Evolution and Optimum Seeking

Multidimensional Strategies 43
e 2
(0,0)
x
Start
(0,1)
x
(0,2) (1,0)
x x
=
(1,1)
x
(1,2) (2,0)
x x
=
End
x (2,1)
x (2,2) (3,0)
= x
e 1
Figure 3.5: Coordinate strategy
Numbering
Iteration index
Direction index
Variable
values
k i x 1 x 2
(0) 0 0 0 9
(1) 0 1 3 9
(2) 0 2 3 5
carried over
(2) 1 0 3 5
(3) 1 1 7 5
(4) 1 2 7 3
carried over
(4) 2 0 7 3
(5) 2 1 9 3
(6) 2 2 9 2
carried over
(6) 3 0 9 2
convergence are so small that the number of signi cant gures to which data are handled
by the computer is insu cient for the variables to be signi cantly altered.
Numerical tests with the coordinate strategy show that an exact determination of the
relative minima is unnecessary, at least at distances far from the objective. It can even
happen that one inaccurate line search can make the next one particularly e ective. This
phenomenon is exploited in the procedures known as under- or overrelaxation (Engeli,
Ginsburg, Rutishauser, and Stiefel, 1959 Varga, 1962 Schechter, 1962, 1968 Cryer,
1971). Although the relative optimum is determined as before, either an increment is
added on in the same direction or an iteration point is de ned on the route between the
start and nish of the one dimensional search. The choice of the under- or overrelaxation
factor requires assumptions about the structure of the problem. The necessary information
is available for the problem of solving systems of linear equations with a positive de nite
matrix of coe cients, but not for general optimization problems.
Further possible variations of the coordinate strategy are obtained if the sequence of
searches parallel to the axes is not made to follow the cyclic scheme. Southwell (1946), for
example, always selects either the direction in which the slope of the objective function
Fx i(x) = @F(x)
is maximum, or the direction in which the largest step can be taken. Toevaluate the choice
@xi