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cautioned that the strategy might be a poor choice when dealing with problems having dynamically changing landscape. 2.8 Parameterization in Evolutionary |
Computations It can be noticed from the foregoing that unlike the gradient-based optimization techniques, evolutionary computation algorithms are involved with a huge number of parameters upon which their overall success relies. Worth noting is that because the performance of an evolutionary algorithm heavily relies on... |
who wishes to derive the optimal parameters of a genetic algorithm for some specified problems put forward a meta-level GA. The design consists of an internal user parameterized genetic algorithm that is used to tune the parameters of a main genetic algorithm. |
The parameters of a typical genetic algorithm can be classified into two main categories i. Structural Parameters : These include choice of data type (i.e. representation scheme) and the types of genetic operators used. |
ii. Numerical Parameters : These include but not limited to the population size, probabilities of mutation and crossover, maximum number of generations and so on. |
2.8.1 The Standard Parameter Settings Early works in genetic algorithms by Holland and later by De Jong have led to the development of the most widely used standard parameter sets. Although mainly obtained from empirical experiments, many other theoretical studies |
have reinforced the validity of these standard parameters. The space of a simple genetic algorithm comprises of at least six basic parameters depicted in table 2.2 . The table shows the ranges and commonly used types of the standard parameters for small to medium sized global optimization problems. |
2.8.2 Adaptive and Dynamic Parameters in EC In order to alleviate the need for tuning and fine tuning of GA parameters as the nature of problems or optimization goals vary, several proposals of adaptive, self-adaptive or dynamic parameters are made. A proposal by |
suggest adapting the probabilities of crossover and mutation based on the population entropy at any stage of the evolution. They described the population entropy as the distribution of individuals in the population. The greater is the difference among the individuals in a population, the higher the entropy and vice ver... |
use varying mutation rates and argue that mutation rates that decrease exponentially as the population average fitness grows over generations have superior performance over their non-adaptive counterparts. |
In a different respect, Niehaus et al. adopt three new methods of dynamically adjusting the probabilities of reproduction operators in GA. The aim was to establish an adaptive system that can not only perform better than randomly chosen settings, but compete with the empirically proven standard settings. The adaptation... |
exhibited similar argument. Elsewhere, Spears studies the role of crossover in GA, EP and other fields of evolutionary computation and noted that despite the wealth of theoretical analysis, it is sometimes difficult to decide a priori which type of crossover to use and at what rate. Spears then suggests an adaptive mec... |
2.8.3 A Parameter-less GA From a somewhat extreme angle of parameter adaptation in evolutionary algorithms, Harik et al. noted that majority of the target end users of EAs barely have sufficient understanding of the EA dynamics. Thus, users may lack the knowhow on manual tuning of its parameters for optimal performance... |
Although it is quite a remarkable idea, the proposed technique can severely increase the computational cost as a result of naively running many instance of the GA at all times. Moreover, the idea used to adapt the crossover probability is based on the Holland’s building-block theory the soundness of which cannot be jus... |
. Finally, the adaptation is limited to crossover operator without addressing the mutation aspect. To sum of, some researchers have a contrary view to the original conception of GAs as been problem-independent. For instance, Boyabaltli et al. |
argue that if viewed from the parameterization perspective, GAs can be very problem specific algorithms. We would like to emphasize here that this is not the view of many EA practitioners. Results from several empirical and analytical experiments |
have shown that the parameters of a GA need not to be overly tuned to suit a particular problem, the algorithm can perform fairly optimal for much wider range of problems so long as its parameters are set to within the range of the standard parameter settings. |
2.9 The Proposed Adaptive Elitism Considering the fact that GAs using the elitist strategy always perform better than most of the generational GAs. However, as described previously, with basic elitism, the single elite individual may lack influence on the fitness growth in the population (i.e., scalability problem). Al... |
i. If the population average fitness value [MATH] in the current generation [MATH] is no better than that at the previous generation [MATH] , and; |
ii. If the population fitness variance [MATH] in the current generation [MATH] is no less than that of the previous generation [MATH] |
The procedure in algorithm elaborates the proposed scheme. Algorithm 2 The Proposed Adaptive Elitism 1: begin 2: [MATH] 3: [MATH] size of [MATH] |
