Lecture11

# Lecture11 - Advanced Operations Research Techniques IE316...

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Unformatted text preview: Advanced Operations Research Techniques IE316 Lecture 11 Dr. Ted Ralphs IE316 Lecture 11 1 Reading for This Lecture Bertsimas 4.1-4.3 IE316 Lecture 11 2 Duality Theory: Motivation Consider the following minimization problem minimize 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? IE316 Lecture 11 3 Lagrange Multipliers The idea is not to strictly enforce the constraints. We associated 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: minimize c T x + p T ( b- Ax ) s.t. x How does the optimal solution of this compare to the original optimum? IE316 Lecture 11 4 Lagrange Multipliers Because we havent changed the cost of feasible solutions to the original problem, this new problem gives a lower bound ....
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## This note was uploaded on 08/06/2008 for the course IE 316 taught by Professor Ralphs during the Fall '08 term at Lehigh University .

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Lecture11 - Advanced Operations Research Techniques IE316...

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