Augmented Lagrangian Methods: Applications to the Numerical by Michel Fortin

By Michel Fortin

The aim of this quantity is to provide the rules of the Augmented Lagrangian procedure, including quite a few functions of this system to the numerical answer of boundary-value difficulties for partial differential equations or inequalities coming up in Mathematical Physics, within the Mechanics of constant Media and within the Engineering Sciences.

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7 0 ) i s n o t . empty, i e . 72) Br(Y,-a,B,)PL+2(Yr-arBr)P I t can e a s i l y be seen t h a t - a, s 0 . 73) y, s a B r r . Before confirming t h i s p o i n t , w e s h a l l conclude t h e proof of converg e n c e ; w e h a v e s e e n i n f a c t , a s s u m i n g ( 4 . 60) w e c a n choose t o have i n t h e l e f t - h a n d s i d e o f 6 ( 4 . 6 6 ) o n l y p o s i t i v e terms. 74) which i m p l i e s t h e c o n v e r g e n c e of t h e s e r i e s w i t h g e n e r a l t e r m IIunll: and t h e r e f o r e t h a t so as It 41 VARIANTS O F METHODS (SEC.

2) = {VER N , Bv = c ) , c~IrnB. 4) P"+l Remark 5 . 2 : Suppose necessariZy symmetric, = pn + p (Bun-c) . A E d(Rn,Rn) i s p o s i t i v e d e f i n i t e , not and suppose t h a t K i s d e f i n e d by ( 5 . 2 ) It (SEC. 6) Bu = c . I n view of ( 5 . 9) P n+l p n + pn(Bun-c) By p r o c e e d i n g a s f o r Theorem 2 . 1 , + sBtc, , pn 2 0 . i t can e a s i l y b e shown t h a t a l g - o r i t h m ( 5 . 7 ) - ( 5 . 11), Au i s t h e symmetric component of A , i . e . of conver- (A+At). gence r a t e s s e e m s much more d i f f i c u l t , s i n c e t h e s p e c t r a l methods of S e c t i o n 2 .

72) Br(Y,-a,B,)PL+2(Yr-arBr)P I t can e a s i l y be seen t h a t - a, s 0 . 73) y, s a B r r . Before confirming t h i s p o i n t , w e s h a l l conclude t h e proof of converg e n c e ; w e h a v e s e e n i n f a c t , a s s u m i n g ( 4 . 60) w e c a n choose t o have i n t h e l e f t - h a n d s i d e o f 6 ( 4 . 6 6 ) o n l y p o s i t i v e terms. 74) which i m p l i e s t h e c o n v e r g e n c e of t h e s e r i e s w i t h g e n e r a l t e r m IIunll: and t h e r e f o r e t h a t so as It 41 VARIANTS O F METHODS (SEC.

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