Abstract
We study the interpolation procedure of Gomory and Johnson (1972), which generates cutting planes for general integer programs from facets of cyclic group polyhedra. This idea has recently been re-considered by Evans (2002) and Gomory, Johnson and Evans (2003). We compare inequalities generated by this procedure with mixed-integer rounding (MIR) based inequalities discussed in Dash and Gunluk (2003). We first analyze and extend the shooting experiment described in Gomory, Johnson and Evans. We show that MIR based inequalities dominate inequalities generated by the interpolation procedure in some important cases. We also show that the Gomory mixed-integer cut is likely to dominate any inequality generated by the interpolation procedure in a certain probabilistic sense. We also generalize a result of Cornuéjols, Li and Vandenbussche (2003) on comparing the strength of the Gomory mixed-integer cut with related inequalities.
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References
Acklam, P.J.: An algorithm for computing the inverse normal cumulative distribution function. http://home.online.no/∼pjacklam/notes/invnorm/index.html
Applegate, D., Cook, W., Dash, S., Mevenkamp, M.: QSopt linear programming solver. http://www.isye.gatech.edu/∼wcook/qsopt
Araoz, J., Gomory, R.E., Johnson, E.L., Evans, L.: Cyclic group and knapsack facets. Math. Prog. 96, 377–408 (2003)
Cornuejols, G., Li, Y., Vandenbussche, D.: K-Cuts: a variation of gomory mixed integer cuts from the LP tableau. INFORMS J. Comput. 15, 385–396 (2003)
Dash, S., Gunluk, O.: Valid inequalities based on simple mixed-integer sets. Math. Prog. To appear
Evans, L.: Cyclic groups and knapsack facets with applications to cutting planes. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, Georgia, 2002
Fishman, G.: Monte Carlo. Springer, New York, 1995
Forrest, J.J.: CLP: Coin LP, a native simplex solver. http://www.ibm.com/developerworks /opensource/coin
Gomory, R.E.: Some polyhedra related to combinatorial problems. J. Linear Algebra and its Applications 2, 451–558 (1969)
Gomory, R.E.: personal communication, 2003
Gomory, R.E., Johnson, E.: Some continuous functions related to corner polyhedra I. Math. Prog. 3, 23–85 (1972)
Gomory, R.E., Johnson, E.: Some continuous functions related to corner polyhedra II. Math. Prog. 3, 359–389 (1972)
Gomory, R.E., Johnson, E.: T-space and cutting planes. Math. Prog. 96, 341–375 (2003)
Gomory, R.E., Johnson, E.L., Evans, L.: Corner polyhedra and their connection with cutting planes. Math. Prog. 96, 321–339 (2003)
Hunsaker, B.: Measuring facets of polyhedra to predict usefulness in branch-and-cut algorithms, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, Georgia, 2003
Kuhn, H.W.: Discussion. In: L. Robinson, (ed.), Proceedings of the IBM Scientific Computing Symposium on Combinatorial Problems: March 16–18, 1964, IBM, White Plains, New York (1966), pp. 118–121
Kuhn, H.W.: On the origin of the Hungarian method. In: J.K. Lenstra, A.H.G. Rinnooy Kan, A. Schrijver, (eds.), History of mathematical programming: a collection of personal reminiscences. Elsevier Science Publishers B.V., The Netherlands (1991), pp. 77–81
Marchand, H., Wolsey, L.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49, 363–371 (2001)
Nemhauser, G.L., Wolsey, L.A.: Integer and combinatorial optimization. Wiley, New York, 1988
Ross, S.: Introduction to probability and statistics. John Wiley and Sons, New York, 1987
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Dash, S., Günlük, O. Valid inequalities based on the interpolation procedure. Math. Program. 106, 111–136 (2006). https://doi.org/10.1007/s10107-005-0600-9
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DOI: https://doi.org/10.1007/s10107-005-0600-9