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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 * + 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 * + z * = c , z * Ax * = b , x * x * z * = 0 define 2 n + m conditions on the 2 n + m unknowns x * , * and z * . UCSD Center for Computational Mathematics Slide 3/29, Friday, March 11th, 2011 We can rewrite the conditions A T * + z * = c Ax * = b x * z * = 0 as a set of 2 n + m equations F ( x ,, z ) = 0, where F ( x ,, z ) = c A T  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 , and z with x and z 0. Basic idea : Define x , and z such that F ( x + x , + , z + z ) with x + x 0 and z + z 0. UCSD Center for Computational Mathematics Slide 5/29, Friday, March 11th, 2011 We want x , and z such that c A T ( + ) ( z + z ) = 0 , z + z A ( x + x ) b = 0 , x + x ( x + x ) ( z + z ) = 0 Moving known quantities to the righthand side, we get A T  z = z c + A T A x = b Ax x z + x z + x z = x z UCSD Center for Computational Mathematics Slide 6/29, Friday, March 11th, 2011 If we assume that the nonlinear term x z 0, then we get linear equations for x , and z , i.e., A T  z = z c + A T A x = b Ax x z + x 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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 Spring '08
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 Linear Programming

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