The Roles of Crossover and Mutation in Real-Coded Genetic Algorithms

Recently many studies on evolutionary algorithms using real encoding have been done. They include ant colony optimization (Socha & Dorigo, 2008), artificial bee colony algorithm (Akay & Karaboga, 2010; Kang et al., 2011), evolution strategies (ES) (Beyer, 2001), differential evolution (Das & Suganthan, 2011; Dasgupta et al., 2009; Kukkonen & Lampinen, 2004; 2005; Mezura-Montes et al., 2010; Noman & Iba, 2005; Ronkkonen et al., 2005; Storn & Price, 1997; Zhang et al., 2008), particle swarm optimization (Chen et al., 2007; Huang et al., 2010; Juang et al., 2011; Krohling & Coelho, 2006; l. Sun et al., 2011), and so on. In particular, in the field of ES, we can find many studies based on self-adaptive techniques (Beyer & Deb, 2001; Hansen & Ostermeier, 2001; Igel et al., 2007; 2006; Jagerskupper, 2007; Kita, 2001; Kramer, 2008a;b; Kramer et al., 2007; Meyer-Nieberg & Beyer, 2007; Wei et al., 2011).

The remainder of this chapter is organized as follows.Traditional and recent genetic operators in real encoding are introduced in Section 2. Previous genetic operators are presented in Section 2.1 and ones we used in real encoding in this study are described in Section 2.2.In Section 3, we describe the concept of bias of genetic operators and analyze that in the case of crossover and mutation for GAs.In Section 4, experimental results for various combinations of crossover and mutation are provided and analyzed.Finally, we make conclusions in Section 5.

Previous operators
The roles of crossover and mutation may change according to the selection of the operators.We reviewed the most frequently used crossover and mutation operators for real-code representation.We are to analyze how the roles of crossover and mutation can change by studying various combinations of crossover and mutation operators.
In literature many crossover operators for real-code representation are found.Traditional crossover operators for the real-code representation are described in (Bäck et al., 2000).The two main families of traditional crossover operators (Mühlenbein & Schlierkamp-Voosen, 1993) are discrete crossovers 1 (Reed et al., 1967) and blend crossovers (Michalewicz, 1996).
Blend crossover operators can be distinguished into line crossovers and box crossovers.Important variations of the last two crossover operators are the extended-line crossover and the extended-box crossover (Mühlenbein, 1994).
The discrete recombination family is the straightforward extension to real vectors of the family of mask-based crossover operators for binary strings including n-point and uniform crossover.
box-crossover(x, y) The mask is still a binary vector dictating for each position of the offspring vector from which parent the (real) value for that position is taken.
The blend recombination family does not exchange values between parents like discrete recombinations but it averages or blends them.Line recombination returns offspring on the (Euclidean) line segment connecting the two parents.Box recombination returns offspring in the box (hyper-rectangle) whose diagonally opposite corners are the parents.Extended-line recombination picks offspring on an extended segment passing through the parent vectors but extending beyond them and not only in the section between them.Analogously extended-box recombination picks offspring on an extended box whose main diagonal passes through the parents but extends beyond them.

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The Roles of Crossover and Mutation in Real-Coded Genetic Algorithms www.intechopen.comline-crossover(x, y) { λ ← a random real number in [0, 1]; The most common form of mutation for real-code vectors generates an offspring vector by adding a vector M of random variables with expectation zero to the parent vector.There are two types of mutations bounded and unbounded depending on the fact that the range of the random variable is bounded or unbounded.The most frequently used bounded mutations are the creep mutation and the single-variable mutation and for the unbounded case is the Gaussian mutation.For the creep (or hyper-box) mutation M ∼ U([−a, a] n ) is a vector of uniform random variables, where a is a parameter defining the limits of the offspring area.This operator yields offspring within a hyper-box centered in the parent vector.For the single-variable mutation M is a vector in which all entries are set to zero except for a random entry which is a uniform random variable ∼ U ([−a, a]).Bounded mutation operators may get stuck in local optima.In contrast, unbounded mutation operators guarantee asymptotic global convergence.The primary unbounded mutation is the Gaussian mutation for which M is a multivariate Gaussian distribution.

Adopted operators for this study
As crossover operators for our analysis, we adopted four representative crossovers: box, extended-box, line, and extended-line crossovers.Their pseudo-codes are shown in Figures 2,  3, 4, and 5, respectively and the possible range for each crossover is represented in Figure 1.
And, as mutation operators for our analysis, we adopted two kinds of mutation: Gaussian mutation and fine mutation.Their pseudo-codes are shown in Figures 6 and 7, respectively.The Gaussian mutation is a simple static Gaussian mutation, the same as in Tsutsui & Goldberg (2001).The i-th parameter z i of an individual is mutated by z i = z i + N(0, σ i ) with a mutation rate p, where N(0, σ i ) is an independent random Gaussian number with the mean of zero and the standard deviation of σ i .In our study, σ i is fixed to (u i − l i )/10 -the tenth of width of given area.The fine mutation is a simple dynamic Gaussian mutation inspired from Ballester & Carter (2004b).In different with Gaussian mutation, it depends on the distance between parents and, as population converges, the strength of the mutation approaches zero.

