By Vanderbei R. J.
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Extra resources for Linear Programming: Foundations and Extensions (2001)(2nd ed.)(en)(450s)
We prove this by assuming that cycling does occur and then showing that this assumption leads to a contradiction. So let’s assume that cycling does occur. Without loss of generality, we may assume that it happens from the beginning. Let D0 , D1 , . . , Dk−1 denote the dictionaries through which the method cycles. That is, the simplex method produces the following sequence of dictionaries: D0 , D1 , . . , Dk−1 , D0 , D1 , . . We say that a variable is fickle if it is in some basis and not in some other basis.
19 Solve the following linear programming problem: n pj xj maximize j=1 n qj xj ≤ β subject to j=1 xj ≤ 1 j = 1, 2, . . , n xj ≥ 0 j = 1, 2, . . , n. Here, the numbers pj , j = 1, 2, . . , n, are positive and sum to one. The same is true of the qj ’s: n qj = 1 j=1 qj > 0. Furthermore (with only minor loss of generality), you may assume that p2 pn p1 < < ··· < . q1 q2 qn Finally, the parameter β is a small positive number. 3 for the motivation for this problem. B. Dantzig in 1949. His monograph (Dantzig 1963) is the classical reference.
Let D0 , D1 , . . , Dk−1 denote the dictionaries through which the method cycles. That is, the simplex method produces the following sequence of dictionaries: D0 , D1 , . . , Dk−1 , D0 , D1 , . . We say that a variable is fickle if it is in some basis and not in some other basis. Let xt be the fickle variable having the largest index and let D denote a dictionary in D0 , D1 , . . , Dk−1 in which xt leaves the basis. Again, without loss of generality we may assume that D = D0 . Let xs denote the corresponding entering variable.