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Unformatted text preview: Integer Programming IE418 Lecture 19 Dr. Ted Ralphs IE418 Lecture 19 1 Reading for This Lecture • Nemhauser and Wolsey Sections II.2.1 • Wolsey Chapter 9 IE418 Lecture 19 2 Generating Stronger Valid Inequalities • We have now seen some “generic” methods of generating valid inequalities. • In general, these methods are not capable of generating strong inequalities (facets). • To generate such inequalities, we must use our knowledge of the problem structure. • This is more of an art than a science . IE418 Lecture 19 3 The Strength of a Valid Inequality • Roughly speaking, for an inequality to be strong, the face it defines should have as high a dimension as possible. • The facetdefining inequalities are those of maximal dimension, i.e., dimension one less than the dimension of the polyhedron. • The facetdefining inequalities dominate all others and are the only ones necessary in a complete description of a polyhedron. • To know which inequalities are facets, we use the following result based on methods for determining the dimension of polyhedra. Proposition 1. If ( π, π ) defines a face of dimension k 1 of conv( S ) , then there are k affinely independent points x 1 , . . . , x k ∈ S such that πx i = π for i = 1 , . . . , k . IE418 Lecture 19 4 Facet Proofs • How do we prove an inequality is facetdefining? • Straightforward approach: use the definition . • First, we need to find the dimension d of the polyhedron. • Then, we need to exhibit a set of d affinely independent points in S satisfying the given inequality at equality. • Example : – Set S = { ( x, y ) ∈ R m + × B  ∑ m i =1 x i ≤ my } . – We want to show that the valid inequality x i ≤ y is facetdefining for conv( S ) . – First, we show that dim (conv( S )) = m + 1 ( how? )....
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This note was uploaded on 08/06/2008 for the course IE 418 taught by Professor Ralphs during the Spring '08 term at Lehigh University .
 Spring '08
 Ralphs

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