Lecture5

Lecture5 - Introduction to Mathematical Programming IE406...

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Unformatted text preview: Introduction to Mathematical Programming IE406 Lecture 5 Dr. Ted Ralphs IE406 Lecture 5 1 Reading for This Lecture Bertsimas 2.5-2.7 IE406 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 + d P 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. IE406 Lecture 5 3 Optimality of Extreme Points Theorem 2. Let P R n be a polyhedron and consider the problem min x P c 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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Lecture5 - Introduction to Mathematical Programming IE406...

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