Abstract
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 [22] (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.
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References
Achterberg, T.: Constraint Integer Programming. Ph.D. thesis, Technische Universität Berlin (2007)
Beasley, J.: OR-Library: Distributing test problems by electronic mail. J. Oper. Res. Soc. 41(11), 1069–1072 (1990)
Bergner, M., Caprara, A., Ceselli, A., Furini, F., Lübbecke, M., Malaguti, E., Traversi, E.: Automatic Dantzig-Wolfe reformulation of mixed integer programs. Math. Prog. 149(1–2), 391–424 (2015)
Bertsimas, D., Tsitsiklis, J.: Introduction to Linear Optimization. Athena Scientific, Belmont (1997)
Bixby, R., Rothberg, E.: Progress in computational mixed integer programming - A look back from the other side of the tipping point. Annals of Operations Research 149(1), 37–41 (2007). http://dx.doi.org/10.1007/s10479-006-0091-y
Bode, C., Irnich, S.: Cut-first branch-and-price-second for the capacitated arc-routing problem. Oper. Res. 60(5), 1167–1182 (2012)
Caprara, A., Furini, F., Malaguti, E.: Uncommon Dantzig-Wolfe reformulation for the temporal knapsack problem. INFORMS J. Comput. 25(3), 560–571 (2013)
Cattrysse, D.G., Salomon, M., Wassenhove, L.N.V.: A set partitioning heuristic for the generalized assignment problem. European J. Oper. Res. 72(1), 167–174 (1994)
Chu, P.C., Beasley, J.E.: A genetic algorithm for the generalised assignment problem. Comput. Oper. Res. 24(1), 17–23 (1997)
Dash, S., Goycoolea, M.: A heuristic to generate rank-1 GMI cuts. Math. Program. Comput. 2(3–4), 231–257 (2010)
Desaulniers, G., Desrosiers, J., Spoorendonk, S.: Cutting planes for branch-and-price algorithms. Networks 58(4), 301–310 (2011)
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)
Galassi, M., et al.: GNU scientific library reference manual. ISBN 0954612078
Galati, M.: Decomposition methods for integer linear programming. Ph.D. thesis, Lehigh University (2010)
Gamrath, G., Lübbecke, M.E.: Experiments with a generic Dantzig-Wolfe decomposition for integer programs. In: Festa, P. (ed.) SEA 2010. LNCS, vol. 6049, pp. 239–252. Springer, Heidelberg (2010)
Goncalves, A.S.: Basic feasible solutions and the Dantzig-Wolfe decomposition algorithm. J. Oper. Res. Soc. 19(4), 465–469 (1968)
Irnich, S., Desaulniers, G., Desrosiers, J., Hadjar, A.: Path-reduced costs for eliminating arcs in routing and scheduling. INFORMS J. Comput. 22(2), 297–313 (2010)
Osman, I.H.: Heuristics for the generalised assignment problem: Simulated annealing and tabu search approaches. OR Spectrum 17(4), 211–225 (1995)
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)
Puchinger, J., Stuckey, P., Wallace, M., Brand, S.: Dantzig-Wolfe decomposition and branch-and-price solving in G12. Constraints 16(1), 77–99 (2011)
Ralphs, T., Galati, M.: DIP - Decomposition for integer programming (2009). https://projects.coin-or.org/Dip
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)
Rios, J., Ross, K.: Converging upon basic feasible solutions through Dantzig-Wolfe decomposition. Optim. Lett. 8(1), 171–180 (2014)
Tempelmeier, H., Derstroff, M.: A lagrangean-based heuristic for dynamic multilevel multiitem constrained lotsizing with setup times. Management Science 42(5), 738–757 (1996)
Vanderbeck, F.: BaPCod - A generic branch-and-price code (2005). https://wiki.bordeaux.inria.fr/realopt/pmwiki.php/Project/BaPCod
Vanderbeck, F., Savelsbergh, M.: A generic view of Dantzig-Wolfe decomposition in mixed integer programming. Oper. Res. Lett. 34(3), 296–306 (2006)
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Lübbecke, M.E., Witt, J.T. (2015). Separation of Generic Cutting Planes in Branch-and-Price Using a Basis. In: Bampis, E. (eds) Experimental Algorithms. SEA 2015. Lecture Notes in Computer Science(), vol 9125. Springer, Cham. https://doi.org/10.1007/978-3-319-20086-6_9
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