Constraint Handling Heuristics for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 110

7.2.5 Multiobjective Optimization

Multiobjective optimization techniques are based on the idea of handling each
constraint as an objective. Under this assumption many multiobjective opti-
mization methods can be applied. Such approaches were used by Parmee and
Purchase [104], Jimenez and Verdegay [65], Coello [23] and Surry et al. [150]. In
the behavioral memory-method by Schoenauer and Xanthakis [130] the EA con-
centrates on minimizing the constraint violation of each constraint in a certain
order and optimizing the objective function in the last step.

7.2.6 Constraint Preserving Operators and Representations

A further method is to avoid infeasible solutions by special constraints pre-
serving representations and operators. An example is the GENOCOP-algorithm
[94] that reduces the problem to convex search spaces and linear constraints. A
predator-prey approach to handle constraints is proposed by Paredis [103] using
two separate populations. Schoenauer and Michalewicz [129] propose special op-
erators that are designed to search regions in the vicinity of active constraints.
A comprehensive overview to constraint handling techniques is given by Coello
[25] and also by Michalewicz [94].

7.2.7 Recent Developments

Recently, Coello [96] introduced a technique based on a multimembered ES
combining a feasibility comparison mechanism with several modifications of the
standard ES. At the CEC 2006 special session on constrained real-parameter op-
timization a number of interesting methods was introduced. The constrained
differential evolution approach by Takahama [152] combines the usage of an
for equality constraints with the differential evolution approach. It furthermore
uses a gradient-based mutation to find feasible points considering the gradient
of infeasible points. Additionally, a scheme of feasible elites is used to improve
the search for feasible solutions. The dynamic multi-swarm particle optimizer
by Liang and Suganthan [84] makes use of a set of sub-swarms concentrating
on different constraints. It is combined with sequential quadratic programming
as a local search method. The approach of Mezura-Montes et al. [92] combines
differential evolution, different mutation operators to increase the probability of
producing better offspring, three selection criteria and a diversity mechanism.
Another interesting approach is the population-based method of Sinha et al.
[140] using a parent centric procedure. It obtained successful results on the CEC
test problems.

7.3 Premature Step Size Reduction

ES on constrained optimization problems suffer from premature step size reduc-
tion in case of active inequality constraints. This results in premature convergence.
Broadly speaking, the reason for the premature step size reduction is the fact that