lecture13 - The Canonical Problem IE417: Nonlinear...

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IE417: Nonlinear Programming: Lecture 13 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 21st March 2006 Jeff Linderoth IE417:Lecture 13 The Canonical Problem 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 13 Recall! Lagrangian L ( x, λ ) = f ( x ) - ± E∪I λ i c i ( x ) Active Set A ( x ) = E ∪ { i ∈ I | c i ( x ) = 0 LICQ Given x * and active set A ( x * ) , the linear indepence constraint qualification (LICQ) holds if the set of vectors c i ( x * ) , i ∈ A ( x * ) is linearly independent Jeff Linderoth IE417:Lecture 13 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 13
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Example minimize ( x 1 - 2) 2 + x 2 2 subject to c 1 ( x 1 , x 2 ) = x 1 0 ( λ 1 ) c 2 ( x 1 , x 2 ) = x 2 0 ( λ 2 ) c 3 ( x 1 , x 2 ) = (1 - x 1 ) 3 - x 2 0 ( λ 3 ) Jeff Linderoth IE417:Lecture 13 Just Some Notation Feasible Sequence Given x * Ω , { z k } k =0 is a feasible sequence if 1 z k ± = x * k 2 lim k →∞ z k = x * 3 z k is feasible for all k “sufficiently” large Local Solutions are points x at which all
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This note was uploaded on 02/29/2008 for the course IE 417 taught by Professor Linderoth during the Spring '08 term at Lehigh University .

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lecture13 - The Canonical Problem IE417: Nonlinear...

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