Evolution and Optimum Seeking

Multidimensional Strategies 53
Numbering Iteration Test index/ Variable Step Remarks
index iteration values lengths
k nl + i x 1 x 2 s 1 s 2
(0) 0 0 0 9 2 2 starting point
(1) 0 1 2 9 2 - success
(2) 0 2 2 11 - 2 failure
(3) 0 3 8 9 6 - failure
(4) 0 4 2 8 - ;1 success
(5) 0 5 ;1 8 ;3 - failure
(6) 0 6 2 5 - ;3 failure
(4) 1 0 2 8 2 2 transformation
and orthogonalization
(7) 1 1 3.8 7.1 2 - success
(8) 1 2 2.9 5.3 - 2 success
(9) 1 3 8.3 2.6 6 - success
(10) 1 4 5.6 ;2.7 - 6 failure
(11) 1 5 24.4 ;5.4 18 - failure
(9) 2 0 8.3 2.6 2 2 transformation
and orthogonalization
In Figure 3.7, including the following table, a few iterations of the Rosenbrock strategy
for n = 2 are represented geometrically. At the starting point x (00) the search directions
are the same as the unit vectors. After three runs through (6 trials), the trial steps in each
direction have led to a success followed by a failure. At the best condition thus attained,
are generated. Five further trials
(4) at x (04) = x (10) , new direction vectors v (1)
1
and v (1)
2
lead to the best point, (9) at x (13) = x (20) , of the second iteration, at which a new choice
of directions is again made. The complete sequence of steps can be followed, if desired,
with the help of the accompanying table.
Numerical experiments show that within a few iterations the rotating coordinates
become oriented such that one of the axes points along the gradient direction. The
strategy thus allows sharp valleys in the topology of the objective function to be followed.
Like the method of Hooke and Jeeves, Rosenbrock's procedure needs no information about
partial derivatives and uses no line search method for exact location of relative minima.
This makes it very robust. It has, however, one disadvantage compared to the direct
pattern search: The orthogonalization procedure of Gram and Schmidt is very costly.
It requires storage space of order O(n 2 ) for the matrices A = faijg and V = fvijg,
and the number of computational operations even increases with O(n 3 ). At least in
cases where the objective function call costs relatively little, the computation time for
the orthogonalization with many variables becomes highly signi cant. Besides this, the
number of parameters is in any case limited by the high storage space requirement.
If there are constraints, care must be taken to ensure that the starting point is inside
the allowed region and su ciently far from the boundaries. Examples of the application of