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. However, in addition to the overhead involved, they principally fail to eliminate the sampling error inherent in FPS selection schemes. This is no doubt one of the reasons why, we can roughly argue from our literature survey that whenever an FPS selection method like RWS is utilized, elitist strategy will be explicit... |
2.5.2 Ranking methods Ranking is another method developed to tackle the problems in FPS selection schemes. Originally proposed by Baker |
it entails ranking the chromosomes in order of their fitness values. Although ranking can lead to loss of some information regarding the actual fitness of the chromosomes, it successfully eliminates the need of rescaling and yields a simple and efficient selection algorithm. Ranking technique assigns new fitness value ... |
[EQUATION] Apart from linear ranking, exponential ranking is another variant of ranked selection method designed to give more diverse population. The method favours worst individuals at the expense of the above average ones. Both linear and exponential ranking methods facilitate maintenance and tunability of the select... |
of ranking methods is the loss of information about the actual fitness values of the chromosomes. This can negatively alter the correlation between fitness and chances of reproductive success. |
2.5.3 Tournament selection methods Tournament selection method is a non-direct fitness proportionate selection scheme. It was inspired by the natural mating contest in which a group of individuals compete for reproduction. Out of a population of size [MATH] [MATH] individuals are selected at random for the contest. An ... |
It can be observed that this technique may have several varieties since for instance; selecting the contestants can be carried out with or without replacement. It is worth noting that running a series of tournaments with replacement risks having the higher (best) fit individuals not been selected for any of the contest... |
A complete tournament selection cycle that yields [MATH] chosen individuals will require each chromosome taking part in a contest [MATH] times. Naturally, the best individual will win all its [MATH] contests, an average individual is expected to win half of its [MATH] contest and at the other extreme, the worse individ... |
A tournament selection of size [MATH] is called binary tournament and it was analytically shown that , the dynamics of a binary strict tournament selection resembles that of a specific form of linear ranking where the best ranked individual is assigned a new fitness value of [MATH] (i.e. [MATH] in equation ( 2.15 )). O... |
Other selection schemes include truncation selection , steady state GAs or Genitor , and the [MATH] and [MATH] methods originally inherited from evolutionary programming and evolutionary strategies |
. Discussions on elitism and its varieties will follow in section 2.7.1 of this chapter. As previously highlighted, when investigating selection schemes it is imperative to understand the crucial parameters that govern the choice of appropriate selection scheme. Some of these parameters are said to have originated from... |
and are presented below: i. Response to selection [MATH] : As the name implies, this measure quantifies the difference in the average fitness of population between two successive generations. In other words, the difference between the average fitness of population at generation [MATH] and its average fitness at generat... |
[EQUATION] where [MATH] is the average fitness of the population at generation [MATH] ii. Selection Differential [MATH] : Conversely, this measures the difference between the average fitness of the parent set [MATH] (i.e. individuals selected for reproduction) at generation [MATH] and the average fitness of the entire ... |
[EQUATION] where [MATH] and [MATH] are the average fitness of the entire population and that of the parent population respectively. |
iii. Selection Intensity [MATH] : This is a dimensionless quantity derived by taking the ratio of the selection differential [MATH] with the population standard deviation also at generation [MATH] such that: |
[EQUATION] where [MATH] is the population’s standard deviation at generation [MATH] Although this definition restricts the use of selection intensity to analyse the effect of selection to a specific generation in the evolution, it can be deployed to investigate the effect of selection for the entire evolution run if th... |
). In this case, [MATH] . Goldberg et al. argued that when the population vector has a standard normal fitness distribution, the selection intensity is basically the expected average fitness of the population under the influence of the selection algorithm. |
iv. Selection Pressure [MATH] : Notice that although selection intensity [MATH] can be useful for comparison purposes, it is application is limited to generational GAs where there is a clear set of parent population, i.e. [MATH] cannot be used to analyse elitist or steady state GAs. Thus, selection pressure is another ... |
. A straight forward definition for [MATH] as given in is in terms of the ratio of probabilities: [EQUATION] v. Takeover time : This is a measure that estimates how long it will take the best individual to take over the entire population. For any selection method applied on a population of size [MATH] consisting of a s... |
