Convergent lower bounds for packing problems via restricted dual polytopes

Daniel Cosmin Porumbel (1, 2) , François Clautiaux (3, 4)

1 LINA - Laboratoire d'Informatique de Nantes Atlantique
2 LERIA - Laboratoire d'Etudes et de Recherche en Informatique d'Angers
3 DOLPHIN - Parallel Cooperative Multi-criteria Optimization
4 LIFL - Laboratoire d'Informatique Fondamentale de Lille

Cutting-stock and bin-packing problems have been widely studied in the operations research literature for their large range of indus- trial applications. Many integer programming models have been proposed for them. The most famous is from Gilmore and Gomory [5] and relies on a column generation scheme. Column generation methods are known to have convergence issues, and (when minimizing) no useful lower bounds are produced before a possibly large number of iterations. We propose a new approach that converges to the optimum through a series of dual- feasible solutions, and therefore produces a series of iteratively improving lower bounds. Each dual-feasible solution is obtained by optimizing over an inner approximation of the dual polytope. This approximation is ob- tained by linking groups of dual variables by linear constraints, leading to a problem of smaller dimension. The inner approximation is iteratively refined by splitting the groups into smaller groups until an optimal dual solution is found.

Mots clés: packing; cutting-stock; column generation; duality

Domaines: Recherche opérationnelle [math.OC]

Daniel Cosmin Porumbel, François Clautiaux. Convergent lower bounds for packing problems via restricted dual polytopes. 2012. ⟨hal-00747375v1⟩