Abstract
We discuss an implementation of the lexicographic version of Gomory’s fractional cutting plane method and of two heuristics mimicking the latter. In computational testing on a battery of MIPLIB problems we compare the performance of these variants with that of the standard Gomory algorithm, both in the single-cut and in the multi-cut (rounds of cuts) version, and show that they provide a radical improvement over the standard procedure. In particular, we report the exact solution of ILP instances from MIPLIB such as stein15, stein27, and bm23, for which the standard Gomory cutting plane algorithm is not able to close more than a tiny fraction of the integrality gap. We also offer an explanation for this surprizing phenomenon.
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References
Arthur, J.L., Ravindran, A.: PAGP, a partitioning algorithm for (linear) goal programming problems. ACM Trans. Math. Softw. 6(3), 378–386 (1980)
Balas, E., Ceria, S., Cornuéjols, G., Natraj, N.: Gomory cuts revisited. Operations Research Letters 19, 1–9 (1996)
Balinski, M.L., Tucker, A.W.: Duality theory of linear programs: A constructive approach with applications. SIAM Review 11(3), 347–377 (1969)
Borg, I., Groenen, P.J.F.: Modern Multidimensional Scaling: Theory and Applications. Springer, Heidelberg (2005)
Balas, E., Saxena, A.: Optimizing over the split closure. Mathematical Programming (2006)
Fischetti, M., Lodi, A.: Optimizing over the first Chvátal closure. Mathematical Programming B 110(1), 3–20 (2007)
Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bulletin of the American Society 64, 275–278 (1958)
Gomory, R.E.: An algorithm for the mixed integer problem. Technical Report RM-2597, The RAND Cooperation (1960)
Gomory, R.E.: An algorithm for integer solutions to linear programming. In: Graves, R.L., Wolfe, P. (eds.) Recent Advances in Mathematical Programming, pp. 269–302. McGraw-Hill, New York (1963)
Tamiz, M., Jones, D.F., El-Darzi, E.: A review of goal programming and its applications. Annals of Operations Research (1), 39–53 (1995)
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Zanette, A., Fischetti, M., Balas, E. (2008). Can Pure Cutting Plane Algorithms Work?. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2008. Lecture Notes in Computer Science, vol 5035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68891-4_29
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DOI: https://doi.org/10.1007/978-3-540-68891-4_29
Publisher Name: Springer, Berlin, Heidelberg
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