Evolution and Optimum Seeking

44 Hill climbing Strategies
of direction, Synge (1944) uses the ratio Fx i=Fx ix i of rst to second partial derivatives at
the point x (k) . Whether or not the additional e ort for this scheme is worthwhile depends
on the particular topology of the contour surface. Adding directions other than parallel
to the axes is also often found to accelerate the convergence (Pinsker and Tseitlin, 1962
Elkin, 1968).
Its great simplicity has always made the coordinate strategy attractive, despite its
sometimes slow convergence. Rules for handling constraints{not counting here penalty
function methods{have been devised, for example, by Singer (1962), Murata (1963), and
Mugele (1961, 1962, 1966). Singer's maze method departs from the coordinate directions
as soon as a constraint is violated and progresses into the feasible region or along the
boundary. For this, however, the gradient of the active constraints must be known.
Mugele's poor man's optimizer, a discrete coordinate strategy without line searches, not
only handles active constraints, but can also cope with narrow valleys that do not run
parallel to the coordinate axes. In this case diagonal steps are permitted. Similar to this
strategy is the direct search methodofHooke and Jeeves, which because it has become
very widely used will be treated in detail in the following chapter.
3.2.1.2 Strategy of Hooke and Jeeves: Pattern Search
The direct pattern search of Hooke and Jeeves (1961) was originally devised as an automatic
experimental strategy (see Hooke, 1957 Hooke andVanNice, 1959). It is nowadays
much more widely used as a numerical parameter optimization procedure.
The method by which the direct pattern search works is characterized by two types
of move. At each iteration there is an exploratory move, which represents a simpli ed
Gauss-Seidel variation with one discrete step per coordinate direction. No line searches
are made. On the assumption that the line joining the rst and last points of the exploratory
move represents an especially favorable direction, an extrapolation is made along
it (pattern move) before the variables are varied again individually. The extrapolations
do not necessarily lead to an improvement in the objective function value. The success of
the iteration is only checked after the following exploratory move. The length of the pattern
step is thereby increased each time, while the optimal search direction only changes
gradually. Thispays o to most advantage where there are narrow valleys. An ALGOL
implementation of the strategy is due to Kaupe (1963). It was improved by Bell and Pike
(1966), as well as by Smith (1969) (see also DeVogelaere, 1968 Tomlin and Smith, 1969).
In the rst case, the sequence of plus and minus exploratory steps in the coordinate directions
is modi ed to suit the conditions at any instant. The second improvement aims
at permitting a retrospective scaling of the variables as the step lengths can be chosen
individually to be di erent from each other.
The algorithm runs as follows:
Step 0: (Initialization)
Choose a starting point x (00) = x (;1n) , an accuracy bound ">0, and initial
step lengths s (0)
i 6= 0 for all i = 1(1)n (e.g., s (0)
1 = 1 if no more plausible
values are at hand).
Set k =0andi =1.