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Lecture5

Lecture5 - Advanced Operations Research Techniques IE316...

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Advanced Operations Research Techniques IE316 Lecture 5 Dr. Ted Ralphs

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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.

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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. Proof :
IE316 Lecture 5 4 Optimality in Linear Programming For linear optimization, a finite optimal cost is equivalent to the

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