Evolution and Optimum Seeking

Multidimensional Strategies 79
Step 0:
Step 1:
Step 2:
(
Set =max"b
"c
jjx (k)
i j
F (x (k) )
jg (k;1)
i
)
("b represents an estimate of the error in the calculation of F (x). Stewart sets
"b =10 ;10 and "c =5 10 ;13 :)
If g (k;1)
i
2
de ne 0 i =2
otherwise de ne
0
vu
u
i =2 3
Set d 0 (k)
and d (k)
i
If h(k;1) i
h (k;1)
i F (x (k) )
v uuut F (x (k) )
h (k;1)
i
t F (x(k) ) g (k;1)
i
0
and i = 0 @1 i ;
(h (k;1)
i ) 2 and i = 0 i
i = i sign(h (k;1)
i
=
d (k)
i
2 g (k;1)
i
( 0
(k)
di if d 0 (k)
i
replace d (k)
i 1
h (k;1)
i
) sign(g (k;1)
)
6= 0
d (k;1)
i if d 0 (k)
i =0:
i
0
@1 ;
3 0 i h (k;1)
i
3 0 i h (k;1)
i
10 ; , use Equation (3.32) otherwise
; g (k;1)
i
and use Equation (3.33). (Stewart chooses =2.)
Stewart's main algorithm takes the following form:
+
0
i h (k;1)
i
2 g (k;1)
i
+4 g (k;1)
i
+4 g (k;1)
i
1
A :
1
A
r
(g (k;1)
i ) 2 +2 10 F (x (k) ) h (k;1)
i
Step 0: (Initialization)
Choose an initial value x (0) , accuracy requirements "ai > 0i = 1(1)n, and
initial step lengths d (0)
for the gradient determination, e.g.,
d (0)
i
=
8
<
: 0:05 x(0)
i if x (0)
i
0:05 if x (0)
i
i
6= 0
=0:
Calculate the vector g (0) from Equation (3.32) using the step lengths d (0)
i :
Set H (0) = Ih (0)
i =1foralli = 1(1)n and k =0:
Step 1: (Prepare for line search)
Determine v (k) = ;H (k) g (k) :
If k = 0, go to step 3.
If g (k)T v (k) < 0, go to step 3.
If h (k)
i > 0 for all i = 1(1)n, go to step 3.