2_basis_art_rev

2_basis_art_rev - Working with the Basis Inverse over a...

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Working with the Basis Inverse over a Sequence of Iterations Robert M. Freund February, 2004 c ± 2004 Massachusetts Institute of Technology. 1
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± ² ³ ³ ³ ³ ³ ³ ³ ³ 1 Equations Involving the Basis Matrix At each iteration of the simplex method, we have a basis consisting of an index of variables: B (1) ,...,B ( m ) , from which we form the basis matrix B by collecting the columns A B (1) ,...,A B ( m ) of A into a matrix: . B := A B (1) ³ A B (2) ³ ... ³ A B ( m 1) ³ A B ( m ) In order to execute the simplex method at each iteration, we need to be able to compute: T T x = B 1 r 1 and/or p = r 2 B 1 , (1) for iteration-specific vectors r 1 and r 2 , which is to say that we need to solve equation systems of the type: T Bx = r 1 and/or p T B = r 2 (2) for x and p . 2 LU Factorization One way to solve (2) is to factor B into the product of a lower and upper triangular matrix L, U : B = LU , and then compute x and/or p as follows. To compute x , we solve the fol- lowing two systems by back substitution: First solve Lv = r 1 for v Then solve Ux = v for x . 2
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To compute p , we solve the following two systems by back substitution: T First solve u T U = r 2 for u Then solve p T L = u T for p . It is straightforward to verify that these procedures yield x and p that satisfy (2). If we compute according to these procedures, then: T Bx = LUx = Lv = r 1 and p T B = p T LU = u T U = r 2 . 3 Updating the Basis and its Inverse As the simplex method moves from one iteration to the next, the basis matrix B changes by one column. Without loss of generality, assume that the columns of A have been re-ordered so that B := [ A 1 | ... | A j 1 | A j | A j +1 | | A m ] at one iteration. At the next iteration we have a new basis matrix ˜ B of the form: ˜ B := [ A 1 | | A j 1 | A k | A j +1 | | A m ] . Here we see that column A j has been replaced by column A k in the new basis. Assume that at the previous iteration we have B and we have computed an LU factorization of B that allows us to solve equations involving B 1 .A t the current iteration, we now have ˜ B and we would like to solve equations involving ˜ B 1 . Although one might think that we might have to compute ˜ an LU factorization of B , that is not the case. Herein we describe how the linear algebra of working with ˜ B 1 is computed in practice. Before we describe the method, we first need to digress a bit to discuss rank-1 matrices and rank-1 updates of the inverse of a matrix. 3
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± ² 3.1 Rank- 1 Matrices Consider the following matrix: 2 2 0 3 4 4 0 6 W = 14 14 0 21 .
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2_basis_art_rev - Working with the Basis Inverse over a...

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