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simplex

# simplex - IEOR 162 NOTES SPRING 2007 1 The Algebra of the...

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IEOR 162 NOTES SPRING 2007 1. The Algebra of the Simplex Algorithm Consider the linear program (LP) max cx s.t. Ax = b x 0 in standard form, where A is an m × n –matrix ( m n ), b is an m –column vector, and c is an n –row vector. Without loss of generality, we may assume that rank ( A ) = m , since otherwise we can remove the linearly dependent rows of A without changing the feasible set of solutions. Let x = ( x B , x N ) be a basic feasible solution of LP, i.e., A = ( A B , A N ), where A B is an invertible m × m submatrix of A , x B = A - 1 B b 0 and x N = 0. So x i , i B are the basic variables and x i , i N are the nonbasic variables. Using the partitioning of the variables into B (basic) and N (nonbasic), we rewrite LP as max c B x B + c N x N st A B x B + A N x N = b x B , x N 0 . Equivalently, solving for x B , we can rewrite LP as max c B A - 1 B b + ( c N - c B A - 1 B A N ) x N st x B = A - 1 B b - A - 1 B A N x N x B , x N 0 . This form of the LP is called the reduced LP. Since x N = 0, we have x B = A - 1 B b and the objective

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simplex - IEOR 162 NOTES SPRING 2007 1 The Algebra of the...

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