are conducted on selection-only genetic algorithms or an elitist GA with mutation. Nevertheless, it remains an important measure that furnishes valuable insight into the complexity, growth rate and many other characteristics of selection methods. |
An early work by Goldberg et al. reviewed various selection methods and compare them based on their time complexities, takeover times, growth ratios and selection pressure. They analytically prove that the complexity of the fitness proportional selection methods is at its best [MATH] and the worse case can be [MATH] . ... |
EC algorithms are increasingly popular because many of the evolution processes involved can easily be executed in parallel. However, selection operators are originally designed to work on the entire population and thus, their traditional implementation is sequential and can severely inhibit performance. Bäck et al. |
suggest applying a parallel algorithm to conduct local selection within subpopulations or neighbourhoods such as in migration or diffusion models. Goldberg et al. |
share the same view with regard to the need for global information by various selection methods, but they argue that tournament selection is an exception. Tournament selection scheme can easily be parallelized since it naturally works by setting up contests on subpopulations rather than the entire population as a whole... |
With regard to selection pressure, while on one hand noted that fitness proportionate methods are slower and have the least selection pressure as compared to ranking and tournament selection methods. Although this may occasionally be beneficial on the quality of convergence (since less pressure is applied to force bad ... |
thereby forestalling convergence to promising solutions. On the other hand, steady state genetic algorithms (such as genitor) have the highest selection pressure. |
stressed that the selection pressure of genitor is twofold (one for always selecting the best and the other for always having to replace the worse individual). The consequence of this is intense susceptibility to premature convergence to suboptimal solutions as a result of rapid loss of diversity. We therefore argue th... |
Moreover, successful choice of a selection method requires analysing their convergence characteristics. Based on the normal distribution theory, Thierens et al. |
developed elegant convergence models for the fitness proportional, truncation, tournament and elitist recombination selection schemes. Their analysis reveals that selection essentially leads to build-up of covariance among allele, and, the growth rate in fitness proportional selection is directly proportional to the fi... |
which he described as a milder form of natural selection and is proportional to population variance. Thus explaining why selection progress in FPS methods drops as the average fitness of the population grows. |
Rudolph et al. modelled the takeover times of various tournament methods as Markov chains and cautioned that stochastic tournament methods just like FPS are prone to sampling errors and should be used with care. They also admit that in spite of the significance of takeover time, it is not sufficient to critically decid... |
The effect of noise on various selection schemes was investigated by Hancock . Gaussian noise was added to the evaluation functions and the growth rates in the presence of mutation for various selection methods were observed. Hancock argued that contrary to the traditional judgement that genetic algorithms are immune t... |
Elsewhere, investigates the placement of selection operation in a typical run of evolution algorithms from two different perspectives. In a classic GA setup, the evolution process begins with selection, crossover and then mutation. This setup is called parent selection because it applies the selection operation on the ... |
In the area of hybrid evolutionary computation further investigation of selection techniques yields some novel selection methods. For instance, |
hybridizes a genetic algorithm by embedding a Nelder-Meid simplex algorithm into different niches of the genetic algorithm. The simplex algorithm serves as a crossover operator and was used together with simplex multi direction search to improve the selection process in GA. In a similar fashion, |
utilizes a simulated annealing (SA) algorithm to undertake selection on the population of candidate solutions such that the SA algorithm is integrated as an operator in the parent genetic algorithm. Some other hybrid algorithms that compare various standard selection algorithms include; |
which compares FPS with tournament selection and which compares ranking and RWS selection methods with the view to categorize their proposed hybrid algorithm. |
Detailed and interesting comparison of various tournament selection methods can be found in . Further discussions and comparisons of various selection schemes based on the aforementioned parameters can be found in |
To sum up, tournament selection method strikes a perfect balance between the FPS methods that tend to be too inclined to the biological evolution (i.e. adaptation) and the steady state GAs that tend to be too inclined to the simulated evolution (i.e. optimization). It maintains moderate level of selection pressure and ... |
