Evolution and Optimum Seeking

46 Hill climbing Strategies
Step 10: (Iteration loop)
Increase k k +1, set i = 1, and go to step 1.
Figure 3.6, together with the following table, presents a possible sequence of iteration
points. From the starting point (0), a successful step (1) and (3) is taken in each coordinate
direction. Since the end point of this exploratory move is better than the starting point,
it serves as a basis for the rst extrapolation. This leads to (4). It is not checked here
whether or not any improvement over (3) has occurred. At the next exploratory move,
from (4) to (5), the objective function value can only be improved in one coordinate
direction. It is now checked whether the condition (5) is better than that of point (3).
This is the case. The next extrapolation step, to (8), has a changed direction because
of the partial failure of the exploration, but maintains its increased length. Now it will
be assumed that, starting from (8) with the hitherto constant exploratory step length,
no success will be scored in any coordinate direction compared to (8). The comparison
with (5) shows that a reduction in the value of the objective function has nevertheless
occurred. Thus the next extrapolation to (13) remains the same as the previous one with
respect to direction and step length. The next exploratory move leads to a point (15),
which although better than (13) is worse than (8). Now there is a return to (8). Only
after the exploration again has no success here, are the step lengths halved in order to
make further progress possible. The fact that at some points in this case the objective
function was tested several times is not typical for n>2.
(0)
(2)
(3)
(1)
(4)
(7)
(5)
(21)
(12)
(22)
(18)
(17)
(9)
(6)
(10)
(19)
(23)
(24)
(8)
(25) (11)
(20)
Starting point
Success
Failure
Extrapolation
Final point
(16)
(15)
(13) (14)
Figure 3.6: Strategy of Hooke and Jeeves