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Lecture8 - Advanced Mathematical Programming IE417 Lecture...

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Advanced Mathematical Programming IE417 Lecture 8 Dr. Ted Ralphs
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IE417 Lecture 8 1 Reading for This Lecture Chapter 4, Section 3-4
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IE417 Lecture 8 2 Optimality Conditions Equality Constrained Problems
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IE417 Lecture 8 3 FJ with Equality Constraints Theorem 1. Consider S = { x X : g i ( x ) 0 , 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 0 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 0 μ 6 = 0
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IE417 Lecture 8 4 Constraint Qualification The same development applies here as with just inequality constraints. Constraint qualification : 5 g i ( x * ) , i I , and 5 h i ( x * ) , i [1 , l ] , are linearly independent.
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