Separation of Generic Cutting Planes in Branch-and-Price Using a Basis
Dantzig-Wolfe reformulation of a mixed integer program partially convexifies a subset of the constraints, i.e., it implicitly adds all valid inequalities for the associated integer hull. Projecting an optimal basic solution of the reformulation’s LP relaxation to the original space does in general not yield a basic solution of the original LP relaxation. Cutting planes in the original problem that are separated using a basis like Gomory mixed integer cuts are therefore not directly applicable. Range  (and others) proposed as a remedy to heuristically compute a basic solution and separate this auxiliary solution also with cutting planes that stem from a basis. This might not only cut off the auxiliary solution, but also the solution we originally wanted to separate.
We discuss and extend Range’s ideas to enhance the separation procedure. In particular, we present alternative heuristics and consider additional valid inequalities strengthening the original LP relaxation before separation. Our full implementation, which is the first of its kind, is done within the GCG framework. We evaluate the effects on several problem classes. Our experiments show that the separated cuts strengthen the formulation on instances where the integrality gap is not too small. This leads to a reduced number of nodes and reduced solution times.
KeywordsMaster Problem Valid Inequality Original Objective Linear Programming Relaxation Price Problem
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- 1.Achterberg, T.: Constraint Integer Programming. Ph.D. thesis, Technische Universität Berlin (2007)Google Scholar
- 4.Bertsimas, D., Tsitsiklis, J.: Introduction to Linear Optimization. Athena Scientific, Belmont (1997)Google Scholar
- 12.Desrosiers, J., Lübbecke, M.E.: Branch-price-and-cut algorithms. In: Cochran, J.J., Cox, L.A., Keskinocak, P., Kharoufeh, J.P., Smith, J.C. (eds.) Wiley Encyclopedia of Operations Research and Management Science. John Wiley & Sons, Inc. (2010)Google Scholar
- 13.Galassi, M., et al.: GNU scientific library reference manual. ISBN 0954612078Google Scholar
- 14.Galati, M.: Decomposition methods for integer linear programming. Ph.D. thesis, Lehigh University (2010)Google Scholar
- 19.Poggi de Aragão, M., Uchoa, E.: Integer program reformulation for robust branch-and-cut-and-price. In: Mathematical Programming in Rio: A Conference in Honour of Nelson Maculan, pp. 56–61 (2003)Google Scholar
- 21.Ralphs, T., Galati, M.: DIP - Decomposition for integer programming (2009). https://projects.coin-or.org/Dip
- 22.Range, T.: An integer cutting-plane procedure for the Dantzig-Wolfe decomposition: Theory. Discussion Papers on Business and Economics 10/2006, Dept. Business and Economics. University of Southern Denmark (2006)Google Scholar
- 25.Vanderbeck, F.: BaPCod - A generic branch-and-price code (2005). https://wiki.bordeaux.inria.fr/realopt/pmwiki.php/Project/BaPCod