2.6 Recombination and Mutation Having chosen the parent population via what is basically a biased process (selection) in the first phase, some variation operations are necessary for the evolution to make progress by exploring other areas of the search space. This second phase of evolution is called the reproduction pha... |
Researchers in evolutionary computation categorize parameters in genetic algorithms into structural and numerical parameters (detailed under the parameterization section). The reproduction operators are involved in both category of parameterization and are therefore critical to the successful design of any evolutionary... |
are of the view that crossover is the primary reproduction operator that clearly distinguish genetic algorithms from other types of optimization algorithms because mutation can be regarded as a sort of local search or hill climbing operation. |
Besides crossover and mutation, Holland’s original proposal includes another variation operator called inversion, this operator differs slightly from mutation and has not receive wide acceptance in the evolutionary computation community. Thus, inversion will not be covered in the following background as it is not utili... |
2.6.1 Implementation of Crossover in EAs In its simplest form, crossover operator yields two offspring by exchange of genetic materials between two parent strings subject to probabilistic decisions. In a typical genetic algorithm in which individuals are represented as binary strings, if any two parents [MATH] and [MAT... |
[EQUATION] [EQUATION] This kind of crossover is called single point (or [MATH] ) and can easily be extended to a number of variants by creating a number of random crossover points after which the parent strings exchange their bits. This extended version of one point crossover is therefore called multi-point ( [MATH] po... |
As agreed by many , crossover plays two key roles in evolution. First it provide a chance for further examination of the already available hyperplane, like offspring [MATH] simply continues with the exploitation of the hyperplane [MATH] . Second, it allow exploration of new area of the search space like the hyperplane ... |
to the adopted method of reproduction. An argument reported in reveals that two bias sources (positional and distributional) are central to the implementation of crossover operators in genetic algorithms. The single point crossover operator is said to suffer positional bias because it tends to favour substrings of cont... |
argued that none of these biases can be said to be working for the betterment or otherwise of the search process. Out of the several proposals to tackle the bias in crossover, a popular alternative is the so-called uniform crossover. Uniform crossover seeks to eliminate the bias by making the operation completely rando... |
reveal that the disruption caused by uniform crossover can be tuned by varying the Bernoulli parameter [MATH] associated with the distribution. A value of [MATH] results in a completely random generation of the crossover mask and can eliminate any bias in genetic algorithm due to the crossover. |
Other Implementation issues Not every optimization problem can be adequately represented in binary, even if it is possible to do so, utilizing any of the aforementioned crossover operators on some optimization problems would simply lead to creation of infeasible solutions. This necessitates the need for specially desig... |
and order crossover are designed for TSP and other permutation problems. A generalized [MATH] point crossover (GNX) is designed to be representation-independent and suitable for all problem domains. Thus, it sets up a general framework for recombination and seeks to alleviate the need for several ad hoc crossover imple... |
Nevertheless, many problem dependent crossover operators are still proposed and applied on various problem domains. Man et al. argued that conventional crossover operators do not perform well on complex optimization problems because they lack problem-specific knowledge in their encoding. Thus, elsewhere, an improved ge... |
. The new operator was designed to balance the exploitation-exploration trade off by producing two offspring biased for exploitation and another two biased for exploitation. |
As mentioned earlier, crossover operator is commonly applied across a population of individuals based on some probabilistic decisions. A survey of various theoretical analyses |
have shown that crossover probabilities [MATH] are considered optimal for most global optimization problems. Elsewhere, Grefenstette |
uses a multi-level genetic algorithm where the parameters of an outer GA are optimized by another GA running internally. Experimental results from |
have shown that while smaller population sizes requires high crossover rate, larger sizes do not. In other words, crossover rate can be safely reduced as population size increases. The results suggest that [MATH] are respectively suitable for population sizes of [MATH] |
Although these proposals have proven to be quite effective on wide range of global optimization problems, several recombination operators that automatically adapt their rates |
