Evolution and Optimum Seeking

Random Strategies 99
works entirely with a restricted choice of directions. With a xed step length, a direction
can be randomly selected only from within an n-dimensional hypercone. The angle subtended
by the cone and its height (andthus the overall step length) are controlled in an
adaptiveway. For a spherical objective function, e.g., the model functions F2 (hypercone),
F3 (hypersphere), or F4 (something intermediate between hypersphere and hypercube),
there is no improvement in the convergence behavior. Advantages can only be gained
if the search has to follow a particular direction for a long time along a narrow valley.
Sudden changes in direction present a problem, however, which leads Heydt to consider
substituting for the cone con guration a hyper-parabolic or hyper-hyperbolic distribution,
with which at least small step lengths would retain su cient freedom of direction.
In every case the striving for rapid convergence is directly opposed to the reliabilityof
global convergence. This has led Jarvis (1968, 1970) to investigate a combination of the
method of Matyas (1965, 1967) with that of McMurtry and Fu (1966). Numerical tests
by Cockrell (1969, 1970 see also Fu and Cockrell, 1970) show that even here the basic
strategy of Matyas (1965) or Schumer and Steiglitz (1967) is clearly the better alternative.
It o ers high convergence rates besides a fair chance of locating global optima, at least
for a small number of variables. In the case of many dimensions, every attempt to reach
global reliability isthwarted by the excessive cost. This leaves the globally convergent
stochastic approximation method of Vaysbord and Yudin (1968) far behind the rest of
the eld. Furthermore, the sequential or creeping random search is the least susceptible
if perturbations act on the objective function.
Users of random strategies always draw attention to their simplicity, exibility and
resistance to perturbations. These properties are especially important if one wishes to
construct automatic optimalizers (e.g., Feldbaum, 1958 Herschel, 1961 Medvedev and
Ruban, 1967 Krasnushkin, 1970). Rastrigin actually built the rst optimalizer with a
random search strategy, whichwas designed for automatic frequency control of an electric
motor. Mitchell (1964) describes an extreme value controller that consists of an analogue
computer with a permanently wired-in digital part. The digital part serves for storage and
ow control, while the analogue part evaluates the objective function. The developmentof
hybrid analogue computers, in which the computational inaccuracy is determined by the
system, has helped to bring random methods, especially of the sequential type, into more
general use. For examples of applications besides those of the authors mentioned above,
the following publications can be referred to: Meissinger (1964), Meissinger and Bekey
(1966), Kavanaugh, Stewart, and Brocker (1968), Korn and Kosako (1970), Johannsen
(1970, 1973), and Chatterji and Chatterjee (1971). Hybrid computers can be applied to
best advantage for problems of optimal control and parameter identi cation, because they
are able to carry out integrations and di erentiations more rapidly than digital computers.
Mutseniyeks and Rastrigin (1964) have devised a special algorithm for the dynamic control
problem of keeping an optimum. Instead of the variable position vector x,avelocityvector
with components @xi=@t is varied. A randomly chosen combination is retained as long as
the objective function is decreasing in value (for minimization @F=@t < 0). As soon as
it begins to increase again, a new velocity vector is chosen at random.
It is always striking, if one observes living beings, how well adapted they are in shape,
function, and lifestyle . In many cases, biological structures, processes, and systems even