lecture13a

# lecture13a - CS 435 Linear Optimization Fall 2008 Lecture...

This preview shows pages 1–2. Sign up to view the full content.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 13: Introduction to Duality Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay Let x be an extreme point. Suppose it is given by A x = b ; A 00 x < b 00 . We call A x = b as the de ning hyperplanes of x . We know that the neighbours of x are along the columns of   A   1 . 1 Proof of correctness of Simplex algorithm Theorem 1 If the cost does not increase along any of the columns of   A   1 then x is optimal. Proof: The columns of   A   1 span R n . Let x opt be an optimal point. We need to show that c T x opt c T x . Since the columns of   A   1 form a basis of R n (why?) the vector x opt   x can be represented as a linear combination of them say as given below. x opt   x = X j (   A   1 ) ( j ) (1) Premultiplying both sides with A yields A x opt   A x = X j A (   A   1 ) ( j ) : (2) We know that A x opt b and A x = b hence A ( x opt   x ) 0. That is, the lhs is a vector, all of whose components are non-positive. Also note that A (   A   1 ) ( j ) is an n ¢ 1 vector whose j th element is   1 and remaining elements are 0. Hence A x opt   A x = B B B B B B @   1   2 : : :   n 1 C C C C C C A (3) This implies that j 0 for all j . Now, c T x opt   c T x = X j c T (   A   1 ) ( j ) (4) Since the cost decreases along the columns of   A   1 we have c T (   A   1 ) ( j ) 0 and since j 0 we conclude that P j c T (   A   1 ) j 0. Hence c T x opt c T x , as desired. £ Note: From the above theorem we infer that when the Simplex algorithm terminates it gives us an optimal solution....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

lecture13a - CS 435 Linear Optimization Fall 2008 Lecture...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online