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Our results can be understood in either a kin or group selection framework. In particular, the metric of between-group variance is equivalent to that of genetic relatedness, since relatedness is a measure of the correlation in behaviour between individuals within a social group compared to individuals chosen from the g... |
The joint evolution of population structure and social behaviour has also been considered in the context of the origin of multicellularity. Roze and Michod ( 2001 showed that when selfish mutations occur during the growth of clonal colonies (representing proto-multicellular organisms), it can be selectively advantageou... |
We stress again that our logical argument does not only apply to founding group size. Rather, it applies to any heritable trait that modifies population structure so as to cause a difference in selection pressure on social behaviour between bearers and non-bearers. Our model was thus designed to provide a simple illust... |
Other social systems in which the evolution of founding group size might naturally apply include those where groups reproduce by propagule or fissioning. In these systems, which include social spider colonies (Avilés 1993 , and bacterial micro-colonies within biofilms (Hall-Stoodley et al. 2004 , a new group is founded... |
Empirical hypotheses testable in bacterial biofilms Biofilms are formed when bacteria attach to a surface or interface and form large aggregate structures bound together by a co-produced extra-cellular matrix. They, rather than individual motile cells, are the most common mode of bacterial growth (Ghannoum and O’Toole ... |
Bacteria in biofilms exhibit a periodic dispersal cycle (Hall-Stoodley et al. 2004 , which is reminiscent of the “Haystack” aggregation and dispersal population structure that we have modelled. Micro-colonies form, grow, and disperse to form new colonies either via the shearing off of propagules containing varying numb... |
Micro-colony size and dispersal will be affected by a number of factors including shear stress, resource availability and competition. However, we predict that the creation of specific biofilm structure is at least partially driven by the benefits of cooperation. For example, the model suggests that smaller propagule s... |
If iron is then made limiting once again via the addition of iron-chelating proteins, then propagule size should evolve back downwards, since siderophore production would again become beneficial, and more of this benefit would be bestowed upon individuals with traits leading to smaller propagule sizes and increased bet... |
Conclusions We argue that considering the concurrent evolution of population structure and social behaviour in general provides a fundamental new perspective on the evolution of cooperation. In previous theories, where the population structure is static, cooperation is simply the adaptation of organisms’ social behavio... |
Acknowledgements We thank Stuart West, Claire El Mouden, and an anonymous reviewer for constructive suggestions. We also thank Seth Bullock and Rob Mills for helpful discussions, and Joel Parker for comments on an earlier version of the manuscript. The first author acknowledges funding from an EPSRC PhD Plus award at t... |
Appendix 1: Simulation procedure Our numerical analysis of the model consists of a combination of individual-based and numerical simulation. Specifically, we assume that the migrant pool contains a finite number of discrete individuals. We model haploid genotypes with two loci, which reproduce asexually (i.e., with no ... |
1. Create a list of all individuals in the migrant pool. 2. Sort this list in reverse order of group size preference, such that the individuals with the largest size preference are at the front of the list. Within each sub-list of individuals with the same size preference, randomise their position in the list with resp... |
3. Create a new group, and add the individual at the front of the list to this group. Remove the added individual from the list. |
4. Continue adding individuals in order from the list, while the following condition is met: the mean size preference of the group members is less than the current group size. When this condition does not hold, advance to step 5. |
5. If there are still individuals in the list, go back to step 3, else all groups have been formed. Regarding step 2 in this algorithm, randomising the order of each sub-list of individuals with the same size allele means that the behaviours are assigned to groups according to a hypergeometric distribution, and not ass... |
Once the groups have been formed, equations are iterated recursively for [MATH] timesteps (when calculating this recursion, a fractional number of individuals are allowed within groups). Note that by the above group formation procedure, a group can contain different size preference alleles. However, there is no selecti... |
