Abstract
Most solution methods for Mixed Integer Programming (MIP) problems require repeated solution of continuous linear programming (LP) problems. It is typical that subsequent LP problems differ just slightly. In most cases it is practically the right-hand-side that changes. For such a situation the dual simplex algorithm (DSA) appears to be the best solution method.
The LP relaxation of a MIP problem contains many bounded variables. This necessitates such an implementation of the DSA where variables of arbitrary type are allowed. The paper presents an algorithm called BSD for the efficient handling of bounded variables. This leads to a “mini” nonlinear optimization in each step of the DSA. Interestingly, this technique enables several cheap iterations per selection making the whole algorithm very attractive. The implementational implications and some computational experiences on large scale MIP problems are also reported. BSD is included in FortMP optimization system of Brunel University.
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References
Chvâtal, V., Linear Programming, Freeman and Co., 1983.
Dantzig, G. B., Linear Programming and Extensions, Princeton University Press, Princeton, N.J., 1963.
Ellison, E. F. D., Hajian, M., Levkovitz, R., Maros, I., Mitra, G., Sayers, D., “FortMP Manual”, Brunel, The University of West London, May 1994.
Fourer, R., “Notes on the Dual Simplex Method”, Unpublished, March, 1994.
Greenberg, H. J., “Pivot selection tactics”, in Greenberg, H. J. (ed.), Design and Implementation of Optimization Software, Sijthoff and Nordhoff, 1978.
Lemke, C. E., “The Dual Method of Solving the Linear Programming Problem”, Naval Research Logistics Quarterly, 1, 1954, p. 36–47.
Maros, I., “A general Phase-I method in linear programming”, European Journal of Operational Research, 23 (1986), p. 64–77.
Maros, I., Mitra, G., “Simplex Algorithms”, Chapter 1 in Beasley J. (ed.) Advances in Linear and Integer Programming, Oxford University Press 1996.
Nemhauser, G. L., Wolsey, L. A., Integer and Combinatorial Optimization, John Wiley, 1988.
Orchard-Hays, W., Advanced Linear-Programming Computing Techniques, McGraw-Hill, 1968.
Wolfe, P., “The composite simplex algorithm”, SIAM Review, 7 (1), 1965, p. 42–54.
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© 1998 Springer Science+Business Media Dordrecht
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Maros, I. (1998). A Piecewise Linear Dual Procedure in Mixed Integer Programming. In: Giannessi, F., Komlósi, S., Rapcsák, T. (eds) New Trends in Mathematical Programming. Applied Optimization, vol 13. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2878-1_12
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DOI: https://doi.org/10.1007/978-1-4757-2878-1_12
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