Constraint Handling Heuristics for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 121

Table 7.3. Summary of the experimental results of the DSES on the considered prob-
lems. The heuristic is able to approximate the optima of most of the problems.

DSES best avg worst dev

2.40 [100; 0 . 7] − 4999 . 9999999999 − 4999 . 9999999995 − 4999 . 9999999960 7.99E-10

2.41 [75; 0 . 7] − 17857 . 1428571428 − 17857 . 1428571425 − 17857 . 1428571404 4.71E-10

TR2 [15; 0 . 3] 2 . 0000000008 2 . 0000042774 2 . 0000539455 8.45E-6

g01 [400; 0 . 5] − 14 . 9999999998 − 14 . 9999999991 − 14 . 9999999908 1.78E-9

g02 [15; 0 . 3] − 0 . 8036187549 − 0 . 7658619287 − 0 . 6999587065 0 . 029

g04 [25; 0 . 7] − 30665 . 5386717833 − 30665 . 5386717831 − 30665 . 5386717826 1.60E-10

g06 [10; 0 . 3] − 6961 . 8138755801 − 6961 . 8138755801 − 6961 . 8138755800 1.90E-11

g07 [70; 0 . 7] 24 . 3062377392 24 . 3067230937 24 . 3081991031 4.57E-4

g08 [2; 0 . 9] − 0 . 0958250414 − 0 . 0958250414 − 0 . 0958250414 9.06E-17

g09 [18; 0 . 7] 680 . 6301304921 680 . 6308434198 680 . 6322597787 6.59E-4

g11 [10; 0 . 5] 0 . 7499000007 0 . 7499008144 0 . 7499035419 1.02E-6

g12 [200; 0 . 5] − 0 . 9999999999 − 0 . 9999999999 − 0 . 9999999999 2.05E-12

g16 [100; 0 . 5] − 1 . 9051552585 − 1 . 9051552584 − 1 . 9051552580 9.12E-11

g24 [15; 0 . 3] − 5 . 5080132715 − 5 . 5080132715 − 5 . 5080132714 2.36E-11

7.5 Constraint-Handling with Two Sexes

The idea of the concept called two sexes evolution strategy (TSES) is to handle
the objective function and the constraint functions as separate objectives. The
TSES heuristic is introduced in this section.

7.5.1 Biologically Inspired Constraint-Handling

The TSES works as follows. Every individual of the TSES is assigned to a new
feature called its sex. Similar to nature, individuals with different sexes are se-
lected according to different objectives. Individuals with sex o are selected by
the objective function. Individuals with sex c are selected by the fulfillment
of constraints. The intermediary recombination operator plays a key role. Re-
combination is only allowed between parents of different sex. The treatment of
objective function and constraints as separate objectives sounds similar to the
multiobjective optimization approaches for constraint-handling. But instead of
a multiobjective optimization method, a biologically inspired concept of pairing
two sexes is introduced. Consider the situation presented in Figure 7.5. Again,
the optimum lies at the boundaries of the feasible search space. The optimum of
the unconstrained objective function lies beyond the boundary in the infeasible
search space. In the so-called ( μ o + μ c ,λ o + λ c )-TSES μ o parents are selected out
of λ o individuals with sex o ,whereas μ c parents are selected out of λ c offspring
individuals of the previous generation with sex c . As the individuals with sex o
are selected according to the objective function, they tend to lie finally in the
infeasible search space (pink dots) whereas the c -individuals are selected by the
fulfillment of all constraints and mostly lie in the feasible search space (green