Lecture8 - Advanced Mathematical Programming IE417 Lecture...

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Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 8 Dr. Ted Ralphs IE417 Lecture 8 1 Reading for This Lecture • Chapter 4, Section 3-4 IE417 Lecture 8 2 Optimality Conditions Equality Constrained Problems IE417 Lecture 8 3 FJ with Equality Constraints Theorem 1. Consider S = { x ∈ X : g i ( x ) ≤ ,i ∈ [1 ,m ] ,h i ( x ) = 0 ,i ∈ [1 ,l ] } where X is a nonempty open set in R n and g i : R n → R , i ∈ [1 ,m ] , h i : R n → R , i ∈ [1 ,l ] . Given a feasible x * ∈ S , set I = { i : g i ( x * ) = 0 } . Assume that f and g i are differentiable at x * for i ∈ I,g i is continuous at x * for i / ∈ I,h i is continuously differentiable. If x * is a local minimum, then there exists μ ∈ R m ,v ∈ R l such that μ 5 f ( x * ) + X μ i 5 g i ( x * ) + X v i 5 h i ( x * ) = 0 μ i g i ( x * ) = 0 ∀ i ∈ [1 ,m ] μ ≥ μ 6 = 0 IE417 Lecture 8 4 Constraint Qualification • The same development applies here as with just inequality constraints....
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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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Lecture8 - Advanced Mathematical Programming IE417 Lecture...

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