4: [MATH] 5: initialize [MATH] 6: [MATH] evaluate and rank [MATH] 7: [MATH] Top [MATH] of [MATH] 8: [MATH] average of [MATH] 9: [MATH] variance of [MATH] |
10: while not converged do 11: [MATH] evolve [MATH] 12: [MATH] evaluate and rank [MATH] 13: [MATH] average of [MATH] 14: [MATH] variance of [MATH] |
15: if [MATH] then 16: [MATH] 17: [MATH] Top [MATH] of [MATH] 18: [MATH] 19: else 20: [MATH] 21: end if 22: [MATH] 23: end while |
24: end Notice from algorithm that the size of the elite population is initially set to some percentage (i.e. [MATH] ) of the original population size (line [MATH] ). This however gets continuously halved at the end of every generation so long as the population’s average fitness grows and its fitness variance does not ... |
The proposed idea of the adaptive elitism has two main goals. First, it safeguards individuals with high fitness values from been lost due to potential catastrophic effect of genetic operators at early stages of the evolution. Second, it ensures maintenance of useful diversity by adaptively shrinking the size of the el... |
2.10 Contributions In addition to the wide survey and analysis on small and major aspects of evolutionary computation algorithms, the chapter gives an in-depth treatment on parameterization issues upon which development of successful global optimization methods relied. Thereafter, a new replacement technique (adaptive ... |
2.11 Remarks This chapter commenced with an overview of the historical foundation and investigates the development and simulation of evolutionary computations. Key initialization aspects such as problem representation , creation and sizing of the initial population are reviewed. Major parameterization issues such as th... |
Chapter 3 Convergence Measurement in EC using Price’s Equation This chapter will focus on investigating the convergence of evolutionary algorithms. After a brief introduction, some commonly used stopping and convergence measures will be reviewed. Price’s theorem will be examined and the use of the extended Price’s equa... |
3.1 The Need for Convergence Measure in EC Algorithms Subsequent to the earlier investigation of the dynamics of evolutionary computations, it is imperative to carefully assess the long-term behaviour of these algorithms. Evolutionary computations are naturally inspired stochastic algorithms that by design are capable ... |
3.2 Conventional Measures of EC Convergence Beside terminating evolution based on some user prescribed limits on evaluations, generations or execution time for a run, more sophisticated convergence measures that are mainly based on population diversity are used to automatically terminate the evolution. In this respect,... |
such as the Hamming distance. This agrees well with the notion that GA converges when the candidate solutions in the population become identical. |
While the convergence characteristics of all evolutionary algorithms rely on several factors, the most important factors are the types of selection mechanism, the reproduction operators and the size of the population of candidate solutions. The selection operation being an exploitation process, it mainly favours the fi... |
use the normal fitness distribution method to model the convergence characteristics of various selection methods. Simple yet elegant ordinary differential equations (ode) models that estimate the true convergence behaviours of the fitness proportional selection (FPS), tournament selection, truncation selection and elit... |
Regarding the manner in which the reproduction operators influence GA convergence, it is important to realize that besides the types of the operators, the frequency of their application also play a major role. A new diversity measure that estimates the average Hamming distance in the population was proposed by |
. They noted that while mutation and its probability of application can severely influence the convergence rate, vast majority of the traditional crossover operators such as [MATH] point, uniform and punctuated crossover have little effect on GA convergence. They analytically prove that the new diversity measure (i.e.,... |
Similarly, Sharapov et al. derive the mean convergence rate of genetic algorithms due to various reproduction operators. Although no numerical experimentation was conducted, probabilistic models were used to theoretically analyse the convergence characteristics due to crossover, mutation and inversion operators. |
Elsewhere, Greenhalgh et al. investigate convergence on the ground of population’s fitness. They consider the maximum iteration limit, on-line and off-line performances as the three basic stopping criteria and convergence measures for genetic algorithms. The on-line performance reflects the average of all fitness funct... |
Hybridization approaches also play vital role to improving GA convergence rate. As reported by Miura et al. , for most nonlinear optimization problems, the time required by a genetic algorithm to converge to an optimal solution can be reduced by incorporating some information about the gradients of the problem’s variab... |
where genetic algorithms are combined (in various ways) with various local search methods to speed up convergence. Nearly [MATH] improvement in the convergence rate was reported by Kaur et al. |
as a result of hybridizing a genetic algorithm with a nearest neighbour search in solving a TSP combinatorial optimization problem. |
