affinescaling - CLASS NOTES IOE 310 FALL 1993 AFFINE...

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CLASS NOTES IOE 310 FALL 1993 AFFINE SCALING METHOD ROMESH SAIGAL 1 Introduction Consider the following linear programming problem: 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, let A = a 1 , 1 a 1 , 2 ... a 1 ,n a 2 , 1 a 2 , 2 ... a 2 ,n . . . . . . . . . . . . a m, 1 a m, 2 ... a m,n ,c = c 1 c 2 . . . c n ,b = b 1 b 2 . . . b m . 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 1
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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 . and the diagonal matrix D = 1 1 1 1 . We will show one step of the affine scaling method on this example. We now give the algorithm. 2 Affine Scaling Method We now present the algorithmic procedure of the affine scaling method. We will explain the various steps later in these notes. Step 1 Compute the matrix: B = AD 2 A T Step 2 Compute the tentative solution to the dual: y = B - 1 AD 2 c Step 3 Compute the dual slack: s = c - A T y Step 4 Compute the next interior point: ¯ x = x - r D 2 s k Ds k where r is the radius of the approximating ellipsoid. 2
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Step 5 Next iterate: If x j = 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. 3 Numerical Example We will now apply the first step of the affine scaling method to the numerical example. Step 1: Compute B . B = AD 2 A T = 1 2 - 3 1 0 1 - 1 1 1 1 1 1 1 0 2 1 - 3 - 1 1 1 = 15 6 6 3 . and the inverse of
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This note was uploaded on 04/02/2008 for the course IOE 310 taught by Professor Saigal during the Spring '08 term at University of Michigan.

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affinescaling - CLASS NOTES IOE 310 FALL 1993 AFFINE...

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