alternative replaces the -variable update (8.31) by taking one step in the gradient direction

¶

r 8 $ p§ Rc"¨¨0

’ ’ w© ¥

¡ Y©0'¢ ¦§

¡ Q¡

§

o ¦

for the quadratic function (8.34), usually with ¬xed step-length . The gradient of at

¦`

a

¨ ‚`

§ ¨

the current iterate is . This results in the Arrow-Hurwicz Algorithm. a

¥ ¨ £ !" ¢ ¤¥% £ £ # ! ¡¢( # '! ! ¥ … ¤ ¡w¤h¨ n¦

0 ¦

¤ £¢

¢

%

1. Select an initial guess to the system (8.30)

E

„BD¬B% %

%¬¬

2. For until convergence Do:

w

f G

a §¨¨‚ `

˜

3. Compute

` ¡ w

a ¨ f

4. Compute §

¨ f

5. EndDo

The above algorithm is a block-iteration of the form

8 87

7 7 7 7

@

b¦@ ¨ § §¨ q

8 w

˜ ¢A f

¬

A8 A A A

¨ ¨

§¡ ¨¡

f

Uzawa™s method, and many similar techniques for solving (8.30), are based on solving

the reduced system (8.32). An important observation here is that the Schur complement

˜

matrix need not be formed explicitly. This can be useful if this reduced

§ §

d

f

¢

system is to be solved by an iterative method. The matrix is typically factored by a

Cholesky-type factorization. The linear systems with the coef¬cient matrix can also be

solved by a preconditioned Conjugate Gradient method. Of course these systems must then

be solved accurately.

Sometimes it is useful to “regularize” the least-squares problem (8.28) by solving the

following problem in its place:

` ©w

¨

¦

o f x ”a % `

¨

minimize ¦`

a !% `

a a%

GH

– ˜

subject to § ¨

¨

in which is a scalar parameter. For example, can be the identity matrix or the matrix

§˜§ . The matrix resulting from the Lagrange multipliers approach then becomes

7

˜ §¨ ¬

A

§

The new Schur complement matrix is

v§ 'd ˜ § ¨ ¨

¬f

¢

¥ t¶ § ˜ ©g

¥¨

© §

In the case where , the above matrix takes the form

§

¬u¦§!d ¨ 8 ¨ ` ˜ ©

af §

¢

Assuming that is SPD, is also positive de¬nite when

¢

¨ ¬ b` Gs S

h8 a

However, it is also negative de¬nite for

¨

% b ` e&h 8

3 a piG

r¡¶

¶ 8˜ ’ ”p20("'&}%#”8³ª%¡"p!¡©’ ”W

8¥1 )© ¥ ’$ ©˜ ’8 © 8¥ ˜ 8©

¡¨ ¨¦ ¤

© §¥ &

$

a condition which may be easier to satisfy on practice.

ª“t˜•wS¡

˜

1 Derive the linear system (8.5) by expressing the standard necessary conditions for the problem

(8.6“8.7).

–¢ e e

¨¦£¤

§¥

2 It was stated in Section 8.2.2 that when for , the vector de¬ned in

¢

© !

– $£

#

Algorithm 8.3 is equal to ."