Lecture10 - Introduction to Mathematical Programming IE406...

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Introduction to Mathematical Programming IE406 Lecture 10 Dr. Ted Ralphs
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IE406 Lecture 10 1 Reading for This Lecture Bertsimas 4.1-4.3
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IE406 Lecture 10 2 Duality Theory: Motivation Consider the following minimization problem min x 2 + y 2 s.t. x + y = 1 How could we solve this problem? Idea : Consider the function L ( x, y, p ) = x 2 + y 2 + p (1 - x - y ) What can we do with this?
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IE406 Lecture 10 3 Lagrange Multipliers The idea is not to strictly enforce the constraints. We associate a Lagrange multiplier, or price , with each constraint. Then we allow the constraint to be violated for a price . Consider an LP in standard form. Using Lagrange multipliers, we can formulate an alternative LP: min c x + p ( b - Ax ) s.t. x 0 How does the optimal solution of this compare to the original optimum?
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IE406 Lecture 10 4 Lagrange Multipliers Because we haven’t changed the cost of feasible solutions to the original problem, this new problem gives a lower bound . g ( p ) = min x 0 £ c x + p ( b - Ax ) / c x * + p ( b - Ax * ) = c x * Since each value of p gives a lower bound, we consider maximizing g ( p ) .
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