Evolution and Optimum Seeking

Random Strategies 101
the principles of population, sexual inheritance, recombination, dominance, and recessiveness
to improve the convergence behavior yield the hoped for breakthrough. He thus
eventually resigns himself to a largely deterministic strategy. In the linear programming
problem, he chooses from the starting point several random directions and follows these in
turn up to the boundary of the feasible region. The best states on the individual bounding
hyperplanes are used to determine a new starting point by taking the arithmetic mean
of the component parameters. Because of the convexity of the allowed region, the new
starting point isalways within it. The simultaneous choice of several search directions
is supposed to be the analogue of the population principle and the construction of the
average the analogue of recombination in sexual propagation. To tackle the problem of
nding the minimum or maximum of an unconstrained, non-linear function, Bremermann
even applies a ve point Lagrangian interpolation to determine relative extrema in the
random directions.
Rechenberg's evolution strategy changes all the components of the variable vector at
each mutation. In his case, the low mutation rate for many dimensions is expressed by
choosing small values for the step lengths, or the spread in the random changes. On the
basis of theoretical work with two model functions he nds that the standard deviations
of the random components are set optimally when they are inversely proportional to the
number of parameters. His two membered evolution strategy resembles the scheme of
Schumer and Steiglitz (1968), which isacknowledged to be particularly good, except that
a(0 2 ) normally distributed random quantity replaces the xed step length s. He has
also added to it a step length modi cation rule, again derived from theory, whichmakes
this look a very promising search method. It is re ned in Chapter 5, Section 5.1 to meet
the requirements of numerical optimization with digital computers. Amultimembered
strategy is treated in Section 5.2, which follows the same basic concept however, by imitating
the principles of population and recombination, it can operate without external
control of the step lengths. Incorporating more than one descendant at a time and forgetting
\parental wisdom" at the end of each iteration loop has provoked erce objections
against a more natural evolution strategy.
Box (1957) also considers that his EVOP (evolutionary operation) strategy resembles
the biological mutation-selection process. He regards the vertices of his pattern of
trial points, of which the best becomes the center of the next pattern, as individuals of
a population, of which only the best \survives." The \o spring" are, however, generated
by purely deterministic rules. Random decisions, as used by Satterthwaite (1959a
after Lowe, 1964) in his REVOP (random evolutionary operation) variant, are actually
explicitly rejected by Box(seeYouden et al., 1959 Satterthwaite, 1959b Budne, 1959
Anscombe, 1959).
From a biological or cybernetic pointofview,Pask (1962, 1971), Schmalhausen (1964),
Berg and Timofejew-Ressowski (1964), Dobzhansky (1965), Moran (1967), and Kussul and
Luk (1971) among others have examined the analogy between optimization and evolution.
The fact that no practical algorithms have come out of this is no doubt because the
evolutionary processes are described only verbally. Although they sometimes even include
their more subtle e ects, they have so far not produced a really quantitative, predictive
theory. Exceptions, such as the work of Eigen (1971 see also Schuster, 1972), Merzenich