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sec13_Notes

# sec13_Notes - 13 Fundamental theorems Now that we have made...

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13 Fundamental theorems Now that we have made the simplex method a reliable finite algorithm, we can use it to deduce some striking properties of general linear programming problems. We start by asking when a linear programming problem in stan- dard equality form, ( * ) maximize c T x subject to Ax = b x 0 , has a basic feasible solution. Theorem 13.1 (Existence of basic feasible solutions) The linear pro- gramming problem ( * ) has a basic feasible solution if and only if it is feasible and the columns of the matrix A span. Proof If the linear programming problem ( * ) has a basic feasible solution, then it is feasible, and furthermore, since the corresponding basis matrix is invertible, the columns of A must certainly span. Conversely, suppose ( * ) is feasible and the columns of A span. Applying Phase 1 of the simplex method using the smallest subscript rule, we must terminate with optimal value zero, because ( * ) is feasible. Furthermore, in the body of the final tableau, no row of coefficients of the original variables in ( *

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