Constraint Handling Heuristics for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 128

Table 7.6. The (15,100)-ES with correlated mutations and standard parameter set-
tings on problem 2.40. The algorithm is far away from approximating the optimum in
every run.

best mean worst dev ffc cfc
2.40 -5000.00000000 -4942.9735056 -4704.2122103 88.653 78 292 180 450

In our new nested angle evolution strategy (NAES) the outer ES adapts the
angles for the rotation of the mutation ellipsoid of the inner ES. Here we use
the advanced notation for nested ES introduced by Rechenberg [115]. In the
[ μ /ρ + ,λ ( μ/ρ + ,λ ) γ ]-ES λ inner ES run for γ generations, also called isolation
time. In our approach the inner ES terminates based on fitness stagnation, i.e.
when premature step size reduction occurs.

7.6.2 Experimental Analysis

Table 7.7 shows the experimental results of the [5 / 2 , 50(5 / 2 , 50)]-NAES on prob-
lems 2.40, 2.41, TR2 and g04 after 15 runs. The variables of the outer ES, which
are the angles for the inner ES, are initialized within the interval [0 ,π/ 2]. The
corresponding initial step sizes are σ i = π/ 8 for all i . Mutation and recombina-
tion parameters are chosen as usual. For both ES fitness stagnation is chosen as
termination condition. For all inner ES the setting θ =10 − 9 and for the outer

θ =10 − 7 is chosen.
As the worst fitness values and the standard deviations show, the NAES is able
to find the optimum of all problems with the accuracy the termination condition
allows. The NAES is able to adapt the rotation angles and thus increases the
success probability p s . This even holds for problem TR2, whose angle δ and
p s respectively, decrease rapidly during the convergence to the optimum. The
NAES causes a very high number of constraint and fitness function calls as we
expect from the nature of a meta-evolutionary approach. Every solution of the
outer ES requires a full run of a couple of inner ES. But the NAES is the only
constraint-handling method which is able to approximate the optimum of every
type of constrained problem. In practice, the NAES might be too inecient,
but the performance can be improved by adequate parameter settings for the

Table 7.7. The [5 / 2 , 50(5 / 2 , 50)]-NAES on problems TR2, 2.40, 2.41 and g04. On
all problems and in all runs the NAES is able to find the optimum with the desired
accuracy.

best mean worst dev ffc cfc

2.40 -5000.000 -5000.000 -5000.000 1.2E-8 35 935 916 149 829 046

2.41 -17857.143 -17857.143 -17857.143 3.8E-8 12 821 710 41 720 881

TR2 2.000 2.000 2.000 3.1E-16 927 372 1 394 023

g04 -31025.560 -31025.560 -31025.560 2.1E-8 19 730 306 60 667 984