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Lecture12 - Introduction to Mathematical Programming IE406...

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Unformatted text preview: Introduction to Mathematical Programming IE406 Lecture 12 Dr. Ted Ralphs IE406 Lecture 12 1 Reading for This Lecture • Bertsimas 4.8-4.9 IE406 Lecture 12 2 Polyhedral Cones Definition 1. A set C ⊂ R n is a cone if λx ∈ C for all λ ≥ and all x ∈ C . Definition 2. A polyhedron of the form P = { x ∈ R n | Ax ≥ } is called a polyhedral cone . Theorem 1. Let C ⊂ R n be the polyhedral cone defined by the matrix A . Then the following are equivalent: 1. The zero vector is an extreme point of C . 2. The cone C does not contain a line. 3. The rows of A span R n . IE406 Lecture 12 3 Comments on Polyhedral Cones • Notice that the origin is a member of every polyhedral cone. • Furthermore, the origin is the only possible extreme point. • A polyhedral cone that has the origin as an extreme point is called pointed . • Graphically, a pointed cone looks like what we would ordinarily call a cone. IE406 Lecture 12 4 The Recession Cone • Consider a nonempty polyhedron P = { x ∈...
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Lecture12 - Introduction to Mathematical Programming IE406...

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