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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 ShermanMorrison formula The ShermanMorrison formula is useful in proving theoretical results (it is unreliable for practical computation). First, consider the case in which ¯ A is a general rankone modification of the nonsingular matrix A : ¯ A = A + uv T where u and v are nvectors. 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 rankone 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 rankone 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 ShermanMorrison 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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 Winter '08
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 Math, Linear Programming, Optimization, Analysis of algorithms, Computational complexity theory, Best, worst and average case, UCSD Center for Computational Mathematics

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