Abstract
Cutting planes have been used with great success for solving mixed integer programs. In recent decades, many contributions have led to successive improvements in branch-and-cut methods which incorporate cutting planes in branch and bound algorithm. Using advances that have taken place over the years on 0–1 knapsack problem, we investigate an efficient approach for 0–1 programs with knapsack constraints as local structure. Our approach is based on an efficient implementation of knapsack separation problem which consists of the four phases: preprocessing, row generation, controlling numerical errors and sequential lifting. This approach can be used independently to improve formulations with cutting planes generated or incorporated in branch and cut to solve a problem. We show that this approach allows us to efficiently solve large-scale instances of generalized assignment problem, multilevel generalized assignment problem, capacitated \(p\)-median problem and capacitated network location problem to optimality.
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Ceselli, A.: Two exact algorithms for the capacitated p-median problem. 4OR 1, 319–340 (2003)
Ceselli, A., Righini, G.: A branch and price algorithm for the capacitated \(p\)-median problem. Networks 45(3), 125–142 (2004)
Pigatti, A., Poggi de Aragao, M., Uchoa, E.: Stabilized branch-and-cut-and-price for the generalized assignment problem. In: 2nd Brazilian Symposium on Graphs, Algorithms and Combinatorics. Electronic Notes in Discrete Mathematics vol. 19, pp. 389–395 (2005)
Avella, P., Boccia, M., Vasilyev, I.: A computational study of exact knapsack separation for the generalized assignment problem. Comput. Optim. Appl. 45, 543–555 (2010)
Boccia, M., Sforza, A., Sterle, C., Vasilyev, I.: A cut and branch approach for the capacitated \(p\)-median problem based on fenchel cutting planes. J. Math. Model. Algorithms 7, 43–58 (2007)
Vasilev, I.: A cutting plane method for knapsack polytope. J. Comput. Syst. Sci. Int. 48, 70–77 (2009)
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)
Boyd, E.A.: Generating fenchel cutting planes for knapsack polyhedra. SIAM J. Optim. 3, 734–750 (1993)
Boyd, E.A.: Solving integer programs with cutting planes and preprocessing. IPCO 1993, 209–220 (1993)
Boyd, E.A.: Fenchel cutting planes for integer programming. Oper. Res. 42, 53–64 (1994)
Boyd, E.A.: On the convergence of fenchel cutting planes in mixed-integer programming. SIAM J. Optim. 5, 421–435 (1995)
Fukasawa, R., Goycoolea, M.: On the exact separation of mixed integer knapsack cuts. Math. Program. 128(1–2), 19–41 (2011)
Kaparis, K., Letchford, A.N.: Separation algorithms for 0–1 knapsack polytopes. Math. Program. 124, 69–91 (2010)
Bonami, M.P.: Étude et mise en oeuvre dapproches polyédriques pour la résolution de programmes en nombres entiers ou mixtes généraux. Ph.D. thesis, L’Université Paris 6 (2003)
Espinoza, D.G.: On Linear Programming, Integer Programming and Cutting Planes. PhD thesis, Georgia Institute of Technology, School of Industrial and Systems Engineering (2006)
Avella, P., Boccia, M., Vasilyev, I.: Computational testing of a separation procedure for the knapsack set with a single continuous variable. INFORMS J. Comput. 24(1), 165–171 (2012)
Avella, P., Boccia, M., Vasilyev, I.: Computational experience with general cutting planes for the set covering problem. Oper. Res. Lett. 37(1), 16–20 (2009)
Applegate, D., Bixby, R., Chvatal, V., Cook, W.: Implementing the dantzig–fulkerson–johnson algorithm for large traveling salesman problems. Math. Program. 97, 91–153 (2003)
Applegate, D.L., Bixby, R.E., Chvatal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study (Princeton Series in Applied Mathematics). Princeton University Press, Princeton (2007)
Cornuejols, G., Lemarechal, C.: A convex-analysis perspective on disjunctive cuts. Math. Program. 106(3), 567–586 (2006)
Pisinger, D.: A minimal algorithm for the 0–1 knapsack problem. Oper. Res. 46, 758–767 (1995)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin (2005)
Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley, London (1990)
Posta, M., Ferland, J.A., Michelon, P.: An exact method with variable fixing for solving the generalized assignment problem. Comput. Optim. Appl. 52, 629–644 (2012)
Beasley, J.E.: Or-library: distributing test problems by electronic mail. J. Oper. Res. Soc. 41(11), 1069–1072 (1990)
Glover, F., Hultz, J., Klingman, D.: Improved computer-based planning techniques. part ii. Interfaces 9(4), 12–20 (1979)
French, A.P., Wilson, J.M.: Heuristic solution methods for the multilevel generalized assignment problem. J. Heuristics 8, 143–153 (2002)
Laguna, M., Kelly, J.P., Gonzlez-Velarde, J.L., Glover, F.: Tabu search for the multilevel generalized assignment problem. Eur. J. Oper. Res. 82(1), 176–189 (1995)
Osorio, M.A., Laguna, M.: Logic cuts for multilevel generalized assignment problems. Eur. J. Oper. Res. 151(1), 238–246 (2003)
Ceselli, A., Righini, G.: A branch-and-price algorithm for the multilevel generalized assignment problem. Oper. Res. 54(6), 1172–1184 (2006)
Baldacci, R., Hadjiconstantinou, E., Maniezzo, V., Mingozzi, A.: A new method for solving capacitated location problems based on a set partitioning approach. Comput. Oper. Res. 29, 365–386 (2002)
Lorena, L., Senne, E.: A column generation approach to capacitated p-median problems. Comput. Oper. Res. 31(6), 863–876 (2004)
Ceselli, A., Liberatore, Federico, Righini, Giovanni: A computational evaluation of a general branch-and-price framework for capacitated network location problems. Ann. Oper. Res. 167, 209–251 (2009)
Holmberg, K., Rnnqvist, M., Yuan, D.: An exact algorithm for the capacitated facility location problems with single sourcing. Eur. J. Oper. Res. 113(3), 544–559 (1999)
Daz, J.A., Fernndez, E.: A branch-and-price algorithm for the single source capacitated plant location problem. J. Oper. Res. Soc. 53(7), 728–740 (2002)
Dolan, E.D., More, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)
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The authors wish to thank Alberto Ceselli, Marcus Posta for their codes and test instances and Pasquale Avella for helpful comments.
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Vasilyev, I., Boccia, M. & Hanafi, S. An implementation of exact knapsack separation. J Glob Optim 66, 127–150 (2016). https://doi.org/10.1007/s10898-015-0294-3
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DOI: https://doi.org/10.1007/s10898-015-0294-3