Lecture 24 - Complexity of the Simplex Method

Lecture 24 - Complexity of the Simplex Method - Lecture 24...

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 Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 24 Complexity of the simplex method UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Wednesday, March 11th, 2009 Recap: complexity of the simplex method Given a particular linear program, can we predict how long it will take to be solved? There are two issues: How much time does it take to do one iteration? How many iterations will be required overall? UCSD Center for Computational Mathematics Slide 2/26, Wednesday, March 11th, 2009 Recap: Column replacement To replace column t (i.e., a t ) of B by zeros , we form B- a t e T t To add column ¯ a in place of the zeros we form B- a t e T t + ¯ ae T t We can combine these steps as ¯ B = B + (¯ a- a t ) e T t which means that column t of B is replaced by ¯ a . UCSD Center for Computational Mathematics Slide 3/26, Wednesday, March 11th, 2009 Recap: The Sherman-Morrison formula The Sherman-Morrison formula is useful in proving theoretical results (it is unreliable for practical computation). First, consider the case in which ¯ A is a general rank-one modification of the nonsingular matrix A : ¯ A = A + uv T where u and v are n-vectors. Then ¯ A- 1 = ( A + uv T )- 1 = A- 1- 1 τ A- 1 uv T A- 1 , with τ = 1 + v T A- 1 u ¯ A is nonsingular if and only if 1 + v T A- 1 u 6 = 0, and its inverse is a rank-one modification of the inverse of A : UCSD Center for Computational Mathematics Slide 4/26, Wednesday, March 11th, 2009 From the previous slide: ¯ A- 1 = A- 1- 1 τ A- 1 uv T A- 1 , with τ = 1 + v T A- 1 u The crucial feature is that, given A- 1 , only n 2 operations are needed to update ¯ A- 1 . Then solutions of ¯ Ax = b and ¯ A T y = c are x = ¯ A- 1 b and y = ( ¯ A- 1 ) T c Column replacement is a special case of rank-one modification. UCSD Center for Computational Mathematics Slide 5/26, Wednesday, March 11th, 2009 ¯ B = B + (¯ a- a t ) e T t The inverse of ¯ B is given by the Sherman-Morrison formula with u = ¯ a- a t and v = e t By definition, Be t = a t , so that B- 1 a t = e t ....
View Full Document

{[ snackBarMessage ]}

Page1 / 7

Lecture 24 - Complexity of the Simplex Method - Lecture 24...

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