Accelerating Column Generation in Highly Degenerate Integer Programming Problems with Template Pricing

TL;DR

This paper introduces Template pricing, a novel column generation acceleration technique that improves convergence and solution quality in highly degenerate integer programming problems, demonstrated on the Generalized Assignment Problem.

Contribution

The paper presents a new Template pricing strategy for column generation, combining similarity-based column selection with exact and heuristic methods to enhance performance.

Findings

Template pricing significantly accelerates CG, solving some instances over 1000x faster.

Achieved CG optimal bounds on all 1735 ISA instances, with stronger bounds in 43%.

Improved integer solutions in 9% of the tested instances.

Abstract

We propose a new pricing strategy for column generation (CG), referred to as Template pricing. This method is motivated by the desire to coordinate solutions of different pricing subproblems in order to accelerate the convergence of the CG process and simultaneously obtain good quality integer feasible solutions. Instead of finding a column with the optimal reduced cost, Template pricing tries to maximize the similarity of columns with a given template vector, while restricting the search to columns with suitable reduced cost. We present an exact and heuristic method (based on Lagrangian relaxation) to efficiently solve the Template pricing problem. We conduct extensive computational experiments on benchmark instances of the Generalized Assignment Problem (GAP). Our results demonstrate that Template pricing can significantly accelerate the CG algorithm, especially in the presence of significant degeneracy, where several benchmark GAP instances solved over 1000x faster than Dantzig pricing, and over 100x with adaptive dual-smoothing. Template pricing allows us to achieve CG optimal bounds on all 1735 ISA instances, finding stronger bounds in 43% and improved integer solutions in 9% of these instances than previously released.