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6 Review Plyhedron Theory_1

# 6 Review Plyhedron Theory_1 - IEE 598 6(I.4 Review...

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IEE 598 - 6 (I.4) Review: Polyhedron Theory Muhong Zhang DEPARTMENT OF INDUSTRIAL ENGINEERING Feb. 11, 2010

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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

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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 0 - x 2 0 x 3 2 } What is the dimension of P ? Consider dim ( P ) = n - rank ( A = , b = ) = max number of affinely independent points in P - 1. Zhang IEE 376 Introduction to OR 5 / 18

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Review: Polyhedral Theory Definition: The inequality π x π 0 or ( π, π 0 ) is called a valid inequality for P if it is satisfied by all points in P .
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6 Review Plyhedron Theory_1 - IEE 598 6(I.4 Review...

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