have been proposed. In this regard, a statistics-based adaptive non-uniform crossover was proposed by . They argue that there is an implicit convergence trend that leads to the build-up of building blocks by alleles of each string which the conventional crossover operators fail to exploit. Hence, they suggest using som... |
is another typical example of [MATH] point crossover where both the positions and number of crossover points can be adapted. 2.6.2 Implementation of Mutation in EAs |
Commonly, evolutionary algorithms rely on two variation operators for the reproduction of new individuals. Besides crossover, mutation is the second variation operator that also works in the genotype space and is capable of producing new individual(s) from a single parent string. Unlike the crossover operator that requ... |
Consider the following binary represented parent string [MATH] of length [MATH] in a hyperplane [MATH] . Mutating bits [MATH] and [MATH] yields an offspring [MATH] located at an entirely different hyperplane [MATH] . Thus, as accorded by many |
mutation principally enables the search process to escape from sub-optimal regions by jumping across hyperplanes thereby preserving the required diversity in a population. |
[EQUATION] Notice that applying mutation to a [MATH] ary (where [MATH] ) represented population will require a more careful treatment because each mutated bit can take up to [MATH] possible allelic values. This may somewhat introduce some complications in the mutation operation. Thus, decisions must be made as to wheth... |
In addition to the aforementioned structural parameterization issues, the numerical aspect of the mutation operator involves its application subject to a probabilistic decision. For a simple genetic algorithm having individuals represented as binary strings of length [MATH] , the rate of application of mutation is usua... |
have shown that a range of mutation probabilities of [MATH] is sufficient for wide range of global optimization problems. A universal setting |
for adapting the probability of mutation is to set [MATH] . This has received wider acceptance in the genetic algorithms community. However these settings are not without some criticism from some researchers. Salomon |
argued that small mutation rates are mostly suitable for problems of unimodal or (pseudo-unimodal) type, but multimodal problems will require larger rates. Also, Bäck |
criticised the universal rate of [MATH] as being too independent and unaware of the fitness landscape. There have been several proposals |
for some kind of adaptive mutation with the aim of reducing the burden of parameterization in GAs. Adapting the probability of mutation [MATH] is mainly aimed at eliminating the tuning problem associated with this parameter in order to make genetic algorithms more parameter-independent. Queiroz et al. |
proposed an excellent adaptive mutation strategy that assesses the current population diversity prior to setting the mutation rate. Elsewhere, a new mutation operator for permutation problems was proposed by |
that allows every individual and gene to mutate at different mutation rates. In a similar trend, Bassett et al. applied an adaptive Gaussian mutation operator to a real valued optimization problem. They argue that the adaptive mutation is more disruptive, but it can contribute more to the search process. Bäck |
suggests a new measure called effective fitness which helps to categorize the fitness landscape prior to adapting the mutation rate. |
Furthermore, in the field of hybrid optimization algorithms, proposals by suggest annealing the rate of mutation as the population nears convergence. In a hybrid setup, a simulated annealing algorithm is integrated into a GA to serve as the mutation operator. Elsewhere, Hong et al. |
use various mutation operators simultaneously and argue that dynamically choosing an appropriate mutation operator for a given problem at a given stage of optimization can significantly enhance the robustness of the search process. Thus, although many other strategies are possible, it is evident that many authors use v... |
As can be noticed from the foregoing, in spite of the wide acceptance of the standard reproduction operators, there have been quite a large number of adaptive, dynamic and problem-dependent operators that have been tried in the literature. A somewhat strange proposal by |
in a set up of a hybrid genetic algorithm features a number of specially designed pairs of reproduction operators. The design entails restarting the evolution process with a new pair of operators whenever progress stalls by a given pair of operators. And the process continues until all operator pairs are exhausted. |
To sum up, without loss of generality, we argue that crossover and mutation are the main reproduction means in genetic algorithm. Thus, if appropriately set, they can facilitate the exploration and reachability of the entire search space even in problems having rough and complicated landscapes. Possible consequences of... |
, it is not the average individuals that drive the evolution forward, but the occasionally exceptional individuals created by crossover and mutation that keep the population improving over generations. |
2.7 Replacement Strategies The field of evolutionary computation as highlighted earlier has three major sub-fields: genetic algorithms, evolutionary strategies and evolutionary programming. The standard replacement strategy in genetic algorithms recommended by Holland |
