Lecture16 - Integer Programming IE418 Lecture 16 Dr. Ted...

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Unformatted text preview: Integer Programming IE418 Lecture 16 Dr. Ted Ralphs IE418 Lecture 16 1 Reading for This Lecture • Wolsey Section 9.1 • N&W Sections I.4.4 and I.4.6 IE418 Lecture 16 2 Describing Polyhedra Back in Lecture 4, we derived the following fundamental results. Theorem 1. 1. Every full-dimensional polyhedron P has a unique (up to scalar multiplication) representation that consists of one inequality representing each facet of P . 2. If dim ( P ) = n- k with k > , then P is described by a maximal set of linearly independent rows of ( A = , b = ) , as well as one inequality representing each facet of P . Theorem 2. If a facet F of P is represented by ( π, π ) , then the set of all representations of F is obtained by taking scalar multiples of ( π, π ) plus linear combinations of the equality set of P . IE418 Lecture 16 3 Descriptions and Formulations • Recall that the reason we are interested in facets is because they are the “ strongest ” valid inequalities. • We have shown that facet-defining inequalities can never be dominated. • Although necessary for describing the convex hull of feasible solutions, they do not have to appear in the formulation of an integer program. • Adding a facet-defining inequality (that is not already represented) to a formulation necessarily increases its strength. For the remainder of this lecture, let P = { x ∈ R n | Ax ≥ b } for A ∈ Q m × n , b ∈ Q m . IE418 Lecture 16 4 Extreme Points Definition 1. x is an extreme point of P if there do not exist x 1 , x 2 ∈ P such that x = 1 2 x 1 + 1 2 x 2 . Proposition 1. x is an extreme point of P if and only if x is a zero- dimensional face of P . Proposition 2. If P = ∅ and rank ( A ) = n- k , then P has a face of dimension k and no proper face of lower dimension....
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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 .

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Lecture16 - Integer Programming IE418 Lecture 16 Dr. Ted...

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