Worth mentioning is that all the aforementioned approaches for estimating GA convergence share their merits and demerits. For instance, in a typical binary coded GA, monitoring the similarity among chromosomes using a distance measure such as Hamming distance can allow effective analysis of the diversity profile in a g... |
We further observe that estimating GA convergence by marrying the two diversity measures that monitor similarity in both the encoding and solution spaces might not totally eliminate dissension between the methods which could trigger false alarms for convergence. Moreover, the combination can severely increase the compu... |
We therefore argue that since progress in evolutionary search is internally governed by the actions of genetic operators (selection and reproduction operators like crossover and mutation), building a convergence measure that assesses the activities of these operators can be a promising alternative to tackling this dile... |
3.3 Monitoring the Effect of EC Operators with Price’s Equation In order to examine the individual effect of evolution operators while in interaction, George Price |
formulated a theorem that permits decomposition of the evolutionary process to separate the genetic effect (or contribution) of the selection operator from that of other reproduction operators (i.e. crossover and mutation). Although Price’s work was mainly in the field of evolutionary genetics, the proposed equation ( ... |
Price’s equation states that: [EQUATION] where [MATH] is the change in the measured characteristics (such as fitness), [MATH] is the number of individuals in the parent population (i.e., population size), [MATH] is the number of offspring of parent [MATH] , and [MATH] is the average number of the offspring produced. Al... |
The two terms in the Price’s equation ( 3.1 ) represent the contribution of different operators as the mean of the characteristic being measured. Since we deploy the Price’s equation to analyse convergence in EC, the measured characteristic [MATH] in this case will be fitness. The first term represents the contribution... |
that sees the covariance between the phenotypic values of individuals and their fitness as the cause of differential productivity that leads to the change in phenotype. |
The following lemmas 1 and 2 will elaborate how Price’s equation decomposes the fitness progress into separate contributions from the selection operator and other reproduction operators. |
Lemma 1: Supposing the fitness [MATH] of each member of the parent population in equation ( 3.1 ) is represented by a vector such that: |
[EQUATION] Suppose also the number of offspring [MATH] produced by each one of the [MATH] parent is represented by a vector such that: |
[EQUATION] Then, the following expansion of the two terms in equation ( 3.1 ) will demonstrate that it is only the first term that represents the contribution of the selection operator. |
First term [EQUATION] where [MATH] is the mean of the number of offspring of parent [MATH] and [MATH] is the mean fitness of the parents. |
Second term [EQUATION] Recall that following any typical evolutionary selection process, the resulting offspring have the same fitness as their parents (i.e. selection process adds no new solutions to the population), therefore, |
[EQUATION] Consequently, the value of equation ( 3.5 ) (i.e., the second term of Price’s equation) sums to zero. Therefore, the contribution of selection operator is only in the first term of the Price’s equation. |
[MATH] Lemma 2: Unlike in the above case for selection process, the following examination of the Price’s equation ( 3.1 ) will show that the second term of the Price’s equation uniquely represents the contribution of the reproduction operators. |
Assuming any traditional [MATH] point crossover operator is employed; crossing any two parents yields two offspring. Therefore, if all parents undergo the crossover operation (i.e. [MATH] ) and the population size [MATH] is even, then, the number of offspring produced will always be equal to the number of their parents... |
A worth noting exception here is that if the population size [MATH] is odd or in the case where not all chromosomes undergo the crossover operation (i.e., when [MATH] ), then [MATH] and thus [MATH] . This is contrary to the traditional notion of decomposing Price’s equation in the literature |
where it is often assumed that the contribution of the reproduction operators in the first term of Price’s equation is always zero. This investigation reveals the contrary if taking into account the special cases mentioned above. |
Now, expanding the second term as in equation ( 3.5 ) reveals that the summation is non-zero since the fitness [MATH] of any parent [MATH] is different from that of all its offspring , and hence, [MATH] (at least in most cases). Hence the contribution of crossover and mutation operators is mainly from the second term o... |
[MATH] 3.3.1 Extension of Price’s Equation Bassett et al. extend the second term ( 3.5 ) of Price’s equation to allow monitoring the contribution of individual reproduction operators so as to ascertain which among them is more effective at various stages of the evolution process. It was argued that utilizing fitness av... |
