6 Review Plyhedron Theory

# 6 Review Plyhedron Theory - IEE 598 6(I.4 Review Polyhedron...

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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 / 14 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 / 14 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 ....
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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 - IEE 598 6(I.4 Review Polyhedron...

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