Lecture 25 - Interior Methods for Linear Programming

Lecture 25 - Interior Methods for Linear Programming -...

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Unformatted text preview: Lecture 25 Interior Methods UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Friday, March 13th, 2009 Recap: using the optimality conditions as equations We are considering the problem in all-inequality form: minimize x ∈ R n c T x subject to Ax ≥ b The optimality conditions Ax- r = b , r ≥ A T λ = c , λ ≥ r · λ = 0 define n + 2 m equations in the n + 2 m unknowns x , r and λ . UCSD Center for Computational Mathematics Slide 2/22, Friday, March 13th, 2009 Recap: using the optimality conditions as equations The equations are F ( x , r ,λ ) = 0, where F ( x , r ,λ ) = Ax- r- b A T λ- c r · λ These equations are almost linear . The only nonlinear terms come from the nonlinear equations r · λ = 0 . UCSD Center for Computational Mathematics Slide 3/22, Friday, March 13th, 2009 Suppose that we are given an approximate solution x , r and λ with r ≥ 0 and λ ≥ 0. We seek changes Δ x , Δ r and Δλ to x , r and λ such that F ( x + Δ x , r + Δ r ,λ + Δλ ) ≈ with r + Δ r ≥ 0 and λ + Δλ ≥ 0. UCSD Center for Computational Mathematics Slide 4/22, Friday, March 13th, 2009 Find Δ x , Δ r and Δλ such that A ( x + Δ x )- ( r + Δ r ) = b , r + Δ r ≥ A T ( λ + Δλ ) = c , λ + Δλ ≥ ( r + Δ r ) · ( λ + Δλ ) = 0 If move the known quantities to the right-hand side, we get A Δ x- Δ r = b- Ax + r A T Δλ = c- A T λ λ · Δ r + r · Δλ + Δ r · Δλ =- r · λ If we ignore the nonlinear term Δ r · Δλ , then we get a set of linear equations for Δ x , Δ r and Δλ . UCSD Center for Computational Mathematics Slide 5/22, Friday, March 13th, 2009 The linear equations are: A Δ x- Δ r = b- Ax + r A T Δλ = c- A T λ λ · Δ r + r · Δλ =- r · λ To help simplify the notation we define diagonal matrices from the elements of r and λ ....
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This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.

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Lecture 25 - Interior Methods for Linear Programming -...

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