Evolution and Optimum Seeking

90 Random Strategies
in order to prevent convergence to inferior local optima.
The rigidity of an algorithm based on a xed internal model of the objective function,
with which the information gathered during the iterations is interpreted, is advantageous
if the objective function corresponds closely enough to the model. If this is not the case,
the advantage disappears and may even turn into a disadvantage. Second order methods
with quadratic models seem more sensitive in this respect than rst order methods with
only linear models. Even more robust are the direct search strategies that work without
an explicit model, such as the strategy of Hooke and Jeeves (1961). It makes no use of
the sizes of the changes in the objective functionvalues, but only of their signs.
A method that uses a kind of minimal model of the objective function is the stochastic
approximation (Schmetterer, 1961 see also Chap. 2, Sect. 2.3). This purely deterministic
method assumes that the measured or calculated function values are samples of a normally
distributed random quantity, of which the expectation value is to be minimized or
maximized. The method feels its way to the optimum with alternating exploratory and
work steps, whose lengths form convergent series with prescribed bounds and sums. In
the multidimensional case this standard concept can be the basis of various strategies for
choosing the directions of the work steps (Fabian, 1968). Usually gradient methods show
themselves to best advantage here. The stochastic approximation itself is very versatile.
Constraintscanbetaken into account (Kaplinskii and Propoi, 1970), and problems of
functional optimization can be treated (Gersht and Kaplinskii, 1971) as well as dynamic
problems of maintaining or seeking optima (Chang, 1968). Tsypkin (1968a,b,c, 1970a,b
see also Zypkin, 1966, 1967, 1970) discusses these topics very thoroughly. There are also,
however, arguments against the reliability of convergence for certain types of objective
function (Aizerman, Braverman and Rozonoer, 1965). The usefulness of the strategy in
the multidimensional case is limited by its high cost. Hence there has been no shortage
of attempts to accelerate the convergence (Fabian, 1967 Berlin, 1969 Saridis, 1968,
1970 Saridis and Gilbert, 1970 Janac, 1971 Kwatny, 1972 see also Chap. 2, Sect. 2.3).
Ideas for using random directions look especially promising some of the many investigations
of this topic which have been published are Loginov (1966), Stratonovich (1968,
1970), Schmitt (1969), Ermoliev (1970), Svechinskii (1971), Tsypkin (1971), Antonov and
Katkovnik (1972), Berlin (1972), Katkovnik and Kulchitskii (1972), Kulchitskii (1972),
Poznyak (1972), and Tsypkin and Poznyak (1972).
The original method is not able to determine global extrema reliably. Extensions of
the strategy in this direction are due to Kushner (1963, 1972) and Vaysbord and Yudin
(1968). The sequence of work steps is so designed that the probability of the following
state being the global optimum is maximized. In contrast to the gradient concept, the
information gathered is not interpreted in terms of local but of global properties of the
objective function. In the case of two local minima, the e ort of the search is gradually
concentrated in their neighborhood and only when one of them is signi cantly better is
the other abandoned in favor of the one that is also a global minimum. In terms of the
cost of the strategy, the acceleration of the local search and the reliability of the global
search are diametrically opposed. Hill and Gibson (1965) show that their global strategy
is superior to Kushner's, as well as to one of Bocharov and Feldbaum. However, they only
treat cases with n 2 parameters. More recent research results have been presented by