Lecture6 - Advanced Mathematical Programming IE417 Lecture...

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Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 6 Dr. Ted Ralphs IE417 Lecture 6 1 Reading for This Lecture Chapter 4, Sections 1-2 IE417 Lecture 6 2 Optimality Conditions Unconstrained Problems IE417 Lecture 6 3 First-order Necessary Conditions Theorem 1. Let f : R n R be differentiable at x * . If there is a vector d such that f ( x * ) T d < , then there exists a > such that f ( x * + d ) < f ( x * ) for each (0 , ) . Corollary 1. Let f : R n R be differentiable at x * . If x * is a local minimum, then f ( x * ) = 0 . The direction d is called a descent direction . IE417 Lecture 6 4 Second-order Necessary Conditions Theorem 2. Let f : R n R be twice differentiable at x * . If x * is a local minimum, then f ( x * ) = 0 and H ( x * ) is positive semi-definite. IE417 Lecture 6 5 Sufficient Conditions Theorem 3. Let f : R n R be twice differentiable at x * . If f ( x * ) = 0 and H ( x * ) is positive definite, then x * is a local minimum.is a local minimum....
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This note was uploaded on 08/06/2008 for the course IE 417 taught by Professor Linderoth during the Fall '08 term at Lehigh University .

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Lecture6 - Advanced Mathematical Programming IE417 Lecture...

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