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An implementation of exact knapsack separation. (English) Zbl 1355.90051
Summary: 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.

##### MSC:
 90C10 Integer programming 90C27 Combinatorial optimization 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
##### Software:
Tabu search; Knapsack; OR-Library; Concorde
Full Text:
##### References:
 [1] Ceselli, A, Two exact algorithms for the capacitated p-Median problem, 4OR, 1, 319-340, (2003) · Zbl 1109.90054 [2] Ceselli, A; Righini, G, A branch and price algorithm for the capacitated $$p$$-Median problem, Networks, 45, 125-142, (2004) · Zbl 1101.68722 [3] 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) · Zbl 1203.90137 [4] 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) · Zbl 1190.90153 [5] 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) · Zbl 1170.90521 [6] Vasilev, I, A cutting plane method for knapsack polytope, J. Comput. Syst. Sci. Int., 48, 70-77, (2009) · Zbl 1269.90097 [7] Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988) · Zbl 0652.90067 [8] Boyd, EA, Generating Fenchel cutting planes for knapsack polyhedra, SIAM J. Optim., 3, 734-750, (1993) · Zbl 0797.90067 [9] Boyd, EA, Solving integer programs with cutting planes and preprocessing, IPCO, 1993, 209-220, (1993) · Zbl 0923.90118 [10] Boyd, EA, Fenchel cutting planes for integer programming, Oper. Res., 42, 53-64, (1994) · Zbl 0809.90104 [11] Boyd, EA, On the convergence of Fenchel cutting planes in mixed-integer programming, SIAM J. Optim., 5, 421-435, (1995) · Zbl 0834.90095 [12] Fukasawa, R; Goycoolea, M, On the exact separation of mixed integer knapsack cuts, Math. Program., 128, 19-41, (2011) · Zbl 1218.90126 [13] Kaparis, K; Letchford, AN, Separation algorithms for 0-1 knapsack polytopes, Math. Program., 124, 69-91, (2010) · Zbl 1198.90297 [14] 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) · Zbl 0923.90118 [15] Espinoza, D.G.: On Linear Programming, Integer Programming and Cutting Planes. PhD thesis, Georgia Institute of Technology, School of Industrial and Systems Engineering (2006) [16] 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, 165-171, (2012) · Zbl 1462.90101 [17] Avella, P; Boccia, M; Vasilyev, I, Computational experience with general cutting planes for the set covering problem, Oper. Res. Lett., 37, 16-20, (2009) · Zbl 1154.90603 [18] 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) · Zbl 1106.90369 [19] 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) [20] Cornuejols, G; Lemarechal, C, A convex-analysis perspective on disjunctive cuts, Math. Program., 106, 567-586, (2006) · Zbl 1149.90175 [21] Pisinger, D, A minimal algorithm for the 0-1 knapsack problem, Oper. Res., 46, 758-767, (1995) · Zbl 0902.90126 [22] Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin (2005) · Zbl 1103.90003 [23] Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley, London (1990) · Zbl 0708.68002 [24] Posta, M; Ferland, JA; Michelon, P, An exact method with variable fixing for solving the generalized assignment problem, Comput. Optim. Appl., 52, 629-644, (2012) · Zbl 1259.90062 [25] Beasley, JE, Or-library: distributing test problems by electronic mail, J. Oper. Res. Soc., 41, 1069-1072, (1990) [26] Glover, F; Hultz, J; Klingman, D, Improved computer-based planning techniques. part ii, Interfaces, 9, 12-20, (1979) [27] French, AP; Wilson, JM, Heuristic solution methods for the multilevel generalized assignment problem, J. Heuristics, 8, 143-153, (2002) · Zbl 1013.90087 [28] Laguna, M; Kelly, JP; Gonzlez-Velarde, JL; Glover, F, Tabu search for the multilevel generalized assignment problem, Eur. J. Oper. Res., 82, 176-189, (1995) · Zbl 0905.90122 [29] Osorio, MA; Laguna, M, Logic cuts for multilevel generalized assignment problems, Eur. J. Oper. Res., 151, 238-246, (2003) · Zbl 1033.90066 [30] Ceselli, A; Righini, G, A branch-and-price algorithm for the multilevel generalized assignment problem, Oper. Res., 54, 1172-1184, (2006) · Zbl 1167.90531 [31] 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) · Zbl 0994.90087 [32] Lorena, L; Senne, E, A column generation approach to capacitated p-Median problems, Comput. Oper. Res., 31, 863-876, (2004) · Zbl 1048.90132 [33] 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) · Zbl 1172.90007 [34] Holmberg, K; Rnnqvist, M; Yuan, D, An exact algorithm for the capacitated facility location problems with single sourcing, Eur. J. Oper. Res., 113, 544-559, (1999) · Zbl 0947.90059 [35] Daz, JA; Fernndez, E, A branch-and-price algorithm for the single source capacitated plant location problem, J. Oper. Res. Soc., 53, 728-740, (2002) · Zbl 1130.90354 [36] Dolan, ED; More, JJ, Benchmarking optimization software with performance profiles, Math. Program., 91, 201-213, (2002) · Zbl 1049.90004
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