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Lecture9

Lecture9 - Advanced Mathematical Programming IE417 Lecture...

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Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 9 Dr. Ted Ralphs IE417 Lecture 9 1 Reading for This Lecture Chapter 6, Section 1-2 IE417 Lecture 9 2 Lagrangian Duality IE417 Lecture 9 3 The Primal Problem Given functions f : R n R , g : R n R m , and h : R n R l , consider the constrained optimization problem P , which we will now call the primal problem : min f ( x ) s.t. g ( x ) h ( x ) = 0 x X Here, X is a set that implicitly enforces additional constraints. There is usually more than one way to define X and this choice can be important, as we will see. IE417 Lecture 9 4 The Dual Problem We can now formulate the following dual problem D : max ( ,v ) s.t. where ( ,v ) = inf { ( x,,v ) : x X } . How do we interpret this? IE417 Lecture 9 5 Weak Duality Theorem 1. Let x be a feasible solution to the primal problem P and let ( ,v ) be a solution to the dual problem D . Then f ( x ) ( ,v ) ....
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Lecture9 - Advanced Mathematical Programming IE417 Lecture...

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