Lecture 13

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

This preview shows pages 1–7. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4 Searching Method for Nonlinear Programming when f ` ( x ) and f `` ( x ) is hard to calculate
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 *

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 23

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

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online