Lecture 24 - Complexity of the Simplex Method

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

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

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