lecture03

# lecture03 - E C O 3 1 L e c t u r e 3 O p t i m i z a t i o...

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Unformatted text preview: E C O 3 1 : L e c t u r e 3 O p t i m i z a t i o n R e v i e w S a t o r u T a k a h a s h i S e p t e m b e r 2 5 , 2 7 L a g r a n g e ’ s M e t h o d R e a l P r o b l e m : M a x i m i z e f ( x , y ) s u b j e c t t o g ( x , y ) = c . L a g r a n g i a n : “ M a x i m i z e ” L ( x , y , λ ) = f ( x , y ) + λ [ c − g ( x , y ) ] w i t h r e s p e c t t o ( x , y ) w i t h o u t c o n s t r a i n t s : “ F O N C s ” a r e L x ( x , y , λ ) = f x ( x , y ) − λ g x ( x , y ) = , L y ( x , y , λ ) = f y ( x , y ) − λ g y ( x , y ) = . T h u s , L x = L y = i m p l i e s f x / g x = f y / g y = λ : t a n g e n c y c o n d i t i o n . A l s o , L λ = i s e q u i v a l e n t t o g ( x , y ) = c . I f λ i s c h o s e n a p p r o p r i a t e l y , t h e F O N C s h o l d s a s i f ( x ∗ , y ∗ ) m a x i m i z e d L , a l t h o u g h i t d o e s n o t n e c e s s a r i l y d o s o . 1 L a g r a n g e ’ s T h e o r e m S u p p o s e t h a t ( x ∗ , y ∗ ) i s n o t o n t h e b o u n d a r y . I f ( x ∗ , y ∗ ) s o l v e s m a x x , y f ( x , y ) s . t . g ( x , y ) = c , f a n d g a r e c o n t i n u o u s l y d i ff e r e n t i a b l e a t ( x ∗ , y ∗ ) , a n d e i t h e r g x ( x ∗ , y ∗ ) o r g y ( x ∗ , y ∗ ) i s n o n z e r o , t h e n t h e r e e x i s t s λ ∈ R s u c h t h a t f x ( x ∗ , y ∗ ) − λ g x ( x ∗ , y ∗ ) = , f y ( x ∗ , y ∗ ) − λ g y ( x ∗ , y ∗ ) = ....
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## This note was uploaded on 04/07/2008 for the course ECO 310 taught by Professor Stephene.morris during the Fall '08 term at Princeton.

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lecture03 - E C O 3 1 L e c t u r e 3 O p t i m i z a t i o...

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