Constraint Handling Heuristics for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 109

The parameter t represents the actual generation, the parameters C and α must
be defined by the user. Typical settings are C =0 . 5, α = 1 or 2. Penalties can be
adapted according to an external cooling scheme [66] or by adaptive heuristics
[11]. In the death penalty approach [75] infeasible solutions are rejected and new
solutions are created until enough feasible ones exist.

7.2.2 Penalty-Related Methods

Many methods revert to the penalty principle. In the segregated genetic algo-
rithm by Le Riche et al. [118] two penalty functions, a weak and an intense one,
are calculated in order to surround the optimum. In the coevolutionary penalty
function approach by Coello [24] the penalty factors of the inner EA are adapted
by an outer EA. Some methods are based on the assumption that any feasible
solution is better than any infeasible [109], [32]. Examples are the metric penalty
functions by Hoffmeister and Sprave [57]. Feasible solutions are compared using
the objective function while infeasible solutions are compared considering the
satisfaction of constraints.

7.2.3 Repair Algorithms

Repair algorithms either replace infeasible solutions or only use the repaired
solutions for evaluation of their infeasible pendants [25], [12]. Repair algorithms
can also be seen as local search methods which reduce the constraint violation.
The repair algorithm generates a feasible solution from an infeasible one. In
the Baldwinian case, the fitness of the repaired solution replaces the fitness of
the original solution. In the Lamarckian case, the feasible solution overwrites
the infeasible one. In general defining a repair algorithms can be as complex as
solving the problem itself.

7.2.4 Decoder Functions

Decoders build up a relationship between the constrained search space and an
artificial search space easier to handle [94], [71]. They map a genotype into a
feasible phenotype. By this means even quite different genotypes may be mapped
onto the same phenotype. Eiben and Smith [38] define decoders as a class of
mappings from the genotype space

S to the feasible regions F of the solution

space S with the properties:

∈S must map to a single solution s

• Every z ∈F .

must have at least one representation s ∈S . • Every solution s ∈F

S (this need • Every s ∈F must have the same number of representations in

not be one).

Michalewicz [94] defined decoder functions for continuous problems.