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Unformatted text preview: 12 Termination In order for the simplex method to be a foolproof algorithm, we must be sure that, starting from a feasible tableau, it terminates after a finite number of iterations. In this section we shall see that the smallest subscript rule ensures termination. We need one last ingredient. We consider the standard equalityform linear programming problem maximize c T x = z subject to Ax = b x , where A is an mby n matrix whose columns span (if they fail to do so, the results below hold trivially). Proposition 12.1 (uniqueness of tableau) There is exactly one tableau cor responding to each (ordered) basis. Proof Let B be a basis, and let N be a list of the indices not in B . By definition, the basis matrix A B with columns from A indexed by B is invertible. Recall from Section 5 that we also write A N for the matrix with columns from A indexed by N . Analogously, we write x B , x N for vectors with entries from x indexed by B and N respectively, with similar definitions for the...
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 '08
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