Consensus-Based Dantzig-Wolfe Decomposition

TL;DR

This paper introduces a fully distributed version of the Dantzig-Wolfe decomposition algorithm that uses consensus-based ADMM, enabling decentralized solving of large-scale linear programs while maintaining privacy and efficiency.

Contribution

It proposes a novel distributed Dantzig-Wolfe decomposition algorithm utilizing consensus ADMM, with theoretical error bounds and preliminary computational validation.

Findings

Achieved high-quality solutions on benchmark instances.

Derived error bounds for optimality gap and feasibility.

Demonstrated feasibility with MPI implementation.

Abstract

Dantzig-Wolfe decomposition (DWD) is a classical algorithm for solving large-scale linear programs whose constraint matrix involves a set of independent blocks coupled with a set of linking rows. The algorithm decomposes such a model into a master problem and a set of independent subproblems that can be solved in a distributed manner. In a typical implementation, the master problem is solved centrally. In certain settings, solving the master problem centrally is undesirable or infeasible, such as in the case of decentralized storage of data, or when independent agents who are responsible for the subproblems desire privacy of information. In this paper, we propose a fully distributed DWD algorithm which relies on solving the master problem using a consensus-based Alternating Direction Method of Multipliers (ADMM) method. We derive error bounds on the optimality gap and feasibility violation of the proposed approach. We provide preliminary computational results for our algorithm using a Message Passing Interface (MPI) implementation on cutting stock instances from the literature and synthetic instances where we obtain high quality solutions.