Lecture24 - IE 495 Lecture 24 Reading for This Lecture...

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Unformatted text preview: IE 495 Lecture 24 November 28, 2000 Reading for This Lecture Primary ¡ Bazaraa, Sherali, and Sheti, Chapter 2. ¡ Chvatal, Chapters 6 and 7. Linear Programming Introduction Consider again the system Ax = b, A ∈ R m × n , b ∈ R m . In this problem, there are either ¡ no solutions ¡ one solution ¡ infinitely many solutions (if n > m) The problem of linear programming is min c T x s.t. Ax = b x ≥ 0 Applications of Linear Programming Linear programming is central to much of operations research. Many resource allocation problems can be described as linear programs. Example: The Diet Problem ¡ We have a set of nutrients with RDAs. ¡ We have a set of available foods. ¡ We have preference constraints which limit the intake of particular foods. ¡ We want to minimize our cost. Convex Sets A set S is convex ⇔ x 1 , x 2 ∈ S, λ ∈ [0,1] ⇒ λ x 1 + ( 1- λ29 x 2 ∈ S If x = Σλ i x i , where λ i ≥ and Σλ i = 1 , then x is a convex combination of the x i 's . If the positivity restriction on λ is removed, then y is an affine combination of the x i 's ....
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This note was uploaded on 08/06/2008 for the course IE 495 taught by Professor Linderoth during the Fall '08 term at Lehigh University .

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Lecture24 - IE 495 Lecture 24 Reading for This Lecture...

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