Abstract
In this paper we use facets of simple mixed-integer sets with three variables to derive a parametric family of valid inequalities for general mixed-integer sets. We call these inequalities two-step MIR inequalities as they can be derived by applying the simple mixed-integer rounding (MIR) principle of Wolsey (1998) twice. The two-step MIR inequalities define facets of the master cyclic group polyhedron of Gomory (1969). In addition, they dominate the strong fractional cuts of Letchford and Lodi (2002).
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References
Aardal, K., Weismantel, R., Wolsey, L.A.: Non-standard approaches to integer programming. Discrete Applied Mathematics 123, 5–74 (2002)
Agra, A., Constantino, M. F.: Description of 2-integer continuous knapsack polyhedra. working paper 2003
Araoz, J., Gomory, R.E., Johnson, E.L., Evans, L.: Cyclic group and knapsack facets. Mathematical Programming 96, 377–408 (2003)
Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Mathematical Programming 58, 295–324 (1993)
Balas, E., Ceria, S., Cornuéjols, G., Natraj, G.: Gomory cuts revisited. Operations Research Letters 19, 1–9 (1996)
Bixby, R. E., Fenelon, M., Gu, Z., Rothberg, E., Wunderling, R.: MIP: Theory and practice – closing the gap. In: Powell, M. J. D., Scholtes, S. (eds), System Modelling and Optimization: Methods, Theory, and Applications, pages 19–49. Kluwer Academic Publishers, 2000
Cornuejols, G., Li, Y., Vandenbussche, D.: K-Cuts: A variation of Gomory mixed integer cuts from the LP tableau. INFORMS Journal on Computing 15, 385–396 (2003)
Dash, S., Günlük, O.: Comparing valid inequalities for cyclic group polyhedra. In preparation 2003
Evans, L.: Cyclic groups and knapsack facets with applications to cutting planes. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, Georgia, 2002
Gomory, R.E.: Some polyhedra related to combinatorial problems. Journal of Linear Algebra and its Applications, 2, 451–558 (1969)
Gomory, R.E., Johnson, E.: Some continuous functions related to corner polyhedra I. Mathematical Programming 3, 23–85 (1972)
Gomory, R.E., Johnson, E.: Some continuous functions related to corner polyhedra II. Mathematical Programming 3, 359–389 (1972)
Gomory, R.E., Johnson, E.: T-space and cutting planes. Mathematical Programming 96, 341–375 (2003)
Gomory, R.E., Johnson, E.L., Evans, L.: Corner Polyhedra and their connection with cutting planes. Mathematical Programming 96, 321–339 (2003)
Letchford, A. N., Lodi, A.: Strengthening Chvátal-Gomory cuts and Gomory fractional cuts. Operations Research Letters 30 (2), 74–82 (2002)
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
Wolsey, L.A.: Integer Programming. Wiley, New York 1998
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Dash, S., Günlük, O. Valid inequalities based on simple mixed-integer sets. Math. Program. 105, 29–53 (2006). https://doi.org/10.1007/s10107-005-0599-y
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DOI: https://doi.org/10.1007/s10107-005-0599-y