Row-reduced column generation for degenerate master problems

Jacques Desrosiers, Jean Bertrand Gauthier and Marco E. Lübbecke

European Journal of Operational Research, 2014, vol. 236, issue 2, 453-460

Abstract: Column generation for solving linear programs with a huge number of variables alternates between solving a master problem and a pricing subproblem to add variables to the master problem as needed. The method is known to often suffer from degeneracy in the master problem. Inspired by recent advances in coping with degeneracy in the primal simplex method, we propose a row-reduced column generation method that may take advantage of degenerate solutions. The idea is to reduce the number of constraints to the number of strictly positive basic variables in the current master problem solution. The advantage of this row-reduction is a smaller working basis, and thus a faster re-optimization of the master problem. This comes at the expense of a more involved pricing subproblem, itself eventually solved by column generation, that needs to generate weighted subsets of variables that are said compatible with the row-reduction, if possible. Such a subset of variables gives rise to a strict improvement in the objective function value if the weighted combination of the reduced costs is negative. We thus state, as a by-product, a necessary and sufficient optimality condition for linear programming.

Keywords: Column generation; Degeneracy; Dynamic row-reduction (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (9)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:ejores:v:236:y:2014:i:2:p:453-460

DOI: 10.1016/j.ejor.2013.12.016

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