Lecture5 - Advanced Operations Research Techniques IE316...

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Unformatted text preview: Advanced Operations Research Techniques IE316 Lecture 5 Dr. Ted Ralphs IE316 Lecture 5 1 Reading for This Lecture • Bertsimas 2.5-2.7 IE316 Lecture 5 2 Existence of Extreme Points Definition 1. A polyhedron P ∈ R n contains a line if there exists a vector x ∈ P and a nonzero vector d ∈ R n such that x + λd ∈ P ∀ λ ∈ R . Theorem 1. Suppose that the polyhedron P = { x ∈ R n | Ax ≥ b } is nonempty. Then the following are equivalent: • The polyhedron P has at least one extreme point. • The polyhedron P does not contain a line. • There exist n rows of A that are linearly independent. IE316 Lecture 5 3 Optimality of Extreme Points Theorem 2. Let P ⊆ R n be a polyhedron and consider the problem min x ∈P c T x for a given c ∈ R n . If P has at least one extreme point and there exists an optimal solution, then there exists an optimal solution that is an extreme point....
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This note was uploaded on 08/06/2008 for the course IE 316 taught by Professor Ralphs during the Fall '08 term at Lehigh University .

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Lecture5 - Advanced Operations Research Techniques IE316...

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