Evolution and Optimum Seeking

The Two Membered Evolution Strategy 115
5.1.4 The Treatment of Constraints
Inequality constraints Gj(x) 0 for all j = 1(1)m are quite acceptable. Sign conditions
may be formulated in the same manner and do not receive anyspecial treatment. In
contrast to linear and non-linear programming, no sign conditions need to be set in order
to keep within a bounded region. If a mutation falls in the forbidden region it is assessed
as a worsening (in the sense of a lethal mutation) and the variation of the variables is not
accepted.
No particular penalty function, such asRosenbrock chooses for his method of rotating
coordinates, has been developed for the evolution strategy. The user is free to use the
techniques for example of Carroll (1961), Fiacco and McCormick (1968), or Bandler and
Charalambous (1974), to construct a suitable sequence of substitute objective functions
and to solve the original constrained problem as a sequence of unconstrained problems.
This, however, can be done outside the procedure.
It is sometimes di cult to specify an allowed initial vector of the variables. If one were
to wait until by chance a mutation satis ed all the constraints, it could take avery long
time. Besides, during this search period the success checks could not be carried out as
described above. It would nevertheless be desirable to apply the normal search algorithm
e ectively to nd an allowed state. Box (1965) has given in the description of his complex
method a simple way of proceeding from a forbidden starting point. He constructs an
auxiliary objective function from the sum of the constraint function values of the violated
constraints:
mX
~F (x) = Gj(x) j(x)
where
j(x) =
j=1
( ;1 if Gj(x) < 0
0 otherwise
(5.7)
Each decrease in the value of ~F(x) represents an approach to the feasible region. When
eventually ~ F(x) = 0, then x satis es all the constraints and can serve as a starting vector
for the optimization proper. This procedure can be taken over without modi cation for
the evolution strategy.
5.1.5 Further Details of the Subroutine EVOL
In Appendix B, Section B.1 a complete FORTRAN listing is given of a subroutine corresponding
to the two membered evolution scheme that has been described. Thus no
detailed algorithm will be formulated here, but a few further programming details will be
mentioned.
In nearly all digital computers there are library subroutines for generating uniformly
distributed pseudorandom numbers. They work, as a rule, according to the multiplicative
or additive congruence method (see Johnk, 1969 Niederreiter, 1992 Press et al., 1992).
From any two numbers taken at random from a uniform distribution in the range [0 1], by
using the transformation rules of Box and Muller (1958) one can generate two independent,