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

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 resource consumption and all feasible configurations have to verify a 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.