mthsc810-lecture05-2x2

# mthsc810-lecture05-2x2 - MthSc 810: Mathematical...

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Unformatted text preview: MthSc 810: Mathematical Programming Lecture 5 Pietro Belotti Dept. of Mathematical Sciences Clemson University September 8, 2011 Reading for today: Sections 2.5-2.7 (2.8 bonus . . . ) Sep. 13: AMPL . Reading: Bob Fourer’s notes, Chapters 1-2 (see web page) Existence of extreme points Def.: A polyhedron P ⊆ R n contains a line if there exists x ∈ P and d ∈ R n \ { } such that x + λ d ∈ P ∀ λ ∈ R ex. A bounded polyhedron, i.e., a P ∈ R n such that ∃ M : ∀ x ∈ P , ∀ i = 1 , 2 . . . , n , | x i | ≤ M , contains no line Q.: Does a polyhedron in standard form contain a line? Lines, extreme points, and linear independence Consider a nonempty polyhedron P = { x ∈ R n : a ⊤ i x ≥ b i , ∀ i = 1 , 2 . . . , m } . Recall: x ∈ P is an extreme point of P if ∄ y , z ∈ P \ { x } , λ ∈ [ , 1 ] : x = λ y + ( 1 − λ ) z Theorem The following are equivalent: (a) P has at least one extreme point (b) P does not contain a line (c) ∃ n linearly independent vectors in { a 1 , a 2 . . . a m } So if there is ≥ 1 extreme point ( ≡ contains no line), m ≥ n Also, a bounded polyhedron has at least one BFS.Also, a bounded polyhedron has at least one BFS....
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## This note was uploaded on 03/14/2012 for the course MTHSC 810 taught by Professor Staff during the Fall '08 term at Clemson.

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mthsc810-lecture05-2x2 - MthSc 810: Mathematical...

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