Abstract
Gomory mixed-integer (GMI) cuts generated from optimal simplex tableaus are known to be useful in solving mixed-integer programs. Further, it is well-known that GMI cuts can be derived from facets of Gomory’s master cyclic group polyhedron and its mixed-integer extension studied by Gomory and Johnson. In this paper we examine why cutting planes derived from other facets of master cyclic group polyhedra (group cuts) do not seem to be as useful when used in conjunction with GMI cuts. For many practical problem instances, we numerically show that once GMI cuts from different rows of the optimal simplex tableau are added to the formulation, all other group cuts from the same tableau rows are satisfied.
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Achterberg, T., Kock, T., Martin, A.: MIPLIB 2003. ZIB Technical report 05-28
Araoz J., Gomory R.E., Johnson E.L. and Evans L. (2003). Cyclic group and knapsack facets. Math. Programm. 96: 377–408
Atamtürk A. (2003). On the facets of the mixed-integer knapsack polyhedron. Math. Programm. 98: 145–175
Balas E., Ceria S., Cornuéjols G. and Natraj G. (1996). Gomory cuts revisited. Oper. Res. Lett. 19: 1–9
Bixby R.E., Ceria S., McZeal C.M. and Savelsbergh M.W.P. (1998). An updated mixed integer programming library: MIPLIB 3.0. Optima 58: 12–15
Bixby R.E., Fenelon M., Gu Z., Rothberg E. and Wunderling R. (2000). MIP: Theory and practice—closing the gap. In: Powell, M.J.D. and Scholtes, S. (eds) System Modelling and Optimization: Methods, Theory, and Applications., pp 19–49. Kluwer Academic, Dordrecht
Cornuejols G., Li Y. and Vandenbussche D. (2003). K-Cuts: a variation of Gomory mixed integer cuts from the LP tableau. INFORMS J. Comput. 15: 385–396
Dash S. and Günlük O. (2006). Valid inequalities based on simple mixed-integer sets. Math. Programm. 105: 29–53
Dash S. and Günlük O. (2006). Valid inequalities based on the interpolation procedure. Math. Programm. 106: 111–136
Dash, S., Günlük, O., Goycoolea, M.: Two step MIR inequalities for mixed-integer programs. Manuscript (2005)
Fischetti M., Glover F. and Lodi A. (2004). Feasibility pump. Math. Programm. 104: 91–104
Fischetti, M., Lodi, A.: Local branching, mathematical programming (in press)
Fischetti, M., Saturni, C.: Mixed integer cuts from cyclic groups. Math. Programm. (in press)
Forrest, J.J.: CLP, an open source code for solving linear programming problems, http://www.coin-or.org/clp/index.html
Gomory R.E. (1969). Some polyhedra related to combinatorial problems. J. Linear Algebra Appl. 2: 451–558
Gomory R.E. and Johnson E. (1972). Some continuous functions related to corner polyhedra I. Math. Programm. 3: 23–85
Gomory R.E. and Johnson E. (1972). Some continuous functions related to corner polyhedra II. Math. Programm. 3: 359–389
Gomory R.E., Johnson E.L. and Evans L. (2003). Corner polyhedra and their connection with cutting planes. Math. Programm. 96: 321–339
http://miplib.zib.de
http://www.or.deis.unibo.it/research_pages/ORinstances/MIPs.html
Marchand H. and Wolsey L. (2001). Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49: 363–371
Mittelmann, H.: MILPLib. http://plato.asu.edu/topics/testcases.html
Schrijver A. (1986). Theory of Linear and Integer Programming. Wiley, New York
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Dash, S., Günlük, O. On the strength of Gomory mixed-integer cuts as group cuts. Math. Program. 115, 387–407 (2008). https://doi.org/10.1007/s10107-007-0179-4
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DOI: https://doi.org/10.1007/s10107-007-0179-4