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Unformatted text preview: IE417: Nonlinear Programming: Lecture 19 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University April 27, 2006 Jeff Linderoth IE417:Lecture 19 Stuff Left To Do Your Final Assignment Code one of the following methods for constrained optimization: 1 An augmented Lagranigan Code 2 A sequential quadratic programming code 3 An interiorpoint based method Include a short writeup of things that worked, things that didn’t work, and the number of problems from the CUTER (or other) test set that you were able to solve. Final Exam Final Exam: Thursday May 4, 15PM. You get three sheets of 8 . 5 × 11 paper. Jeff Linderoth IE417:Lecture 19 The Canonical Problem min x ∈ R n f ( x ) subject to c i ( x ) = ∀ i ∈ E c i ( x ) ≤ ∀ i ∈ I Or if Ω = { x ∈ R n  c i ( x ) = 0 ,i ∈ E ,c i ( x ) ≤ ,i ∈ I} then min x ∈ Ω f ( x ) (NLP) Jeff Linderoth IE417:Lecture 19 Recall! Local Solution ˆ x is a local solution of (NLP) if ˆ x ∈ Ω and ∃ a neighborhood N (ˆ x ) such that f ( x ) ≥ f (ˆ x ) ∀ x ∈ N ∩ Ω Strict Local Solution ˆ x is a strict local solution of (NLP) if ˆ x ∈ Ω and ∃ a neighborhood N (ˆ x ) such that f ( x ) > f (ˆ x ) ∀ x ∈ N ∩ Ω with ˆ x 6 = x Jeff Linderoth IE417:Lecture 19 Lagrangians Lagrangian L ( x,λ ) = f ( x ) 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 qualifi cation (LICQ) holds if the set of vectors ∇ c i ( x * ) ,i ∈ A ( x * ) is linearly independent Jeff Linderoth IE417:Lecture 19 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 ∪ I such that KKT ∇ x L ( x * ,λ * ) = (1) c i ( x * ) = ∀ i ∈ E (2) c i ( x * ) ≥ ∀ i ∈ I (3) λ * i ≥ ∀ i ∈ I (4) λ * i c i ( x * ) = ∀ i ∈ I ∪ E (5) Can write (1) as 0 = ∇ x L ( x * ,λ * ) = ∇ f ( x * ) X i ∈A ( x * ) λ * i ∇ c i ( x * ) Jeff Linderoth IE417:Lecture 19 Examples Geometrically, if ( ˆ x ) is an optimal solution, then we must be able to write ˆ x as an appropriate linear combination of the binding constraints. If a constraint is not binding, it’s “weight” must be 0. Also, first order conditions are only necessary if a constraint qualification holds. LICQ, MFCQ, Linear. Jeff Linderoth IE417:Lecture 19 Example? maximize f ( x ) = 2 x 1 + 3 x 2 + 4 x 2 1 + 2 x 1 x 2 + x 2 2 subject to x 1 x 2 ≥ x 1 + x 2 ≤ 4 x 1 ≤ 3 Jeff Linderoth IE417:Lecture 19 KKT Conditions Primal Feasibility x 1 x 2 ≥ x 1 + x 2 ≤ 4 x 1 ≤ 3 Dual Feasibility 2 + 8 x 1 + 2 x 2 = λ 1 + λ 2 + λ 3 3 + 2 x 1 + 2 x 2 = λ 1 + λ 2 λ 1 , λ 2 , λ 3 ≥ Complementary Slackness λ 1 ( x 1 x 2 ) = 0 λ 2 (4 x 1 x 2 ) = 0 λ 3 (3 x 1 ) = 0 Jeff Linderoth IE417:Lecture 19 Checking Some Points Is x 1 = 2 ,x 2 = 2 an optimal point?...
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 Spring '08
 Linderoth
 Optimization, Systems Engineering, Mathematical optimization, Jeff Linderoth

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