handout19

handout19 - Math 171A: Linear Programming Lecture 19...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 171A: Linear Programming Lecture 19 Finding a Feasible Point II Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Friday, February 18th, 2011 Recap: The phase-1 Linear Program To find a feasible x for minimize x R n c T x subject to Ax b | {z } general constraints , x | {z } simple bounds Solve the linear program : minimize R , x R n subject to a T 1 x + b 1 a T 2 x + b 2 . . . . . . a T m x + b m x , UCSD Center for Computational Mathematics Slide 2/27, Friday, February 18th, 2011 Phase-1 LP example Example: min c T x subject to Ax b , x , for c = 2 1 ! , A = 1 1 1 2- 1 1 , b = 1 1- 2 UCSD Center for Computational Mathematics Slide 3/27, Friday, February 18th, 2011 x 2 x 1 #1 #2 #3 #5 #4 The point x = 0 is infeasible, with r ( x ) = Ax- b = 1 1 1 2- 1 1 x- 1 1- 2 = - 1- 1 2 UCSD Center for Computational Mathematics Slide 5/27, Friday, February 18th, 2011 = max { b i } = max { 1 , 1 ,- 2 } = 1 The phase-1 problem is: minimize R , x R n subject to x 1 + x 2 + 1 x 1 + 2 x 2 + 1- x 1 + x 2 + - 2 x 1 , x 2 , with initial vertex x = 0 , = 1 UCSD Center for Computational Mathematics Slide 6/27, Friday, February 18th, 2011 x 2 x 1 #1 #2 #3 #5 #4 x 2 x 1 #1 #2 #3 #5 #4 It is not necessary to shift feasible general constraints. e.g., constraint #3 (- x 1 + x 2 - 2) is feasible at x = 0. x 2 x 1 #1 #2 #3 #5 #4 If constraint #3 is unshifted, the phase-1 LP is: minimize R , x R n subject to x 1 + x 2 + 1 x 1 + 2 x 2 + 1- x 1 + x 2 - 2 x 1 , x 2 , The definition of x and does not change.does not change....
View Full Document

Page1 / 7

handout19 - Math 171A: Linear Programming Lecture 19...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online