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

3 Realopt - Reformulations based algorithms for Combinatorial Optimization
LaBRI - Laboratoire Bordelais de Recherche en Informatique, IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-00747375
Contributor : François Clautiaux <>
Submitted on : Monday, December 16, 2013 - 10:27:15 AM
Last modification on : Thursday, May 16, 2019 - 3:54:05 PM
Long-term archiving on : Tuesday, March 18, 2014 - 12:30:18 PM

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• HAL Id : hal-00747375, version 2

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

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