affinescaling_slides

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

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Unformatted text preview: ALGORITHMS FOR LP PROBLEMS I 1 2 3 4 Simplex Method - Dantzig 1947. Affine Scaling Method - Dikin 1967 Ellipsoidal Method (Polynomial time algorithm)- Kachian 1979. Projective Transformation Method (Polynomial time algorithm) Karmarkar 1984. Path Following Method (Polynomial time algorithm)- Various 1986. 5 Saigal (U of M) IOE 310 1/8 INTERIOR POINT METHOD: THE FORM OF LP I Consider the following linear programming problem in standard form: minimize n j=1 cj xj n j=1 ai,j xj = bi xj 0 i = 1, , m j = 1, , n and assume that it has an interior point solution 0 0 0 x1 > 0, x2 > 0, , xn > 0. Also, the lp in matrix form: minimize cT x Ax = b x 0 IOE 310 2/8 Saigal (U of M) INTERIOR POINT METHOD: THE FORM OF LP II Given the interior point x 0 , define the diagonal matrix 0 x1 0 x2 D= . .. . 0 xn As an example, consider the following linear program: minimize 6x1 x1 x1 0 Saigal (U of M) - + 2x2 + x3 2x2 - 3x3 + x4 = 1 x2 - x3 + x4 = 1 x2 0 x3 0 x4 0 IOE 310 3/8 INTERIOR POINT METHOD: THE FORM OF LP III with the interior point 0 0 0 0 x1 = 1, x2 = 1, x3 = 1, x4 = 1. Note that 1 2 -3 1 0 1 -1 1 6 -2 ,c = 1 ,b = 0 1 1 A= . and the diagonal matrix D= Saigal (U of M) 1 1 1 1 . IOE 310 4/8 THE METHOD - Primal Affine Scaling Method Step 1 Compute the matrix: B = AD 2 AT Step 2 Compute: y = B -1 AD 2 c Step 3 Compute: s = c - AT y Step 4 Compute the next interior point: x =x -r D 2s Ds I where r is the radius of the approximating ellipsoid. Saigal (U of M) IOE 310 5/8 THE METHOD - Primal Affine Scaling Method II Step 5 Next iterate: If xj = 0 for some j, then STOP. x is the optimal solution to the primal and y is the optimal solution of the dual. Otherwise, set x = x , and D the appropriate diagonal matrix, and go to Step 1. Saigal (U of M) IOE 310 6/8 MIN-MAX ON CIRCLE cx II c x^2 + y^2 10 cx Saigal (U of M) IOE 310 8/8 MIN-MAX ON CIRCLE II Saigal (U of M) IOE 310 10 / 8 ...
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affinescaling_slides - ALGORITHMS FOR LP PROBLEMS I 1 2 3 4...

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