Evolution and Optimum Seeking

A Multimembered Evolution Strategy 149
5.2.6 Global Convergence
In our discussion of deterministic optimization methods (Chap. 3) we haveestablished that only simultaneous strategies are capable of locating with certainty global minima
of arbitrary objective functions. The computational cost of their application increases
with the volume of the space under consideration and thus with the power of n. The
dynamic programming technique of Bellman allows the reliabilityofglobalconvergence to
be maintained at less cost, but only if the objective function has a rather special structure,
such that only a part of the space IR n needs to be investigated. Of the stochastic search
procedures, the Monte-Carlo method has the best chance of global convergence it o ers a
high probability rather than certainty of nding the global optimum. If one requires a 90%
probability, its cost is greater than that of the equidistant grid search. However, the (1+1)
evolution strategy can also be credited with a nite probability of global convergence if the
step lengths (variances) of the random changes are held constant (see Rechenberg, 1973
Born, 1978 Beyer, 1989, 1990). How great the chance is of nding an absolute minimum
among several local minima depends on the topology, in particular on the disposition and
\width" of the minima.
If the user wishes to realize the possibility of a jump from a local to a global extremum,
it requires a trial of patience. The requirement of approaching an optimum as quickly and
as accurately as possible is always diametrically opposed to maintaining the reliability of
global convergence. In the formulation of the algorithms of the evolution strategies we
have mainly strived to satisfy the rst requirement of rapid convergence, by adaptation
of the step lengths. Thus for both strategies no claims can be made for good global
convergence properties.
With > 1 in the multimembered evolution scheme, several state vectors x (g)
k 2
IR n k = 1(1) are stored in each generation g. If the x (g)
k are very di erent, the
probability is greater that at least one point is situated near the global optimum and that
the others will approach it in the process of generation. The likelihood of this is less if
the x (g)
k fall close together, with the associated reduction in the step lengths. It always
remains nite, however, and increases with ,thenumber of parents. This advantage
over the (1+1) strategy is best exploited if one starts the search with initial vectors x (0)
k
roughly evenly distributed over the whole region of interest, and chooses fairly large initial
(0)
values of the standard deviations k 2 IRn k = 1(1) . Here too the ( )scheme is
preferable to the ( + ) because concentration at a locally very favorable position is at
least delayed.
5.2.7 Program Details of the ( + ) ES Subroutines
Appendix A, Section A.2 contains FORTRAN listings of the multimembered ( + )
evolution strategy developed here, with the alternatives
GRUP without recombination
REKO with recombination (intermediary recombination for the step lengths)
KORR the so far most general form with correlated mutations as well as ve
di erent recombination types (see Chap. 7)