The extended Price’s equation can be defined as: [EQUATION] where [MATH] is the number of genetic operators, [MATH] is the average value of the fitness of all the offspring of parent [MATH] after the application of operator [MATH] , and [MATH] is the difference between the average value [MATH] (i.e., fitness) of the of... |
For the proposed hybrid evolution algorithm, we deploy only the mutation and crossover operators, thus, [MATH] . Therefore, the following proposed extension to the Price’s equation ( 3.8 ) contains only three terms; a term for the selection, crossover and mutation respectively. |
[EQUATION] Hence, each of the terms in this equation estimates the changes in the mean of the population’s fitness due to one of the three genetic operators. |
3.4 Analysing Evolution Progress with Extended Price’s Equation In order to analyse the effect of the proposed extension of Price’s equation ( 3.8 ), we conduct a number of experiments with a genetic algorithm having some standard parameters previously highlighted in table 2.2 and some newly proposed parameters previou... |
The objectives for this experiment are: To observe and analyse the growth in the fitness of the population of candidate solutions as the evolution progresses; and |
• To investigate the change in fitness that can be attributed to the selection operator, and changes due to the crossover and the mutation operator respectively. |
Ultimately, the outcome of this experiment is aimed at providing better insight through observation of the manner in which the selection, crossover and mutation operators collectively move the evolutionary search forward. It will also give insight on their individual effect on ensuring sustainable balance for the explo... |
Experiment 1: Visualizing Evolution with Roulette Wheel Selection (RWS) In this experiment, RWS is used to investigate the individual effects of the genetic operators over the entire period of evolution. Table 3.2 shows the various parameter sets for this experiment. |
Experiment 2: Visualizing Evolution with Binary Tournament Selection (BTS) Here a binary tournament selection without replacement is used to investigate the effects of the genetic operators during the evolution process. Similar parameter sets are used as in table 3.2 |
Notice from table 3.2 that there are six parameter sets for testing each of the two selection methods. The first half will be tested with a population size of [MATH] while for the other half, a population size of [MATH] will be used. Note that with the parameter sets in tables 3.1 and 3.2 , a population size of [MATH] ... |
3.4.1 Results: Comparing RWS and BTS methods The results obtained for the above experiments 1 and 2 are summarized in table 3.3 . The table compares RWS and BTS methods from two perspectives. First, on their performance under two different population sizes [MATH] |
[MATH] and [MATH] . Second, on their performance against three different settings for the probabilities of crossover [MATH] and mutation [MATH] |
Notice from table 3.3 that for the two different population sizes, the results obtained with the BTS clearly outperform those when RWS selection method is used. Also, when BTS method is utilized, the results obtained with a population size of [MATH] are fairly close to when the population size is doubled to [MATH] . Of... |
Recall that the overall goal is to explore which among the setting will lead to highest fitness and at the same time ensure convergence detection with ease. Therefore, with the results presented in table 3.3 and the analysis of the following figures 3.1 and 3.2 for the Price’s plots, we infer that setup (i) (i.e., [MAT... |
First: With setup (i), convergence to a good approximation of the optimum solution is possible even with low population sizes. Thus, it is computationally cheap since the required function evaluations can be minimized. |
Second: As will be seen from figures 3.1 and 3.2 , setup (i) is more suitable for convergence detection via monitoring the effect of genetic operators. |
Therefore, setup (i) is the best candidate for achieving the proposed objective. The plots shown in figure 3.1 are for setup (i) (see table 3.3 ) where [MATH] and [MATH] , the remaining setups are relatively similar and therefore figures omitted. The fitness comparison plots in figures 3.1 (a and c) compare the best wo... |
Discussion of Results for Experiments 1: RWS selection A quick glimpse at the curves in the Price’s plot in figure 3.1 (b) for the contribution of operators barely allows any meaningful conclusion. One might naively infer that the idea of decomposing the fitness progress to investigate the individual effect of the gene... |
i. It can be noticed from figure 3.1 (b) that the curves for all the three operators fluctuate above and below zero on the fitness axis in a somewhat random manner. Thus, the plot provides no vital information as to which among the three operators ensures continues growth in fitness (i.e. exploitation) and which is res... |
ii. A careful look at the initial shape of the curve for the selection operator in figure 3.1 (b) reveals that there is a huge selection pressure at the early generations. This corresponds to the steep gradient in the best and average fitness curves (figure 3.1 (a)) during the early generations. However, the selection ... |
