Evolution and Optimum Seeking

174 Comparison of Direct Search Strategies for Parameter Optimization
hold for the idealized concept of an algorithm, not for a particular computer program.
The susceptibility of a strategy to rounding errors depends on how it is coded. Thus, for
this reason too there is a need to check the convergence properties of numerical methods
experimentally.
Because of the nite word length of a digital computer the number range is also
limited. If it is exceeded, the program that is running normally terminates. Such fatal
execution errors ( oating over ow, oating divide check), are usually the consequence of
rounding errors in previous steps if the error is in going below the absolutely smallest
number value ( oating under ow) it is not regarded as fatal. Only few algorithms, e.g.,
Brent (1973), take special account of nite machine accuracy.
In spite of the frequent mention of the importance of numerical comparisons of strategies,
few publications to date have reported results on several di erent test problems using
a large number of minimization methods. By virtue of its scope, the work of Colville
(1968, 1970) stands out among the older studies by Brooks (1959), Spang (1962), Dickinson
(1964), Leon (1966a), Box (1966), and Kowalik and Osborne (1968). It included
30 strategies and 8 di erent problems, but not many direct search methods compared to
gradient methods. In some other tests by Jacoby, Kowalik, and Pizzo (1972), Himmelblau
(1972a), Smith (1973), and others in the collection of Lootsma (1972a), derivative-free
strategies receive much moreattention. The comparisons of Gorvits and Larichev (1971)
and Larichev and Gorvits (1974) treat only gradient methods, and that of Tapley and
Lewallen (1967) deals with some schemes for the numerical treatment of functional optimization
problems. The huge collection of test problems of Hock andSchittkowski (1981)
is biased towards standard methods of mathematical programming and their capabilities
(Schittkowski, 1980).
6.3.1 Computer Used
The machine on which thenumerical experiments were carried out was a PDP 10 from
the rm Digital Equipment Corporation, Maynard, Massachusetts. It had the following
speci cations:
Core storage area: 64K (1K = 1024 words)
Word length: 36 bits
Cycle time: 1.65 or 1.8 s
The time-sharing operating system accounted for about 34K of core, so that only
30K remained available to the user. To tackle some problems with as many variables
as possible, the computations were generally worked only to single precision. The main
program, which was the same for all strategies, occupied about 5+
2 n
1024
Kwords,
and the FORTRAN library a further 5K. The consequent maximum number nmax of
parameters is given for each search method under test in Table 6.2. The nite word length
of a digital computer means that its number range is limited. The absolute bounds for
oating point arithmetic were given by:
Largest absolute number: 2 127 ' 1:7 10 38
Smallest absolute number: 2 ;128 ' 2:9 10 ;39