Constraint Handling Heuristics for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 117

7.4 The Death Penalty Step Control Approach

In the following we introduce the death penalty step control evolution strategy
(DSES). It is based on a least step size, which has to be controlled, in order to
allow convergence to the optimum.

7.4.1 Basic Minimum Step Size Reduction Mechanism

As mentioned in section 7.3 the death penalty method suffers from premature
step size reduction. The DSES is based on death penalty, i.e. rejection of infea-
sible solutions. For the initialization feasible starting points are required. The
concept of the approach is a minimum step size , a lower bound on the step
sizes σ that prevents the evolutionary process from premature step size reduc-
tion. But it also prevents the optimization process from unlimited convergence
to the optimum when reaching the range of . Consequently, a control mecha-
nism is introduced with the task of reducing when approximating the optimum.
Intuitively, the reduction process depends on the number of infeasible mutations
produced when reaching the area of the optimum at the boundary of the fea-
sible search space. Consider the situation presented in Figure 7.3. For the sake
of better understanding we do as if all mutations fall within a σ -circle around
the parental individual. On the left (a), the individual x has come quite close
to the optimum at a vertex of the feasible search space. Further approximation
(b) with the same minimum step size means an increase of infeasible mutations
which are counted with the parameter z . The reduction process of depends

infeasible infeasible infeasible

f e asible feasi b le feasibl e

optimum optimum optimum

Fig. 7.3. Concept of the DSES. The optimum lies in a vertex of the feasible search
space. For the sake of better understanding we do as if all mutations fall into a σ -circle
around the parental individual instead of using a normal distribution with standard
deviation σ . (a) The individual x approximates the optimum. (b) A further approxima-
tion is possible because of , until the marked region of success becomes considerably
small and many mutations fall into the infeasible region. (c) When the number of infea-
sible trials exceeds the parameter the minimum step size is reduced and a further
approximation of the optimum becomes possible.