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Unformatted text preview: Math 171A: Linear Programming Lecture 26 Interior Methods for Linear Programming Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Friday, March 11th, 2011 Consider an LP in standard form: minimize x ∈ R n c T x subject to Ax = b , x ≥ where A has m rows. The optimality conditions are: A T π * + z * = c , z * ≥ Ax * = b , x * ≥ x * · z * = 0 where x * · z * is a vector with components x * i z * i . UCSD Center for Computational Mathematics Slide 2/29, Friday, March 11th, 2011 The optimality conditions : A T π * + z * = c , z * ≥ Ax * = b , x * ≥ x * · z * = 0 define 2 n + m conditions on the 2 n + m unknowns x * , π * and z * . UCSD Center for Computational Mathematics Slide 3/29, Friday, March 11th, 2011 We can rewrite the conditions A T π * + z * = c Ax * = b x * · z * = 0 as a set of 2 n + m equations F ( x ,π, z ) = 0, where F ( x ,π, z ) = c A T π z Ax b x · z These equations are almost linear . The only nonlinear terms come from the nonlinear equations x · z = 0 . UCSD Center for Computational Mathematics Slide 4/29, Friday, March 11th, 2011 Suppose we have an approximate solution x , π and z with x ≥ and z ≥ 0. Basic idea : Define Δ x , Δπ and Δ z such that F ( x + Δ x ,π + Δπ, z + Δ z ) ≈ with x + Δ x ≥ 0 and z + Δ z ≥ 0. UCSD Center for Computational Mathematics Slide 5/29, Friday, March 11th, 2011 We want Δ x , Δπ and Δ z such that c A T ( π + Δπ ) ( z + Δ z ) = 0 , z + Δ z ≥ A ( x + Δ x ) b = 0 , x + Δ x ≥ ( x + Δ x ) · ( z + Δ z ) = 0 Moving known quantities to the righthand side, we get A T Δπ Δ z = z c + A T π A Δ x = b Ax Δ x · z + x · Δ z + Δ x · Δ z = x · z UCSD Center for Computational Mathematics Slide 6/29, Friday, March 11th, 2011 If we assume that the nonlinear term Δ x · Δ z ≈ 0, then we get linear equations for Δ x , Δπ and Δ z , i.e., A T Δπ Δ z = z c + A T π A Δ x = b Ax Δ x · z + x · Δ z = x · z UCSD Center for Computational Mathematics Slide 7/29, Friday, March 11th, 2011 To simplify the equations, we define X = x 1 x 2 ....
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This note was uploaded on 03/19/2012 for the course MATH 171a taught by Professor Staff during the Spring '08 term at UCSD.
 Spring '08
 staff
 Linear Programming

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