Convergent Dual Bounds Using an Aggregation of Set-Covering Constraints for Capacitated Problems

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

1 LGI2A - Laboratoire de Génie Informatique et d'Automatique de l'Artois
2 IMB - Institut de Mathématiques de Bordeaux
3 Realopt - Reformulations based algorithms for Combinatorial Optimization

Extended formulations are now widely used to solve hard combinatorial optimization problems. Such formulations have prohibitively-many variables and are generally solved via Column Generation (CG). CG algorithms are known to have frequent convergence issues, and, up to a sometimes large number of iterations, classical Lagrangian dual bounds may be weak. This paper is devoted to set-covering problems in which all elements to cover require a given \emph{resource consumption} and all feasible configurations have to verify a \emph{resource constraint}. We propose an iterative aggregation method for determining convergent dual bounds using the extended formulation of such problems. The set of dual variables is partitioned into $k$ groups and all variables in each group are artificially linked using the following groupwise restriction: the dual values in a group have to follow a linear function of their corresponding resource consumptions. This leads to a restricted model of smaller dimension, with only $2k$ dual variables. The method starts with one group ($k=1$) and iteratively splits the groups. Our algorithm has three advantages: (i) it produces good dual bounds even for low $k$ values, (ii) it reduces the number of dual variables, and (iii) it may reduce the time needed to solve sub-problems, in particular when dynamic programming is used. We experimentally tested our approach on two variants of the cutting-stock problem: in many cases, the method produces near optimal dual bounds after a small number of iterations. Moreover the average computational effort to reach the optimum is reduced compared to a classical column generation algorithm.

Mots clés: column generation; duality; aggregation

Domaines: Recherche opérationnelle [math.OC]

Daniel Cosmin Porumbel, François Clautiaux. Convergent Dual Bounds Using an Aggregation of Set-Covering Constraints for Capacitated Problems. 2013. ⟨hal-00747375v2⟩