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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 “mixedconstraint” 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 uppertriangular, 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....
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 Spring '08
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 Linear Programming, Optimization, UCSD Center for Computational Mathematics, Computational Mathematics

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