For computational convenience, we introduce a global population carrying capacity. Thus the total population size, [MATH] , remains fixed. This is achieved by multiplying [MATH] and [MATH] , after each iteration of equations , by a factor [MATH] , where [MATH] is the new total population size after equations have been ... |
After [MATH] iterations of equations , all groups disperse into a new global migrant pool (at which point the number of individuals is rounded to the nearest integer), and mutation occurs as follows. |
A fraction [MATH] of individuals in the migrant pool are randomly chosen to be mutated. Of this subset of the population chosen for mutation, a fraction [MATH] have their size preference allele mutated, the remaining [MATH] fraction their behavioural allele mutated; only one locus is mutated per individual, to illustra... |
After mutation has occurred, the next generation of groups is formed from the migrant pool, and the aggregation and dispersal process is repeated for a sufficient number of cycles for an equilibrium to be reached. |
# Source: arxiv 1301.0929 # Title: Hybridization of Evolutionary Algorithms # Sections: all # Downloaded: 2026-03-03T01:56:50.081387+00:00 |
Hybridization of Evolutionary Algorithms Abstract Evolutionary algorithms are good general problem solver but suffer from a lack of domain specific knowledge. However, the problem specific knowledge can be added to evolutionary algorithms by hybridizing. Interestingly, all the elements of the evolutionary algorithms ca... |
To cite paper as follows: Iztok Fister, Marjan Mernik and Janez Brest (2011). Hybridization of Evolutionary Algorithms, Evolutionary Algorithms, Eisuke Kita (Ed.), ISBN: 978-953-307-171-8, InTech, Available from: |
Introduction Evolutionary algorithms are a type of general problem solvers that can be applied to many difficult optimization problems. Because of their generality, these algorithms act similarly like Swiss Army knife (Michalewicz and Fogel, 2004 that is a handy set of tools that can be used to address a variety of tas... |
Similarly, when a problem to be solved from a domain where the problem-specific knowledge is absent evolutionary algorithms can be successfully applied. Evolutionary algorithms are easy to implement and often provide adequate solutions. An origin of these algorithms is found in the Darwian principles of natural selecti... |
As illustrated in Fig. , evolutionary algorithms operate with the population of solutions. At first, the solution needs to be defined within an evolutionary algorithm. Usually, this definition cannot be described in the original problem context directly. In contrast, the solution is defined by data structures that desc... |
An evolutionary search is categorized by two terms: exploration and exploitation. The former term is connected with a discovering of the new solutions, while the later with a search in the vicinity of knowing good solutions (Eiben and Smith, 2003 ; Liu et al., 2009 . Both terms, however, interweave each other in the ev... |
Exploration and exploitation of evolutionary algorithms are controlled by the control parameters, for instance the population size, the probability of mutation [MATH] , the probability of crossover [MATH] , and the tournament size. To avoid a wrong setting of these, the control parameters can be embedded into the genot... |
Igel and Toussaint ( 2003 , however, widened the notion of self-adaptation with a generalized concept of self-adaptation. This concept relies on the neutral theory of molecular evolution (Kimura, 1968 . Regarding this theory, the most mutations on molecular level are selection neutral and therefore, cannot have any imp... |
Although evolutionary algorithms can be applied to many real-world optimization problems their performance is still subject of the No Free Lunch (NFL) theorem (Wolpert and Macready, 1997 . According to this theorem any two algorithms are equivalent, when their performance is compared across all possible problems. Fortu... |
In Fig. 2 some possibilities to hybridize evolutionary algorithms are illustrated. At first, the initial population can be (Moscato, 1999 art:wilfried2010 . Evolutionary operators (mutation, crossover, parent and survivor selection) can incorporate problem-specific knowledge or apply the operators from other algorithms... |
In this chapter the hybrid self-adaptive evolutionary algorithm (HSA-EA) is presented that is hybridized with: construction heuristic, |
local search, neutral survivor selection, and heuristic initialization procedure. This algorithm acts as meta-heuristic, where the down-level evolutionary algorithm is used as generator of new solutions, while for the upper-level construction of the solutions a traditional heuristic is applied. This construction heuris... |
The chapter is further organized as follows. In the Sect. 2 the self-adaptation in evolutionary algorithms is discussed. There, the connection between neutrality and self-adaptation is explained. Sect. 3 describes hybridization elements of the self-adaptive evolutionary algorithm. Sect. 4 introduces the implementations... |
