6 Review Plyhedron Theory_1

6 Review Plyhedron Theory_1 - IEE 598 - 6 (I.4) Review:...

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Unformatted text preview: IEE 598 - 6 (I.4) Review: Polyhedron Theory Muhong Zhang DEPARTMENT OF INDUSTRIAL ENGINEERING Feb. 11, 2010 Review: Polyhedral Theory Definition: H = { x R n : ax = b } is called a hyperplane . Definition: S = { x R n : ax b } is called a (closed) half space . Definition: A polyhedron P R n is the set of points that satisfy a finite number of linear independent inequalities, i.e. P = { x R n : Ax b } . Definition: A polyhedron P R n is bounded if there exists a constant K such that | x i | K , i = 1 ,..., n for all x P . Definition: A bounded polyhedron is called a polytope . Definition: A set S R n is said to be convex if conv ( S ) = S . Proposition: All polyhedra are convex. Zhang IEE 376 Introduction to OR 2 / 18 Review: Polyhedral Theory Definition: The dimension of a polyhedron P , denoted by dim ( P ) , is k if the maximum number of affinely independent points in P is k + 1. Example: dim ( { x } ) = , where x R n . dim ( {} ) = . Definition: A polyhedron P R n is said to be full-dimensional if dim ( P ) = n . Let P = { x R n : Ax b } = { x R n : a i x b i , i M } , where M = { 1 , ..., m } . M = = { i M : a i x = b i , x P } , Equality set. M = M \ M = , Inequality set. Let ( A = , b = ) , ( A , b ) be the corresponding rows of ( A , b ) . Zhang IEE 376 Introduction to OR 3 / 18 Review: Polyhedral Theory Definition: x P is called an inner point if a i x < b i for all i M . Definition: x P is called an interior point if a i x < b i for all i M . Proposition: Every nonempty polyhedron has an inner point. Proof. Proposition: If P R n and P 6 = , then dim ( P ) + rank ( A = , b = ) = n . Proof. Zhang IEE 376 Introduction to OR 4 / 18 Review: Polyhedral Theory Example: P = { x R 3 : x 1 + x 2 + x 3 1 ,- x 1- x 2- x 3 - 1 x 1 + x 3 1- x 1 - x 2 x 3 2 } What is the dimension of P ?...
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This note was uploaded on 05/01/2011 for the course IEE 598 taught by Professor Hillary during the Spring '10 term at USC.

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6 Review Plyhedron Theory_1 - IEE 598 - 6 (I.4) Review:...

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