abstract

This paper presents centralized and distributed Alternating Direction Method of Multipliers (ADMM) frameworks for solving large-scale nonconvex optimization problems with binary decision variables subject to spanning tree or rooted arborescence constraints. We address the combinatorial complexity by introducing a continuous relaxation of the binary variables and enforcing agreement through an augmented Lagrangian formulation. The algorithms alternate between solving a convex continuous subproblem and projecting onto the tree-feasible set, reducing to a Minimum Spanning Tree or Minimum Weight Rooted Arborescence problem, both solvable in polynomial time. The distributed algorithm enables agents to cooperate via local communication, enhancing scalability and robustness. We apply the framework to multicommodity flow design with hop-constrained spanning trees. Numerical experiments demonstrate that our methods yield high-quality feasible solutions in many cases, achieving near-optimal performance.

Paper Structure

This paper contains 28 sections, 2 theorems, 55 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Problem
  3. Motivation and Applications
  4. Solution Methods
  5. The Alternating Direction Method of Multipliers as a Heuristic
  6. Distributed Optimization
  7. Related Works and Contribution
  8. The Algorithm
  9. Review
  10. Minimum Spanning Tree (MST)
  11. Minimum Weight Rooted Arborescence (MWRA)
  12. The centralized ADMM Algorithm
  13. The Distributed ADMM Algorithm
  14. Execution Details
  15. Initialization
  16. ...and 13 more sections

Key Result

Proposition 2.1

Problem AA2 is equivalent to solving the following discrete optimization problem at each iteration: where the weight vector $\boldsymbol{h}_{k}\in \mathbb{R}^{m}$ is given by

Figures (9)

  • Figure 1: Average execution time and performance comparison of centralized and distributed ADMM with Gurobi for different values of $n$, $p$, and $\rho$.
  • Figure 2: Average optimality gap for the centralized ADMM across various values of $n$, $p$, and $\rho$.
  • Figure 3: Average optimality gap for the distributed ADMM across various values of $n$, $p$, and $\rho$.
  • Figure 4: Residual error evolution for centralized and distributed ADMM with $n=10$ across $\rho \in \{0.1,1,10\}$.
  • Figure 5: Objective value evolution for centralized and distributed ADMM with $n=10$.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4