II The Self-adaptive Evolutionary Algorithms Optimization is a dynamical process, therefore, the values of parameters that are set at initialization become worse through the run. The necessity to adapt control parameters during the runs of evolutionary algorithms born an idea of self-adaptation (Holland, 1992 , where s... |
[EQUATION] [EQUATION] where [MATH] and [MATH] denote the learning rates. To keep the mutation strengths [MATH] greater than zero, the following rule is used |
[EQUATION] Frequently, a crossover operator is used in the self-adaptive Evolutionary Strategies. This operator from two parents forms one offsprings. Typically, a discrete and arithmetic crossover is used. The former, from among the values of two parents [MATH] and [MATH] that are located on [MATH] -th position, selec... |
[EQUATION] where parameter [MATH] captures the values from interval [MATH] . In the case of [MATH] , the uniform arithmetic crossover is obtained. |
The potential benefits of neutrality was subject of many researches in the biological science (Conrad, 1990 ; Hynen, 1996 ; Kimura, 1968 . At the same time, the growing interest for the usage of this knowledge in evolutionary computation was raised (Barnett, 1998 ; Ebner et al., 2001 Toussaint and Igel ( 2002 dealt wit... |
As a result, control parameters in evolutionary strategies represent a search strategy. The change of this strategy enables a discovery of new regions of the search space. The genotype, therefore, does not include only the information addressing its phenotype but the information about further discovering of the search ... |
III How to hybridize the Self-adaptive Evolutionary Algorithms Evolutionary algorithms are a generic tool that can be used for solving many hard optimization problems. However, the solving of that problems showed that evolutionary algorithms are too problem-independent. Therefore, there are hybridized with several tech... |
hybridization between two evolutionary algorithms (Grefenstette, 1986 neural network assisted evolutionary algorithm (Wang, 2005 |
fuzzy logic assisted evolutionary algorithm (Herrera and Lozano, 1996 ; Lee and Takagi, 1993 particle swarm optimization assisted evolutionary algorithm (Eberhart and Kennedy, 1995 ; Kennedy and Eberhart, 1995 |
ant colony optimization assisted evolutionary algorithm (Fleurent and Ferland, 1994 ; Tseng and Liang, 2005 bacterial foraging optimization assisted evolutionary algorithm (Kim and Cho, 2005 ; Neppalli and Chen, 1996 |
hybridization between an evolutionary algorithm and other heuristics, like local search (Moscato, 1999 , tabu search (Galinier and Hao, 1999 , simulated annealing (Ganesh and Punniyamoorthy, 2004 , hill climbing (Koza et al., 2003 , dynamic programming (Doerr et al., 2009 , etc. |
In general, successfully implementation of evolutionary algorithms for solving a given problem depends on incorporated problem-specific knowledge. As already mentioned before, all elements of evolutionary algorithms can be hybridized. Mostly, a hybridization addresses the following elements of evolutionary algorithms (... |
initial population, genotype-phenotype mapping, evaluation function, and variation and selection operators. First, problem-specific knowledge incorporated into heuristic procedures can be used for creating an initial population. Second, genotype-phenotype mapping is used by evolutionary algorithms, where the solutions ... |
The mentioned hybridizations can be used to hybridize the self-adaptive evolutionary algorithms as well. In the rest of chapter, we propose three kinds of hybridizations that was employed to the proposed hybrid self-adaptive evolutionary algorithms: |
the construction heuristics that can be used by the genotype-phenotype mapping, the local search heuristics that can be used by the evaluation function, and |
the neutral survivor selection that incorporates the problem-specific knowledge. Because the initialization of initial population is problem dependent we omit it from our discussion. |
III.1 The Construction Heuristics Usually, evolutionary algorithms are used for problem solving, where a lot of experience and knowledge is accumulated in various heuristic algorithms. Typically, these algorithms work well on limited number of problems (Hoos and Stützle, 2005 . On the other hand, evolutionary algorithm... |
Algorithm 1 The construction heuristic. [MATH] : task, [MATH] : solution. 1: while NOT [MATH] do 2: [MATH] 3: end while III.2 The Local Search |
A local search belongs to a class of improvement heuristics (Aarts and Lenstra, 1997 . In our case, main characteristic of these is that the current solution is taken and improved as long as improvements are perceived. |
The local search is an iterative process of discovering points in the vicinity of current solution. If a better solution is found the current solution is replaced by it. A neighborhood of the current solution [MATH] is defined as a set of solutions that can be reached using an unary operator [MATH] |
