Augmented Lagrangian Methods: Applications to the Numerical by Michel Fortin

By Michel Fortin

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

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45) 1 I 5 ynl ~ ~ ,V n 2 0 . onvergence r a t e We deduce from ( 2 . 46) i t appears t h a t f o r = p = pn r algorithm ( 2 . 1 ) - ( 2 . 3 ) becomes f a s t e r , i t e r a t i v e l y , a s t h e v a l u e o f larger. r << I f , in particular, 1 r;- , it m r gets f o l l o w s from ( 2 . 4 6 ) t h a t a l g o r i t h m ( 2 . 1 ) - ( 2 . 3 ) w i l l i n g e n e r a l be i t e r a t i v e l y slow. 1 note t h a t i f r = - then R 3 7 . n Remark 2 . 4 : Although r e l a t i o n ( 2 . 4 6 ) a p p e a r s t o i n d i c a t e t h a t i t i s advantageous t o work w i t h pn = p = r a s l a r g e a s p o s s i b l e , one m u s t r e a l i s e t h a t a 2 1 o t h e r t h i n g s b e i n g e q u a l t h e d e t e r m i n a t i o n of n u i n ( 2 .

N,, Vnzo. 42) Vi=l r is AUGMENTED LAGRANGIAN METHODS 12 (CHAP. 44) W e d e d u c e from ( 2 . 2 % Am+ 8 (SEC. A FIRST ALGORITHM 2) (iii) The c a s e 13 r. 45) 1 I 5 ynl ~ ~ ,V n 2 0 . onvergence r a t e We deduce from ( 2 . 46) i t appears t h a t f o r = p = pn r algorithm ( 2 . 1 ) - ( 2 . 3 ) becomes f a s t e r , i t e r a t i v e l y , a s t h e v a l u e o f larger. r << I f , in particular, 1 r;- , it m r gets f o l l o w s from ( 2 . 4 6 ) t h a t a l g o r i t h m ( 2 . 1 ) - ( 2 . 3 ) w i l l i n g e n e r a l be i t e r a t i v e l y slow.

6 0 ) by an i t e r a t i v e m e t h o d , t h e convergence, being l i n k e d t o t h e c o n d i t i o n number, w i l l become s l o w e r a s t h e v a l u e of r i n c r e a s e s , and t h i s may l e a d t o a l a r g e number of i t e r a t i o n s t o s o l v e ( 2 . 6 0 ) t o an a p p r o p r i a t e a c c u r a c y even i f , i n t h e obvious n n- 1 u with u . manner, w e i n i t i a l i s e t h e c a l c u l a t i o n o f F u r t h e r m o r e , i f w e s o l v e ( 2 . 6 0 ) by a d i r e c t method, t h e s e n s i t i v i t y t o rounding-error a c c u m u l a t i o n w i l l be g r e a t e r when r is large.

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