Bias of genetic operators
Pre-existing crossovers for the real-coded representation have an inherent bias toward the center of the space.Some boundary extension techniques to reduce crossover bias have been extensively studied (Someya & Yamamura, 2005;Tsutsui, 1998;Tsutsui & Goldberg, 2001).The concept of crossover bias first appeared in (Eshelman et al., 1997) and it has been extensively used in (Someya & Yamamura, 2005;Tsutsui & Goldberg, 2001), in which they tried to remove the bias of real-coded crossover heuristically (and theoretically incompletely).
Notice that the notion of bias of a crossover operator has different definitions depending upon the underlying representation considered.The bias toward the center of the space considered in real-coded crossovers conceptually differs from the crossover biases on binary strings, which focus on how many bits are passed to the offspring and from which positions, which, in turn conceptually differs from the bias considered in Genetic Programming focusing on bloat.
The notion of bias so defined can be understood as being the inherent preference of a search operator for specific areas of the search space.This is an important search property of a search operator: an evolutionary algorithm using that operator, without selection, is attracted to the areas the search operator prefers.Arguably, also when selection is present, the operator bias acts as a background force that makes the search keener to go toward the areas preferred by the search operator.This is not necessarily bad if the bias is toward the optimum or an area with high-quality solutions.However, it may negatively affect performance if the bias is toward an area of poor-quality solutions.If we do not know the spatial distribution of the fitness of the problem, we may prefer not to have any a priori bias of the search operator, and instead use only the bias of selection, which is informed by the fitness of sampled solutions that constitute empirical knowledge about promising areas obtained in the search, and which is better understood.
In this chapter, we investigate the bias caused by crossover itself and crossover combined with mutation in real-coded GAs.Intuitively, box and line crossover are biased toward the center on the Euclidean space.This intuition is easy to verify experimentally by picking a large number of pairs (ideally infinitely many) of random parents and generating offspring uniformly at random in the boxes (or lines) identified by the pairs of parents.
) is the optimal solution, which is randomly located in the domain.
In the Hamming space, the distribution of the offspring of uniform crossover tends in the limit to be uniform on all space, whereas in the Euclidean space the distribution of the offspring tends to be unevenly distributed on the search space and concentrates toward the center of the space.One way to compensate, but not eliminate, such bias is using extended-line and extended-box crossovers.Figure 8 visualizes the crossover bias in the one-dimensional real space by plotting frequency rates of 10 7 offspring randomly generated by each type crossover.
As we can see, box and line crossover are biased toward the center of the domain.We could also observe that extended-box and extended-line crossover largely reduce the bias but they are still biased toward the center.2 For analyzing the effect of mutation in relation with the bias, we also performed the same test using crossover combined with Gaussian mutation.We picked 10 7 pairs of random parents, generated offspring randomly using each type crossover, and then applied Gaussian mutation.The tests are performed for various mutation rates from 0.0 to 1.0.The results for box, extended-box, line, and extended-line crossover are shown in Figures 9, 10, 11, and 12, respectively.Interestingly, for all cases, we could observe that the higher mutation rate reduces the bias more largely.However, even high mutation rates cannot eliminate the bias completely.

Combination of crossover and mutation
In this section, we try to figure out the properties of crossover and mutation through experiments using their various combinations.For our experiments, four test functions are chosen from Suganthan et al. (2005).They are described in Table 1.
We mainly followed the genetic framework by Tsutsui & Goldberg (2001).Its basic evolutionary model is quite similar to that of CHC (Eshelman, 1991) and (µ + λ)-ES (Beyer, 2001).Here, r is a divergence rate and we set it to 0.25 as in Eshelman (1991).The used GA terminates when it finds the global optimum.
For crossover, we used four crossover operators: box crossover, extended-box crossover (extension rate α: 0.5), line crossover, and extended-line crossover (extension rate α: 0.5).After crossover, we either mutate the offspring or do not.We used two different mutation operators; Gaussian mutation and fine mutation.Different mutation rates were applied to each crossover type and the rates decrease as the number of generations increases.
Table 2 shows the results from 30 runs.Each value in 'Ave' means the average function value from 30 runs.The smaller, the better.The limit of function evaluations is 50,000, i.e., the genetic algorithm terminates after 50,000 evaluations and outputs the best solution among evaluated ones so far over generations.In the table, k = 1 + ⌊numberOfGenerations/100⌋ and the rate of fine mutation is 0.5/k.
From these experiments we can obtain the following properties.

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Conclusions
In this chapter, we tried to analyze distinct roles of crossover and mutation when using real encoding in genetic algorithms.We investigated the bias of crossover and mutation.From this investigation, we could know that extended crossover and mutation can reduce the inherent bias of traditional crossover in real-coded genetic algorithms.
We also studied the functions of crossover and mutation operators through experiments for various combinations of both operators.From these experiments, we could know that extended-box crossover is good in the case of using only crossover without mutation.However, it is possible to surpass the performance of extended-box crossover using well-designed combination of crossover and mutation.In the case of other crossover operators, not only the function of perturbation but also that of fine tuning by mutation is important, but extended-box crossover contains the fine tuning function in itself.
There are many other test functions defined on real domains.We conducted experiments with limited test functions.We may obtain more reliable conclusions through experiments with more other functions.So, more extended experiments on more various test functions are needed for future work.We may also find other useful properties from those empirical study.

Fig. 1 .
Fig. 1.The range of possible offspring in two-dimensional bounded real space

Fig. 5 .
Fig. 5. Pseudo-code of extended-line crossover Fig. 8. Crossover bias in one-dimensional bounded real space Fig. 9. Bias of box crossover with mutation