Evolution and Optimum Seeking

Random Strategies 91
Pardalos and Rosen (1987), Torn and Zilinskas (1989), Floudas and Pardalos (1990),
Zhigljavsky (1991), and Rudolph (1991, 1992b). Now thereareeven specialized journals
established in the eld, see Horst (1991).
All the strategies mentioned so far are fundamentally deterministic. They only resort
to chance in dead-end situations, or they operate on the assumption that the objective
function is stochastically perturbed. Jarvis (1968), who compares deterministic and
probabilistic optimization methods, nds that random methods that do not stick toany
particular model are most suitable when an optimum must be located under particularly
di cult conditions, such as a perturbed objective function or a \pathological" problem
structure with several extrema, discontinuities, plateaus, forbidden regions, etc. The
homeostat of Ashby (1960) is probably the oldest example of the application of a random
strategy. Its objective istomaintain a condition of equilibrium against stochastic
disturbances. It may happen that no optimum is sought, but only a point in an allowed
region (today one calls such taskaconstraints satisfaction problem or CSP). Nevertheless,
corresponding solution methods are closely tied to optimization, and there are a series of
various heuristic planning methods available (e.g., Weinberg and Zehnder, 1969). Ashby's
strategy, which he calls a blind homeostatic process, becomes activewhenever the apparatus
strays from equilibrium. Then the controllable parameters are randomly varied until
the desired condition is restored. The nite number (in this case) of discrete settings of
the variables all enter the search process with equal probability. Chichinadze (1960) later
constructed an electronic model on the same principle and used it for synthesizing simple
optimal control systems.
Brooks (1958), probably stimulated by R. L. Anderson (1953), is generally regarded as
the initiator of the use of random strategies for optimization problems. He describes the
simple, later also called blind or pure random search for nding a minimum or maximum
in the experimental eld. In a closed interval a x b several points are chosen at
random. The probability density w(x) is constant everywhere within the region and zero
outside.
(
1=V for all a x b
w(x) =
0 otherwise
V , the volume of the cube with corners ai and bi for i =1(1)n, is given by
V =
nY
i=1
(bi ; ai)
The value of the objective function must be determined at all selected points. The point
that has the lowest or highest function value is taken as optimum. How well the true
extremum is approximated depends on the numberoftrialsaswell as on the actual
random results. Thus one can only give a probability p that the optimum will be found
within a given number N of trials with a prescribed accuracy.
p =1; (1 ; v=V ) N
(4.1)
The volume v < V < 1 contains all points that satisfy the accuracy requirement. By