Lecture 13 - Lecture 13: Solving NLP If an NLP is...

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1 Lecture 13: Solving NLP If an NLP is unconstrained Find stationary points by 1st order conditions Use 2nd order conditions to check if a stationary point is a local maximum or minimum Questions to be addressed NLP with constraints When the derivative is not easy to obtain
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2 NLP with Linear Equality Constraints • Minimize f ( x ) subject to linear g i ( x )=0, i =1, …, m Define Lagrangian function L ( x , λ )= f ( x )+ λ 1 g 1 ( x )+ λ 2 g 2 ( x )+…+ λ m g m ( x ) λ=( λ 1 , λ 2 ,…, λ m ) an unconstrained NLP to minimize L ( x , λ ) The necessary condition of a stationary point is 0 2 1 2 1 = = = = = = = = m n L L L x L x L x L λ
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3 Example • Minimize f ( x 1 , x 2 )=3 x 1 2 + x 2 2 subject to x 1 + x 2 =6 Lagrangian function L ( x 1 , x 2 , λ )=3 x 1 2 + x 2 2 + λ ( x 1 + x 2 – 6) First order conditions – 6 x 1 + λ =0, 2 x 2 + λ =0, x 1 + x 2 – 6=0 – Solution: x 1 =1.5, x 2 =4.5, λ = – 9 • Is ( x 1 =1.5, x 2 =4.5) a local or global minimum? f ( x 1 , x 2 ) is convex, feasible region is a convex region – So , ( x 1 =1.5, x 2 =4.5) is a global minimum
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4 Searching Method for Nonlinear Programming when f ` ( x ) and f `` ( x ) is hard to calculate
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5 Unimodal Functions A function f ( x ) is unimodal in the interval [ a , b ] if it has a unique minimum x *, f ( x ) is strictly increasing in [ x *, b ], and f ( x ) is strictly decreasing in [ a , x *] A convex function is unimodal, but a unimodal function may not be convex f ( x ) f ( x ) x x x * x *
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This note was uploaded on 01/05/2011 for the course IELM IELM202 taught by Professor D during the Fall '10 term at HKUST.

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Lecture 13 - Lecture 13: Solving NLP If an NLP is...

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