L12 - CO350 Linear Programming Chapter 5: Basic Solutions...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
CO350 Linear Programming Chapter 5: Basic Solutions 30th May 2005
Background image of page 1

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

View Full DocumentRight Arrow Icon
Chapter 5: Basic Solutions 1 Recap Last week, we learn Definition of a basis B of matrix A . Definition of basic solution x * of Ax = b determined by basis B . x * is a basic solution if and only if { A j : x * j 6 = 0 } is linearly independent. Definition of basic feasible solution of { Ax = b, x 0 } . Definition of convex sets. Definition of extreme points of convex sets.
Background image of page 2
2 Relating bfs and extreme points Theorem 5.3 (Pg 63) Let A be m by n with rank m . Let F be the set { x : Ax = b, x 0 } . x * is a bfs of { Ax = b, x 0 } ⇐⇒ x * is an extreme point of F . Proof : “ = ” [Contradiction] Suppose x * is a bfs. x * is a feasible solution = x * F . x * is basic solution = x * is determined by some basis B . Suppose x * is not an extreme point of F . x * F = x * lie strictly between two vectors in F . I.e., there are x 1 ,x 2 F with x 1 6 = x 2 and 0 < λ < 1 such that x * = λx 1 + (1 - λ ) x 2 . For each j / B , x * j = 0 . So 0 = x * j = λx 1 j + (1 - λ ) x 2 j x 1 ,x 2 0 and 0 < λ < 1 = λx 1 j and (1 - λ ) x 2 j 0 . So λx 1 j = (1 - λ ) x 2 j = 0 . Hence x 1 j = x 2 j = 0 . Now x 1 j = x 2 j = 0 for all j / B , and also Ax 1 = Ax 2 = b . This means that x 1 and x 2 are both basic solutions of Ax = b determined by B . This contradicts
Background image of page 3

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

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 06/11/2011 for the course C 350 taught by Professor Wolkowicz during the Fall '97 term at Waterloo.

Page1 / 10

L12 - CO350 Linear Programming Chapter 5: Basic Solutions...

This preview shows document pages 1 - 4. Sign up to view the full document.

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