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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 MINMAX ON CIRCLE
cx II c x^2 + y^2 10 cx Saigal (U of M) IOE 310 8/8 MINMAX ON CIRCLE II Saigal (U of M) IOE 310 10 / 8 ...
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 Spring '08
 Saigal

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