handout21

# handout21 - Math 171A Linear Programming Lecture 21 Linear...

This preview shows pages 1–3. Sign up to view the full content.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 171A: Linear Programming Lecture 21 Linear Programs in Standard Form Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Friday, February 25th, 2011 Recap: Linear programs in standard form minimize x ∈ R n c T x subject to Ax = b | {z } equality constraints , x ≥ | {z } simple bounds The matrix A is m × n with shape A = We apply “mixed-constraint” simplex with full matrix A I UCSD Center for Computational Mathematics Slide 2/44, Friday, February 25th, 2011 Example minimize x 1 , x 2 , x 3 , x 4 , x 5- 6 x 1- 9 x 2- 5 x 3 subject to 2 x 1 + 3 x 2 + x 3 + x 4 = 5 x 1 + 2 x 2 + x 3- x 5 = 3 x 1 ≥ x 2 ≥ x 3 ≥ x 4 ≥ x 5 ≥ UCSD Center for Computational Mathematics Slide 3/44, Friday, February 25th, 2011 A I ! = 2 3 1 1 1 2 1- 1 1 1 1 1 1 ← constraint #1 ← constraint #2 ← constraint #3 ← constraint #4 ← constraint #5 ← constraint #6 ← constraint #7 Consider the vertex defined by: the two rows of A rows 3, 4, and 5 of D = I , i.e., rows 5, 6 and 7 of A I UCSD Center for Computational Mathematics Slide 4/44, Friday, February 25th, 2011 A D k ! = 2 3 1 1 1 2 1- 1 1 1 1 ← constraint #1 ← constraint #2 ← constraint #5 ← constraint #6 ← constraint #7 This defines a vertex since rank A D k ! = 5 = n UCSD Center for Computational Mathematics Slide 5/44, Friday, February 25th, 2011 2 3 1 1 1 2 1- 1 1 1 1 ¯ x 1 ¯ x 2 ¯ x 3 ¯ x 4 ¯ x 5 = 5 3 These equations are block upper-triangular, with structure: B N I n- m ! x B x N ! = b ! with B = 2 3 1 2 ! and N = 1 1 1- 1 ! UCSD Center for Computational Mathematics Slide 6/44, Friday, February 25th, 2011 2 3 1 1 1 2 1- 1 1 1 1 ¯ x 1 ¯ x 2 ¯ x 3 ¯ x 4 ¯ x 5 = 5 3 Then, ¯ x 3 = ¯ x 4 = ¯ x 5 = , and 2 3 1 2 ! ¯ x 1 ¯ x 2 ! = 5 3 ! ⇒ ¯ x 1 ¯ x 2 ! = 1 1 ! UCSD Center for Computational Mathematics Slide 7/44, Friday, February 25th, 2011 ¯ x = 1 1 ⇒ ¯ x is a basic solution of Ax = b Every vertex is like this, i.e., it has n- m zero components and m nonnegative components....
View Full Document

{[ snackBarMessage ]}

### Page1 / 11

handout21 - Math 171A Linear Programming Lecture 21 Linear...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online