(Hoos and Stützle, 2005 . In fact, each neighbor [MATH] in neighborhood [MATH] can be reached from current solution [MATH] in [MATH] strokes. Therefore, this neighborhood is called [MATH] neighborhood of current solution [MATH] as well. For example, let the binary represented solution [MATH] and [MATH] operator on it a... |
[EQUATION] where [MATH] denotes a Hamming distance of two binary vectors as follows [EQUATION] where operator [MATH] means [MATH] operation. Essentially, the Hamming distance in Equation is calculated by counting the number of different bits between vectors [MATH] and [MATH] . The [MATH] operator defines the set of fea... |
Algorithm 2 The local search. [MATH] : task, [MATH] : solution. 1: [MATH] 2: repeat 3: [MATH] 4: if [MATH] then 5: [MATH] 6: end |
if 7: until [MATH] As illustrated by Algorithm , the local search can be described as follows (Michalewicz and Fogel, 2004 The initial solution is generated that becomes the current solution (procedure [MATH] ). |
The current solution is transformed with [MATH] strokes and the given solution [MATH] is evaluated (procedure [MATH] ). If the new solution [MATH] is better than the current [MATH] the current solution is replaced. On the other hand, the current solution is kept. |
Lines 2 to 7 are repeated until the set of neighbors is not empty (procedure [MATH] ). In summary, the [MATH] operator represents a basic element of the local search from which depends how exhaustive the neighborhood will be discovered. Therefore, the problem-specific knowledge needs to be incorporated by building of t... |
III.3 The Neutral Survivor Selection A genotype diversity is one of main prerequisites for the efficient self-adaptation. The smaller genotypic diversity causes that the population is crowded in the search space. As a result, the search space is exploited. On the other hand, the larger genotypic diversity causes that t... |
the fittest individual can survive in the struggle for existence, the less fitter individual is eliminated by the natural selection, |
individual with the same fitness undergo an operation of genetic drift, where its survivor is dependent on a chance. Each candidate solution represents a point in the search space. If the fitness value is assigned to each feasible solution then these form a fitness landscape that consists of peeks, valleys and plateaus... |
However, to determine how distant one solution is from the other, some measure is needed. Which measure to use depends on a given problem. In the case of genetic algorithms, where we deal with the binary solutions, the Hamming distance (Equation ) can be used. When the solutions are represented as real-coded vectors an... |
[EQUATION] and measures the root of quadrat differences between elements of vectors [MATH] and [MATH] . The main characteristics of fitness landscapes that have a great impact on the evolutionary search are the following (Merz and Freisleben, 1999 |
the fitness differences between neighboring points in the fitness landscape: to determine a ruggedness of the landscape, i.e., more rugged as the landscape, more difficultly the optimal solution can be found; |
the number of peaks (local optima) in the landscape: the higher the number of peaks, the more difficulty the evolutionary algorithms can direct the search to the optimal solution; |
how the local optima are distributed in the search space: to determine the distribution of the peeks in the fitness landscape; how the topology of the basins of attraction influences on the exit from the local optima: to determine how difficult the evolutionary search that gets stuck into local optima can find the exit... |
existence of the neutral networks: the solutions with the equal value of fitness represent a plateaus in the fitness landscape. When the stochastic fitness function is used for evaluation of individuals the fitness landscape is changed over time. In this way, the dynamic landscape is obtained, where the concept of fitn... |
An operation of the neutral survivor selection is divided into two phases. In the first phase, the evolutionary algorithm from the population of [MATH] offsprings finds a set of neutral solutions [MATH] that represents the best solutions in the population of offsprings. If the neutral solutions are better than or equal... |
[EQUATION] where the ordering relation [MATH] is defined as [EQUATION] Finally, for the next generation the evolutionary algorithm selects the best [MATH] offsprings according to the Equation . These individuals capture the random positions in the next generation. Likewise the neutral theory of molecular evolution, the... |
IV The Hybrid Self-adaptive Evolutionary Algorithms in Practice In this section an implementation of the hybrid self-adaptive evolutionary algorithms (HSA-EA) for solving combinatorial optimization problems is represented. The implementation of this algorithm in practice consists of the following phases: |
finding the best heuristic that solves the problem on a traditional way and adapting it to use by the self-adaptive evolutionary algorithm, |
defining the other elements of the self-adaptive evolutionary algorithm, defining the suitable local search heuristics, and including the neutral survivor selection. |
