New Lagrangian Relaxation Approach for the Discrete Cost Multicommodity Network Design Problem

TL;DR

This paper introduces a novel path-based Lagrangian relaxation method with multiple variants for the Discrete Cost Multicommodity Network Design Problem, demonstrating improved lower bounds through extensive computational experiments.

Contribution

It proposes a new path-based formulation and six deflected subgradient variants for Lagrangian relaxation, enhancing solution bounds for DCMNDP.

Findings

Six variants of deflected subgradient methods evaluated

Improved lower bounds on random and real-world instances

Enhanced computational performance over traditional arc-based approaches

Abstract

We aim to derive effective lower bounds for the Discrete Cost Multicommodity Network Design Problem (DCMNDP). Given an undirected graph, the problem requires installing at most one facility on each edge such that a set of point-to-point commodity flows can be routed and costs are minimized. In the literature, the Lagrangian relaxation is usually applied to an arc-based formulation to derive lower bounds. In this work, we investigate a path-based formulation and we solve its Lagrangian relaxation using several non-differentiable optimization techniques. More precisely, we devised six variants of the deflected subgradient procedures, using various direction-search and step-length strategies. The computational performance of these Lagrangian-based approaches are evaluated and compared on a set of randomly generated instances, and real-world problems.