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affinescaling_slides - ALGORITHMS FOR LP PROBLEMS I 1 2 3 4...

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ALGORITHMS FOR LP PROBLEMS I 1 Simplex Method - Dantzig 1947. 2 Affine Scaling Method - Dikin 1967 3 Ellipsoidal Method (Polynomial time algorithm)- Kachian 1979. 4 Projective Transformation Method (Polynomial time algorithm) - Karmarkar 1984. 5 Path Following Method (Polynomial time algorithm)- Various 1986. Saigal (U of M) IOE 310 1 / 8

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INTERIOR POINT METHOD: THE FORM OF LP I Consider the following linear programming problem in standard form: minimize n j =1 c j x j n j =1 a i , j x j = b i i = 1 , · · · , m x j 0 j = 1 , · · · , n and assume that it has an interior point solution x 0 1 > 0, x 0 2 > 0, · · · , x 0 n > 0 . Also, the lp in matrix form: minimize c T x Ax = b x 0 Saigal (U of M) IOE 310 2 / 8
INTERIOR POINT METHOD: THE FORM OF LP II Given the interior point x 0 , define the diagonal matrix D = x 0 1 x 0 2 . . . x 0 n . As an example, consider the following linear program: minimize 6 x 1 - 2 x 2 + x 3 x 1 + 2 x 2 - 3 x 3 + x 4 = 1 x 2 - x 3 + x 4 = 1 x 1 0 x 2 0 x 3 0 x 4 0 Saigal (U of M) IOE 310 3 / 8

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INTERIOR POINT METHOD: THE FORM OF LP III with the interior point x 0 1 = 1, x 0 2 = 1, x 0 3 = 1, x 0 4 = 1 . Note that A = 1 2 - 3 1 0 1 - 1 1 , c = 6 - 2 1 0 , b = 1 1

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