IE534_Lecture_17 - IE 534 Linear Programming Lecture 17 Dual Simplex Lizhi Wang October 1 2012 Lizhi Wang([email protected] IE 534 Linear Programming

IE534_Lecture_17 - IE 534 Linear Programming Lecture 17...

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IE 534 Linear Programming Lecture 17: Dual Simplex Lizhi Wang October 1, 2012 Lizhi Wang ([email protected]) IE 534 Linear Programming October 1, 2012 1 / 13
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Primal dictionary max x { ζ = c > x : Ax b, x 0 } . max x,w { ζ = c > x : Ax + w = b, x 0 , w 0 } . max x { ζ = c > x : A x = b, x 0 } . c = c 0 m × 1 , A = A I m × m , and x = x w R n + m . For any fbp ( B , N ) : max x B , x N n ζ = c > B x B + c > N x N : A B x B + A N x N = b ; x B , x N 0 o . Primal dictionary: ζ = c > B A - 1 B b + ( c > N - c > B A - 1 B A N ) x N x B = A - 1 B b - A - 1 B A N x N . Lizhi Wang ([email protected]) IE 534 Linear Programming October 1, 2012 2 / 13
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Dual dictionary min y { ζ = b > y : A > y c ; y 0 } . min z,y { ζ = b > y : A > y - z = c ; y, z 0 } . max ˆ z {- ζ = ˆ b > ˆ z : ˆ A > ˆ z = c, ˆ z 0 } . ˆ b = 0 n × 1 - b , ˆ A = [ - I n × n , A > ] , ˆ z = z y R n + m . For any fbp ( B , N ) : max ˆ z B , ˆ z N n - ζ = ˆ b > B ˆ z B + ˆ b > N ˆ z N : ˆ A B ˆ z B + ˆ A N ˆ z N = c ; ˆ z B , ˆ z N 0 o . Dual dictionary: - ζ = ˆ b > N ˆ A - 1 N c + ( ˆ b > B - ˆ b > N ˆ A - 1 N ˆ A B z B ˆ z N = ˆ A - 1 N c - ˆ A - 1 N ˆ A B ˆ z B . Lizhi Wang ([email protected]) IE 534 Linear Programming October 1, 2012 3 / 13
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Optimality conditions 0 x b - Ax A > y - c y 0 . 0 x w z y 0 . 0 x ˆ z 0 . x N = 0 x B 0 ˆ z N 0 ˆ z B = 0 . Lizhi Wang ([email protected]) IE 534 Linear Programming October 1, 2012 4 / 13
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Example 1 Consider this LP. max ζ = - 1 - 1 x 1 x 2 > (LP0) s . t . - 2 - 1 - 2 4 - 1 3 x 1 x 2 4 - 8 - 7 x 1 x 2 > 0 .
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