Constraint Handling Heuristics for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 127

eciency advantage of the TSES. The experiments on problem g12 have already
been commented in this section. On problem g16 the (8+8,13+87)-TSES is not
more successful, but less ecient than DP with an average accuracy of 3 deci-
mal places. But the (40+40,50+400)-TSES reaches the optimum with arbitrary
accuracy demanding 3-4 times more ffc and cfc . The (8+8,10+90)-TSES is the
best compromise between quality of the results and e ciency on problem g24.
But we recommend to use DP as it demands much less ffc and cfc .

7.6 The Nested Angle Evolution Strategy

The last proposed constraint handling heuristic is the NAES. It is based on a
nested ES and rotates the mutation ellipsoid until the success rates near the
constraint boundary improve.

7.6.1 Meta-evolution for Mutation Ellipsoid Rotation

The success rates at the boundary of the feasible search space will change consid-
erably, if the mutation ellipsoid is rotated by the angle δ , see Figure 7.8. Here,
δ is the smaller angle between one parameter axis and the constraint bound-
ary. The mutation ellipsoid can adapt to a situation, where the infeasible search
space does not cut off the success area. The success rate improves to p s =0 . 5
and consequently prevents premature step size reduction. This consideration
leads to NAES. Figure 7.8 shows the situation when the mutation ellipsoid is
rotated by angle δ . Rotation of the mutation ellipsoid can be achieved by

correlated mutations introduced by Schwefel [132]. But experiments with the
correlated mutations show that the diversity in the population of a (15 , 100)-ES
may be too low to achieve the adaptation of both mean step sizes and angles, see
table 7.6.

−

direction
to optimum

infeasible search space

lines of constant fitness

x

feasible search space

Fig. 7.8. Situation at the boundary of the feasible search space after rotation of the
mutation ellipsoid by the angle − δ : The self-adaptation process enables the step sizes
σ 1 and σ 2 to form an ellipsoid that is not cut off by the constrained search space.