Projected Chvátal–Gomory cuts for mixed integer linear programs
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Recent experiments by Fischetti and Lodi show that the first Chvátal closure of a pure integer linear program (ILP) often gives a surprisingly tight approximation of the integer hull. They optimize over the first Chvátal closure by modeling the Chvátal–Gomory (CG) separation problem as a mixed integer linear program (MILP) which is then solved by a general- purpose MILP solver. Unfortunately, this approach does not extend immediately to the Gomory mixed integer (GMI) closure of an MILP, since the GMI separation problem involves the solution of a nonlinear mixed integer program or a parametric MILP. In this paper we introduce a projected version of the CG cuts, and study their practical effectiveness for MILP problems. The idea is to project first the linear programming relaxation of the MILP at hand onto the space of the integer variables, and then to derive Chvátal–Gomory cuts for the projected polyhedron. Though theoretically dominated by GMI cuts, projected CG cuts have the advantage of producing a separation model very similar to the one introduced by Fischetti and Lodi, which can typically be solved in a reasonable amount of computing time.
KeywordsMixed integer linear program Chvátal–Gomory cut Separation problem Projected polyhedron
Mathematics Subject Classification (2000)90C10 90C11 90C57
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- 1.Ascheuer, N.: Hamiltonian path problems in the on-line optimization of flexible manufacturing systems. PhD Thesis, Technische Universität Berlin, Berlin (1995)Google Scholar
- 4.Balas, E., Saxena, A.: Optimizing over the Split Closure: Modeling and Theoretical Analysis, IMA “Hot Topics” Workshop: Mixed Integer Programming, Minneapolis, 25–29 July 2005Google Scholar
- 5.Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P.: MIPLIB 3.0, http://www.caam. rice.edu/~bixby/miplib/miplib.htmlGoogle Scholar
- 10.COIN-OR. www.coin-or.orgGoogle Scholar
- 14.Dash, S., Günlük, O., Lodi, A.: On the MIR closure of polyhedra. IBM, T.J. Watson Research, Working paper (2005)Google Scholar
- 16.Fischetti, M., Lodi, A. : Optimizing over the first Ch closure. In: Jünger, M., Kaibel, V. (eds.) Integer Programming and Combinatorial Optimization—IPCO 2005, LNCS 3509., pp. 12–22. Springer, Berlin Heidelberg New York (2005)Google Scholar
- 18.Gomory, R.E.: An algorithm for integer solutions to linear programs. In: Graves, R.L., Wolfe, P. (eds.) Recent Advances in Mathematical Programming., pp. 269–302. McGraw-Hill, New York (1963)Google Scholar
- 20.ILOG Cplex 9.0: User’s Manual and Reference Manual, ILOG, S.A., http://www.ilog.com/ (2005)Google Scholar