Recitation9Clarification

Recitation9Clarification - Fall 2010 Optimization I(ORIE...

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Fall 2010 Optimization I (ORIE 3300/5300) Recitation 9 Clariﬁcation We’ll illustrate the parallels between revised simplex method and the ’plain’ simplex method using an example. Before that, we’ll derive the formula for the simplex tableau for a given basis B , which is as follows: z - 0 x B - ( c T N - y T A N ) x N = y T b x B + A - 1 B A N x N = A - 1 B b. (1) Derivation of (1) Given an LP in standard form: min c T x s.t.Ax = b, x 0 , our tableau is initially as follows: z - c T x = 0 Ax = b (2) Suppose that we have a basis B , indicating which variables are basic. Then, let us rewrite the tableau (2) where we group the basic variables together and the nonbasic together: z - c T B x B - c T N x N = 0 A B x B + A N x N = b. (3) Now, the coeﬃcients c B in front of x B are not necessarily zero, but recall that we want the coeﬃcients in front of the basic variables, in the objective row, to be all zeros. So, we add some linear combination of the constraint rows from the objective rows, to make the ﬁrst row all zeroes. Let the vector y indicate this linear combination: z - c T B x B - c T N x N + y T A B x B + y T A N x N = 0 + y T b A B x B + A N x N = b. (4) Rewriting the ﬁrst row, z - ( c T B - y T A B ) x B - ( c T N - y T A N ) x N = y T b A B x B + A N x N = b. (5) We haven’t decided what y is so far, but we need to choose y such that c T B - y T A B = 0. Equivalently, we want y T A B = c T B , or y T = c T B A - 1 B . Taking the transpose of both sides: y = ( A - 1 B ) T c B = ( A T B ) - 1 c B . If we choose y as above, then we obtain z - 0 x B - ( c T N - y T A N ) x N = y T b A B x B + A N x N = b. (6) 1

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Fall 2010 Optimization I (ORIE 3300/5300) Almost there. Now, since we want the columns corresponding to the basic variables to be the columns of the identity matrix, then we multiply the constraint rows by the inverse of A B , such that: A - 1 B A B x B + A - 1 B A N x N = A - 1 B b, or x B + A - 1 B A N x N = A - 1 B b. Hence, replacing the constraint rows in tableau (6) with the above equation, we get z - 0 x B - ( c T N - y T A N ) x N = y T b x B + A - 1 B A N x N = A - 1 B b, (7) which is precisely the formula for the simplex tableau that we saw in (1).
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Recitation9Clarification - Fall 2010 Optimization I(ORIE...

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