is the generational replacement scheme. This scheme mimics the natural evolution in such a way that subsequent to reproduction phase, an offspring population completely replaces the parent population. In the ES and EP communities, this strategy is popularly referred to as [MATH] strategy. In addition to this, a [MATH] ... |
It is quite easy to realize the risk involved in the generational replacement method. It genuinely fits the natural evolution but it is unsuitable for actual optimization purpose where the goal is to explore and keep the best solution found. Realization of the danger of losing the optimum solution |
over generations following disruptions caused by crossover and mutation has led to the proposals of elitism and its variants details of which will be considered in the next section. |
2.7.1 Elitism in Evolutionary Computations The notion of elitism was originally coined by after critical analysis of the behavioural trend in the evolution dynamics based on several empirical experiments. The concept is basically aimed at preserving the candidate solution having the best fitness value as the population... |
Thereafter, many other variants of the elitist strategy quickly surface. A generalized categorization of replacement schemes by is: |
i. Generational : Also called non-elitist simple GA (sGA), it corresponds to the ES [MATH] strategy. ii. Elitist : This follows the standard elitism and mildly corresponds to the ES [MATH] strategy. |
iii. Overlapping : This generalizes other variants of the elitist strategy and can simply be represented as [MATH] strategy where [MATH] is an aging parameter [MATH] signifying an individual’s life span. i.e. the maximum number of generation an individual can survive. |
2.7.2 Overlapping Populations and Steady State GAs As highlighted earlier, overlapping population refers to the replacement strategy in which the parent and offspring populations compete such that some percentage of the parent population survive across generations. In the original elitist strategy, only a single (best)... |
replace only [MATH] worst individuals in the parent population with the [MATH] best new individuals from the offspring population to generate new population. |
Also a product of overlapping population, the steady state GA (ssGA) is a special case that permits replacing only a single or two individual(s) from the parent population by an individual with best fitness value from the offspring population. It is also popularly known as incremental replacement scheme and is what the... |
is uniquely known for. Studies have shown that ssGAs have in many cases outperformed the standard genetic algorithms for reasons attributed to their ability to effectively exploit promising regions of the search space while retaining the necessary diversity. |
Worth noting is that the idea of always replacing the worst by the best individual can severely heighten selection pressure. Thus, most of ssGA implementations require some radical measures to evade premature convergence due to potential loss of diversity. One of the typical ways used to stabilize the selection pressur... |
use an overlapping population with [MATH] overlap for optimization of a TSP problem but adopt a mutation rate of [MATH] and utilize RWS selection method. Recall that fitness proportionate selection methods like RWS inherently have lower selection pressure that continue to fall over generations. Therefore, utilizing RWS... |
that aids balance their disproportionate selection pressure. In agreement with the above issue, Il-Seok et al. recommend setting the crossover probability as [MATH] for the elitists, overlapping replacement schemes and all their variants like ssGA. They however argued that a much lower value like [MATH] is necessary fo... |
Also, in an attempt to preserve population diversity in ssGAs, a recent proposal advocates replacing only the worst individual that has the least contribution to diversity in the parent population. They defined the contribution to diversity as a measure of the similarity between an individual and its nearest neighbour ... |
in which the replacement and recombination operations are interleaved. Thierens justified the effectiveness of the method and derived the expressions for its selection intensity and selection pressure. |
Other strategies include the use of niching and crowding . A niche is a subset of individuals in a given population that share some similarity. The idea is to allow a new individual to replace only those individual in its own niche, rather than potentially any individual in the population. Several other replacement sch... |
such as correlative family-based selection to mention but a few. To sum up, it is worth mentioning that the main purpose of elitist replacement scheme and its variants is not to guard against sampling error due to selection algorithms |
, but as a safeguard to possible disruptions caused by crossover and mutation operators. Moreover, as compared to original elitist strategy, the overlapping populations prove to perform better |
for both GA and ES evolutionary algorithms. Finally, even though elitism has proven effective and widely accepted, Whitley et al. |
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