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lecture14

# lecture14 - The Canonical Problem Problem Day We will do...

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IE417: Nonlinear Programming: Lecture 14 Jeﬀ Linderoth Department of Industrial and Systems Engineering Lehigh University 23rd March 2006 Jeﬀ Linderoth IE417:Lecture 14 The Canonical Problem Problem Day! We will do mostly examples and problems today! min x R n f ( x ) subject to c i ( x ) = 0 i ∈ E c i ( x ) 0 i ∈ I Or if Ω = { x R n | c i ( x ) = 0 , i ∈ E , c i ( x ) 0 , i ∈ I} then min x Ω f ( x ) (NLP) Jeﬀ Linderoth IE417:Lecture 14 First Order Necessary (KKT) Conditions If x * is a local solution NLP, and LICQ holds at x * , then there exists multipliers λ * i , i ∈ E ∪ cI such that KKT x L ( x * , λ * ) = 0 (1) c i ( x * ) = 0 i ∈ E (2) c i ( x * ) 0 i ∈ I (3) λ * i 0 i ∈ I (4) λ * i c i ( x * ) = 0 i ∈ I ∪ E (5) Can write (1) as 0 = x L ( x * , λ * ) = f ( x * ) - ± i ∈A ( x * ) λ * i c i ( x * ) Jeﬀ Linderoth IE417:Lecture 14 Second Order Conditions Assume we have λ * that satisﬁes the KKT conditions. Then deﬁne F 2 ( λ * ) = { d F 1 |∇ c i ( x * ) T d = 0 i A ( x * ) ∩ I with λ * i > 0 } d F 2 ( λ * ) λ * i c i ( x * ) T d = 0

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lecture14 - The Canonical Problem Problem Day We will do...

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