This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 fulldimensional 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 ?...
View
Full
Document
This note was uploaded on 05/01/2011 for the course IEE 598 taught by Professor Hillary during the Spring '10 term at USC.
 Spring '10
 Hillary

Click to edit the document details