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

TL;DR

This paper develops a heuristic ADMM framework for efficiently solving large-scale nonconvex tree-constrained optimization problems, with applications to hop-constrained spanning tree multicommodity flow design, combining continuous relaxations and polynomial-time projections.

Contribution

It introduces a novel heuristic ADMM approach that alternates between convex subproblems and tree projections, enabling scalable distributed solutions for complex combinatorial problems.

Findings

High-quality solutions achieved in many cases

Near-optimal performance demonstrated

Distributed algorithm enhances scalability and robustness

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.