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Lecture19

Lecture19 - Integer Programming IE418 Lecture 19 Dr Ted...

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Integer Programming IE418 Lecture 19 Dr. Ted Ralphs

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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 .

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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 facet-defining inequalities are those of maximal dimension, i.e., dimension one less than the dimension of the polyhedron. The facet-defining 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 0 ) 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 0 for i = 1 , . . . , k .
IE418 Lecture 19 4 Facet Proofs How do we prove an inequality is facet-defining? 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 facet-defining for conv( S ) . First, we show that dim (conv( S )) = m + 1 ( how? ). For a chosen i , we exhibit m +1 affinely independent points in X that satisfy x i = y ( how? ).

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IE418 Lecture 19 5 Facet Proofs: Another Method In the case where P is full-dimensional , another method for proving ( π, π 0 ) defines a facet is the following.
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