Abstract
The generalized lattice point (GLP) problem provides a formulation that accommodates a variety of discrete alternative problems. In this paper, we show how to substantially strengthen the convexity cuts for the GLP problem. The new cuts are based on the identification ofsynthesized lattice point conditions to replace those that ordinarily define the cut. The synthesized conditions give an alternative set of hyperplanes that enlarge the convex set, thus allowing the cut to be shifted deeper into the solution space. A convenient feature of the strengthened cuts is the evidence of linking relationships by which they may be constructively generated from the original cuts. Geometric examples are given in the last section to show how the new cuts improve upon those previously proposed for the GLP problem.
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Balas, E.,The Intersection Cut—A New Cutting Plane for Integer Programming, Operations Research, Vol. 19, pp. 19–39, 1970.
Glover, F.,Convexity Cuts and Cut Search, Operations Research, Vol. 21, pp. 123–134, 1973.
Glover, F., andKlingman, D.,The Generalized Lattice Point Problem, Operations Research, Vol. 21, pp. 141–156, 1973.
Glover, F., Klingman, D., andStutz, J.,The Disjunctive Facet Problem:Formulation and Solution Techniques, Operations Research, Vol. 22, pp. 582–602, 1974.
Young, R. D.,Hypercylindrically Deduced Cuts in Zero-One Integer Programming, Operations Research, Vol. 19, pp. 1393–1405, 1971.
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Communicated by M. Avriel
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Glover, F., Klingman, D. Improved convexity cuts for lattice point problems. J Optim Theory Appl 19, 283–291 (1976). https://doi.org/10.1007/BF00934097
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DOI: https://doi.org/10.1007/BF00934097