section12 - 12 Termination In order for the simplex method...

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12 Termination In order for the simplex method to be a reliable 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. Proposition 12.1 (uniqueness of tableau) Two tableaux corresponding to the same basis can differ only in the order in which the equations are written. Proof Consider two tableaux corresponding to a basis B : (12.2) z - X j 6∈ N ¯ c j x j = ¯ v x i + X j 6∈ N ¯ a ij x j = ¯ b i ( i B ) and (12.3) z - X j 6∈ N c * j x j = v * x i + X j 6∈ N a * ij x j = b * i ( i B ) . These tableaux are equivalent systems of equations. For any nonbasic index q 6∈ B , setting x q = t x j = 0 ( j 6∈ B, j 6 = q ) x i = ¯ b i - ¯ a iq t ( i B ) z = ¯ v + ¯ c q t satisfies tableau (12.2) for any number t , and hence must also satisfy tableau (12.3). Therefore we have, for all t , ¯ v + ¯ c q t - c * q t = v * ¯ b i - ¯ a iq t + a * iq t = b * i ( i B ) . 66
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Equating the constant terms and the coefficients of t on each side, we deduce ¯ v = v * , ¯ c q = c * q , ¯ b i = b * i , and ¯ a iq = a * iq for all basic indices i B . Since this holds for all q 6∈ B , the two tableaux have identical coefficients.
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section12 - 12 Termination In order for the simplex method...

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