Evolution and Optimum Seeking

Multidimensional Strategies 55
The algorithm runs as follows:
Step 0: (Initialization)
Specify a starting point x (00) and an initial step length s (0)
(the same for all directions).
De ne an accuracy requirement ">0.
Choose as a rst set of directions v (0)
i = ei for all i = 1(1)n.
Set k =0andi =1.
Step 1: (Line search)
Starting from x (ki;1) , seek the relative minimum x (ki)
in the direction v (k)
i
such that
F (x (ki) )=F (x (ki;1) + d (k)
i v (k)
i ) = min
d fF (x (ki;1) + dv (k)
i )g:
Step 2: (Main 8 loop)
>< :
= n
= n +1
go to step 3
go to step 4.
Step 3: (Eventually one more line search)
Construct z = x (kn) ; x (k0) .
If kzk > 0, set v (k)
n+1 = z=kzk i = n +1 , and go to step 1
otherwise set x (kn+1) = x (kn) d (k)
n+1 = 0 , and go to step 5.
Step 4: (Check appropriateness of step length)
If kx (kn+1) ; x (k0) k s (k) , go to step 6.
Step 5: (Termination criterion)
Set s (k+1) =0:1 s (k) .
If s (k+1) " end the search
otherwise set x (k+10) = x (kn+1) ,
increase k k +1, set i = 1, and go to step 1.
Step 6: (Check appropriateness of orthogonalization)
Reorder the directions v (k)
i and associated distances d (k)
i such that
jd (k)
(
>"for all i = 1(1)p
i j
" for all i = p +1(1)n:
If p