The Genetic Chromodynamics Framework

The framework assumes the following considerations. Subpopulations that are each connected to a local or global optimum of the problem to be solved are built and maintained by introducing a set of restrictions on how selection is applied or the way recombination takes place [Dumitrescu, 2000]. There is not a typical selection scheme considered, as every individual represents a stepping stone for the forming of the new generation, that is each is taken into account for reproduction. If a parent has no similar individuals to itself, then it mutates. For recombination, a local inter-action principle is considered, meaning that only individuals similar under a given threshold mate. After either recombination or mutation takes place, the offspring fights for survival with the stepping-stone parent. In order for the subpopulations to become better and better separated with each iteration, a new genetic operator, called merging, is introduced. Merging represents a form of selection that shrinks the population, as some individuals are discarded. The operator ”merges” very simi-lar individuals by keeping only the most prolific candidate solutions and it is applied after the perturbation process.

The interplay between the merging and the mating regions strongly impacts con-vergence properties: a large merging radius leads to the deletion of potential mates of the surviving individuals and thus compromises recombination. Therefore, it is usually chosen to be much smaller than the mating radius.

The complete evolutionary process takes place as follows. First, the initial pop-ulation is randomly generated. Next, every individual participates in the forming of the new generation. Mating regions around each individual are determined by a radius and, therefore, only neighboring individuals recombine. When no mate is found in the corresponding region of the current individual, the latter produces one offspring by mutation, with a step size that still keeps the descendant in the mating region of its parent. If there is more than one individual near the current, the mate is determined using proportional selection. Then, if the offspring has better fitness than the current individual, it replaces the latter in the population.

Immediately after variation, the merging operator eliminates similarity from the population. For illustration, let an individual c be given. If the distance between c and another individual is very small, i.e., under a given merging radius, then the latter is considered as part of the merging region of c. From the set of all individuals in the merging region of c, only one is kept. Generally, it is the one with the best fitness evaluation that is preserved. Alternatively, other merging schemes may be used (for instance, the mean of the individuals in the merging region).

The unfolding of the GC steps are outlined by Algorithm 4.1.

By reducing the potential partners for recombination of an individual to those lying in its mating region, only individuals that are close to each other recombine, favoring the appearance and maintenance of subpopulations. The offspring replaces the current individual only if fitter. Thus, after a few generations, the candidate so-lutions will concentrate on the most promising regions of the search space, i.e. those connected to the optima.

4.2 The Genetic Chromodynamics Framework 49

Algorithm 4.1 The GC algorithm.

Require: An optimization problem

Ensure: The set of solutions where every one corresponds to an optimum of the problem begin

t←0;

Initialize population P(t);

while termination condition is not satisfied do Evaluate each individual;

for all individuals c in the population do if mating region of c is empty then

Apply mutation to c;

if obtained individual is fitter than c then Replace c in P(t+1);

end if else

Choose one individual from the mating region of c for recombination;

Obtain and evaluate one offspring;

if the offspring is fitter than c then Replace c in P(t+1);

end if end if end for repeat

Consider an individual c as the current one;

Select all m individuals in the merging region of c, including itself;

Keep only the fittest individual from the selection in the population of the next gen-eration;

until merging cannot be further applied t←t + 1;

end while

return population P(t)containing the optima end

Merging is an useful operator since it triggers a better computational time by reducing the size of the population, as less fitness evaluations are necessary. Con-sequently, subpopulations independently evolve and become better separated with each iteration and lead, at convergence, each one to an optimum.

Courtesy of merging, subpopulations become better separated with each iteration and the population size is reduced in a way that preserves the coordinates of each subpopulation. Along with separation, the worse individuals are removed step by step, and the process gradually transforms from a globally–oriented population–

based methodology to a parallel local search technique. Therefore, in the end of evolution, it is only one individual that remains connected to every optimum. Each such individual now actually corresponds to a single hill–climber that uses only mutation as a variation operator.

Illustrations of the GC radii-based mating and merging are given in Fig. 4.1 and 4.2.

Fig. 4.1 An example of mating within GC.

Individual c1 is alone in its mating region (dotted circle), so it produces one offspring by mutation. Individual c2 selects another parent from its mating region and creates one offspring by recombination that is worse than c2 and consequently does not replace its step-ping stone parent. Crossed lines indicate in-dividuals with worse fitness that are further replaced.

c1 c2

Fig. 4.2 An example of merging within GC that follows the scenario from the mating episode in Fig. 4.1. Individual c2 is deleted because there is another individual with bet-ter fitness in its merging region (solid circle).

c2

GC can be metaphorically viewed as if all the individuals originally have a dis-tinct color. But, once they mate and merge, the colors of the most promising so-lutions spread over the entire search space. In the end, there are as many different colors as the number of distinct subpopulations, each connected to an optimum.

The weak point of the methodology is that it makes use of two parameters – the mating and merging radii – which are, for some problems, hard to parameterize.

Nevertheless, if the mating parameter is computed using formula (4.2) introduced in [Deb and Goldberg, 1989] and smaller comparable values are tried for the merging radius, one could reach configurations that provide consistent results.

The work [Deb and Goldberg, 1989] proposed a manner of computing the value for theσshare radius leading to the formation of subpopulations in niching EAs, that has become very popular among researchers keen to control such parameters. It uses the radius of the smallest hypersphere containing feasible space, which is given by (4.1):

r=1 2

n

i=1

∑

where n represents the number of dimensions of the problem at hand and biand ai

are the upper and the lower bounds of the i-th dimension. Knowing the number of global optima NGand being aware that each niche is enclosed by an n-dimensional hypersphere of radius r, the niche radiusσsharecan be estimated as:

4.3 Crowding Genetic Chromodynamics 51