iii. Another crucial observation is that because all the curves for the three operators in figure 3.1 (b) continuously effect changes in the population’s fitness, the population tends to slowly evolve thereby growing the average fitness (see figure 3.1 (a)) continuously (although at a slower rate) till the evolution is... |
Nevertheless, the above behaviour of the selection operator agrees with the fact that RWS is a fitness proportionate selection method and as previously discussed in section 2.5.1 , it suffers inherent stochastic sampling errors . These sampling errors increase as the average fitness of the population grows. Thus, it is... |
Moreover, as can be seen from the Price’s plot in figure 3.1 (b), the contribution of the selection operator to the fitness progress becomes quite similar to (or no better than) that of the crossover and mutation operators. In other words, all the three operators play the role of exploration and exploitation of the sea... |
Discussion of Results for Experiments 2: BTS The plots shown in figure 3.1 (c and d) convey similar information to those in experiment 1 above but now BTS is employed. Moreover, contrary to the results obtained in experiment 1, an examination of the actions of the individual operators, and comparing their interactions ... |
i. A simple observation of the fitness curves in figure 3.1 (c) reveals that both the average and the best fit individuals in the population rapidly grow with a steep gradient to their peak values within the very early generations of the evolution. It is interesting to note how at the same time useful diversity is main... |
ii. Notice from figure 3.1 (d) that the curve for the selection operator is always above zero on the fitness axis. Thus, the selection operator can now be seen as a biased process that primarily guides the search towards the fitter individuals seen so far. It effectively drives the evolution towards converging to the p... |
iii. Notice also from the same figure that the curve for the crossover operator swing above and below zero on the fitness axis. This is indicative of the fact that the crossover operator has both exploration and exploitation effects. In other words, while crossover improves the population’s average fitness via exploita... |
iv. Contrary to the previous operators, notice that the curve for the mutation operator always lie beneath zero on the fitness axis (see figure 3.1 (d)). This is because mutation operator improves population’s diversity by exploring other parts of the search space. But in so doing, it tends to drastically lower the fit... |
Following the above observations, it is thought that among all the three genetic operators, it is the behaviour of the crossover operator with regard to the fitness growth in the population that is most crucial to effective detection of convergence of the evolution process. This is because both the mutation and selecti... |
Finally, for the same set of parameters (test setup (i)), it is interesting to notice how the best and average fitness curves in the fitness comparison plot in figure 3.1 (c) (where BTS is used) clearly outperform those in figure 3.1 (a) (where RWS is used) by converging rapidly towards the global optimum solution of [... |
In the following section an investigation on how the crossover can be used to detect convergence will be presented. 3.5 Using Extended Price’s Equation to Measure |
Convergence Having decided, during the discussions of the previous section 3.4.1 , that the crossover operator can be utilized as a means for assessing convergence in the evolution, this section will focus on formulating and setting up the parameter for the convergence measurement. A graphical means for evaluating this... |
Recall from equation ( 3.8 ) that [MATH] is only the collective change in the average fitness due to the three genetic operators. As argued by |
observing changes in averages alone does not convey sufficient information on the true effect of an operator. This is evident from the plot of the crossover operator in figure 3.1 (d) where it is hard to appreciate its effect on the fitness even during the early generations of the evolution. Therefore monitoring the sp... |
From equation ( 3.8 ), let the change in fitness due to a genetic operator [MATH] (crossover in this case) be: [EQUATION] Then, since [MATH] is the random variable of interest, the expectation and variance of the crossover term can be obtained by respectively taking the first and second moment of ( 3.9 ) with respect t... |
[EQUATION] where [MATH] is the sum over all the children of parent [MATH] . Typically, there will be two children for each parent when a crossover probability of [MATH] is used. All other parameters are as previously defined in section 3.3 . The variance, [MATH] is: |
[EQUATION] Hence, the standard deviation is: [EQUATION] Now, overlaying the curves for the one standard deviation interval ( [MATH] ) on the plot of the [MATH] changes in the average fitness due to the crossover operator yield the envelope of the shaded areas shown in plots (a and c) of figure 3.2 . The plots give a be... |
Plots (a and b) in figure 3.2 show the results for experiment 1 where RWS is used. The shaded area above the curve for the change in the average fitness due to crossover in figure 3.2 (a) reveal that crossover operator does indeed contribute to the fitness growth, but at the same time, the shaded area underneath it whi... |
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