Lecture24 - Advanced Operations Research Techniques IE316...

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Advanced Operations Research Techniques IE316 Lecture 24 Dr. Ted Ralphs
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IE316 Lecture 24 1 Reading for This Lecture Bertsimas Sections 10.2, 10.3, 11.1, 11.2
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IE316 Lecture 24 2 Preprocessing Often, it is possible to simplify a model using logical arguments. Most commercial LP solvers do preprocessing automatically, but if you are developing a model that will be solved repeatedly, it may be worthwhile. The constraint a 1 x b 1 dominates the constraint a 2 x b 2 if a 1 i a 2 i i, and b 1 b 2 In this case, the dominated inequality could be deleted. We can also derive implied bounds for variables from each constraint ax b . If a 0 > 0 , then x 1 ( b - X j : a j > 0 a j l j - X j : a j < 0 a j u j ) /a 0
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IE316 Lecture 24 3 More Preprocessing The constraint ax b is redundant if X j : a j > 0 a j u j + X j : a j < 0 a j l j b. The LP is infeasible if X j : a j > 0 a j l j + X j : a j < 0 a j u j > b. For an LP of the form min { c T x | Ax b, l x b } , If a ij 0 i [1 ..m ] and c j < 0 , then x j = u j in any optimal solution. If a ij 0 i [1 ..m ] and c j > 0 , then x j = l j in any optimal solution.
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IE316 Lecture 24 4 More Preprocessing More sophisticated rules can also be applied.
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