[Paper Review] Relaxed Dual Optimal Inequalities for Relaxed Columns: with Application to Vehicle Routing

Naveed Haghani, Claudio Contardo+1 more|arXiv (Cornell University)|Apr 11, 2020

Vehicle Routing Optimization Methods13 references4 citations

TL;DR

This paper introduces relaxed dual optimal inequalities (SF-DOI) tailored for column generation in capacitated vehicle routing problems using ng-route relaxations. By adapting smooth and flexible DOI to handle relaxed columns, the method accelerates convergence without weakening the LP relaxation, achieving up to 1.6x speedup on benchmark instances while preserving optimality.

ABSTRACT

We address the problem of accelerating column generation for set cover problems in which we relax the state space of the columns to do efficient pricing. We achieve this by adapting the recently introduced smooth and flexible dual optimal inequalities (DOI) for use with relaxed columns. Smooth DOI exploit the observation that similar items are nearly fungible, and hence should be associated with similarly valued dual variables. Flexible DOI exploit the observation that the change in cost of a column induced by removing an item can be bounded. We adapt these DOI to the problem of capacitated vehicle routing in the context of ng-route relaxations. We demonstrate significant speed ups on a benchmark data set, while provably not weakening the relaxation.

Motivation & Objective

• Address the challenge of accelerating column generation in set covering formulations with relaxed columns, particularly in capacitated vehicle routing.
• Extend the smooth and flexible DOI (SF-DOI) framework to handle relaxed columns such as ng-routes, which permit repeated customer visits.
• Maintain the tightness of the LP relaxation while significantly reducing solution time in column generation for large-scale vehicle routing problems.
• Demonstrate that DOI can be effectively applied to relaxed pricing problems without compromising the optimality of the final solution.

Proposed method

• Adapt the smooth DOI (S-DOI) framework, which enforces similar dual values for spatially or demand-similar customers, to ng-routes.
• Introduce flexible DOI (F-DOI) that bound the cost change when removing a customer from a route, enabling tighter dual bounds.
• Combine S-DOI and F-DOI into a unified SF-DOI framework applicable to relaxed columns in ng-route formulations.
• Apply DOI to the pricing subproblem by constraining the dual space, reducing the search domain without altering the optimal solution.
• Implement a dynamic removal mechanism for active DOI at termination to prevent unbounded primal solutions.
• Use the ng-route relaxation to make pricing tractable by allowing non-elementary routes while forbidding localized short cycles.

Experimental results

Research questions

• RQ1 Can smooth and flexible dual optimal inequalities (DOI) be effectively adapted to work with relaxed columns in column generation?
• RQ2 Does applying SF-DOI to ng-route relaxations in vehicle routing improve convergence speed without weakening the LP relaxation?
• RQ3 How do S-DOI and F-DOI individually and jointly impact the performance of column generation on benchmark CVRP instances?
• RQ4 What is the effect of active DOI removal at termination on convergence and solution quality?
• RQ5 Can DOI be applied to non-elementary route relaxations like ng-routes while preserving dual bound tightness?

Key findings

• The SF-DOI framework achieved an average speedup of 20% across 46 benchmark CVRP instances, with S-DOI alone providing speedups in 44 out of 46 instances.
• The SF-DOI outperformed S-DOI in 21 out of 46 instances, indicating that F-DOI contributes meaningfully to acceleration when combined with S-DOI.
• The F-DOI alone did not yield average speedup, but its integration with S-DOI significantly improved performance in complex instances.
• DOI removal at termination was necessary in 42 out of 46 instances to prevent unbounded primal solutions, highlighting its role in convergence stability.
• No instances exhibited unbounded primal solutions due to DOI, confirming that the method maintains numerical stability.
• The method preserved the tightness of the LP relaxation, as SF-DOI do not weaken the relaxation while accelerating convergence.