mthsc810-lecture05-1x2

mthsc810-lecture05-1x2 - MthSc 810: Mathematical...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 Fourers 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 + d P 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 } , [ , 1 ] : x = y + ( 1 ) 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....
View Full Document

Page1 / 6

mthsc810-lecture05-1x2 - MthSc 810: Mathematical...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online