Tight representation of logical constraints as cardinality rules
John Hooker1999, Mathematical Programming
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Abstract
A mathematical programming model may contain qualitative as well as quantitative elements. One may, for example, wish to combine a rule base with numerical constraints. This raises the issue of how to represent logical constraints in inequality form so that they have a useful linear relaxation. We provide a simple recursive procedure that generates a convex hull description of any logical condition that can be written as a "cardinality rule", which seems to be a form that occurs often in practice. A cardinality rule asserts that if at least k of the propositions A 1 , . . . , A m are true, then at least of the propositions B 1 , . . . , B n are true. The main result of the paper is that the procedure in fact provides a convex hull description.
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