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secn12

# secn12 - 12 Termination In order for the simplex method to...

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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 equality-form linear programming problem maximize c T x = z subject to Ax = b x 0 , where A is an m -by- 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 vectors c B and c N . Using this notation, we can rewrite the system of equations

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