Abstract
We give several results which characterize MIP-representability of sets and functions, and the finite union of MIP-representable sets. These results extend earlier ones due to R.R. Meyer. We provide representations whose linear relaxations have an efficient formulation, while retaining maximum accuracy to the problem modelled. Under mild hypotheses, these properties are retained as integer variables are set to values in nodes of branch-and-bound trees.
This author’s research has been partially supported by grant ECS8001763 of the National Science Foundation, USA and completed while the author was an Alexander-von-Humboldt Senior Scientist at Institute for ökonometrie and Operations Research of the University of Bonn.
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References
E. Balas, “Disjunctive programming: Facets of the convex hull of feasible points,” Report 348, GSIA, Carnegie-Mellon University (Pittsburgh, PA, 1974).
M. Bazaraa and M. Shetty, Nonlinear programming (Wiley, New York, 1979).
E.M.L. Beale, “Branch-and-bound methods for mathematical programming,” in: P.L. Hammer, E.L. Johnson and B.H. Korte, eds., Discrete optimization II (North-Holland, Amsterdam, 1979).
G. Dantzig, Linear programming and extensions (Princeton University Press, Princeton, NJ, 1963).
R.G. Jeroslow, “Representations of unbounded optimizations as integer programs,” Journal of Optimization Theory and its Applications 30 (1980) 339–351.
R.G. Jeroslow, “Cutting-planes for relaxation of integer programs,” MSSR NO. 347, GSIA, Carnegie-Mellon University (Pittsburgh, PA, 1974).
R.G. Jeroslow, “Cutting plane theory: Disjunctive methods,” Annals of Discrete Mathematics 1 (1977) 293–330.
R.G. Jeroslow, “Some basis theorems for integral monoids,” Mathematics of Operations Research 3 (1978) 145–154.
R.G. Jeroslow and J.K. Lowe, “Modelling with integer variables,” College of Management, Georgia Institute of Technology (Atlanta, GA, 1981).
R.R. Meyer, “On the existence of optimal solutions to integer and mixed-integer programming problems,” Mathematical Programming 7 (1974) 223–235.
R.R. Meyer, “Integer and mixed-integer programming models: General properties,” Journal of Optimization Theory and Applications 16 (1975) 191–206.
R.R. Meyer, “Mixed integer minimization models for piecewise linear functions of a single variable,” Discrete Mathematics 16 (1976) 163–171.
R.R. Meyer, “A theoretical and computational comparison of ‘equivalent’ mixed integer formulations,” Naval Research Logistics Quarterly 28 (1981) 115–131.
R.R. Meyer, M.V. Thakkar and W.P. Hallman, “Rational mixed integer and polyhedral union minimization models,” Mathematics of Operations Research 5 (1980) 135–146.
R.T. Rockafeller, Convex analysis (Princeton University Press, Princeton, NJ, 1970).
J. Stoer and C. Witzgall, Convexity and optimization in finite dimensions I (Springer-Verlag, New York, 1970).
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© 1984 The Mathematical Programming Society, Inc.
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Jeroslow, R.G., Lowe, J.K. (1984). Modelling with integer variables. In: Korte, B., Ritter, K. (eds) Mathematical Programming at Oberwolfach II. Mathematical Programming Studies, vol 22. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121015
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DOI: https://doi.org/10.1007/BFb0121015
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