The main idea behind use of the construction heuristics in the HSA-EA is to exploit the knowledge accumulated in existing heuristics. Moreover, this knowledge is embedded into the evolutionary algorithm that is capable to discover the new solutions. To work simultaneously both algorithms need to operate with the same r... |
The genotype-phenotype mapping consists of two phases as follows: decoding, constructing. Evolutionary algorithms operate in genotypic search space, where each genotype consists of real-coded problem variables and control parameters. For encoded solution only the problem variables are taken. This solution is further de... |
The other elements of self-adaptive evolutionary algorithm consists of: evaluation function, population model, parent selection mechanism, |
variation operators (mutation and crossover), and initialization procedure and termination condition. The evaluation function depends on a given problem. The self-adaptive evolutionary algorithm uses the population model [MATH] , where the [MATH] offsprings is [MATH] parents. However, the parents that are selected with... |
Algorithm 3 Hybrid Self-Adaptive Evolutionary Algorithm. 1: [MATH] 2: [MATH] 3: [MATH] 4: while not termination_condition do 5: [MATH] |
6: [MATH] 7: [MATH] 8: [MATH] 9: [MATH] 10: end while In the rest of the chapter we present the implementation of the HSA-EA for the graph 3-coloring. This algorithm is hybridized with the DSatur (Brelaz, 1979 construction heuristic that is well-known traditional heuristic for the graph 3-coloring. |
IV.1 Graph 3-coloring Graph 3-coloring can be informally defined as follows. Let assume, an undirected graph [MATH] is given, where [MATH] denotes a finite set of vertices and [MATH] a finite set of unordered pairs of vertices named edges book:murty2008 . The vertices of graph [MATH] have to be colored with three color... |
Graph 3-coloring can be formalized as constraint satisfaction problem (CSP) that is denoted as a pair [MATH] , where [MATH] denotes a free search space and [MATH] a Boolean function on [MATH] . The free search space denotes the domain of candidate solutions [MATH] and does not contain any constraints, i.e., each candid... |
However, for the 3-coloring of graph [MATH] the free search space [MATH] consists of all permutations of vertices [MATH] for [MATH] . On the other hand, the function [MATH] (also feasibility condition) is composed of constraints on vertices. That is, for each vertex [MATH] the corresponding constraint [MATH] is defined... |
Handling direct constraints in evolutionary algorithms is not straightforward. To overcome this problem, the constraint satisfaction problems are, typically, transformed into unconstrained (also free optimization problem) by the sense of a penalty function. The more an infeasible solution is far away from feasible regi... |
[EQUATION] where the function [MATH] is defined as [EQUATION] Note that all constraints in solution [MATH] are satisfied, i.e., [MATH] if and only if [MATH] . In this way, the Equation 10 represents the feasibility condition and can be used to estimate the quality of solution [MATH] in the permutation search space. The... |
[EQUATION] where [MATH] represents the weight. Higher than the value of weights harder the appropriate vertex is to color. IV.1.1 The Hybrid Self-adaptive Evolutionary Algorithm for Graph 3-coloring |
The hybrid self-adaptive evolutionary algorithm is hybridized with the DSatur (Brelaz, 1979 construction heuristic and the local search heuristics. In addition, the problem specific knowledge is incorporated by the initialization procedure and the neutral survivor selection. In this section we concentrate, especially, ... |
the initialization procedure, the genotype-phenotype mapping, local search heuristics and the neutral survivor selection. The other elements of this evolutionary algorithm, as well as neutral survivor selection, are common and therefore, discussed earlier in the chapter. |
The Initialization Procedure Initially, original DSatur algorithm orders the vertices [MATH] for [MATH] of a given graph [MATH] descendingly according to the vertex degrees denoted by [MATH] that counts the number of edges that are incident with the vertex [MATH] |
book:murty2008 . To simulate behavior of the original DSatur algorithm (Brelaz, 1979 , the first solution in the population is initialized as follows: |
[EQUATION] Because the genotype representation is mapped into a permutation of weights by decoder the same ordering as by original DSatur is obtained, where the solution can be found in the first step. However, the other [MATH] solutions in the population are initialized randomly. |
The Genotype-phenotype mapping As illustrated in Fig. , the solution is represented in genotype search space as tuple [MATH] , where problem variables [MATH] for [MATH] denote how hard the given vertex is to color and control parameters [MATH] for [MATH] mutation steps of normal mutation. A decoder decodes the problem ... |
the heuristic selects a vertex with the highest saturation, and colors it with the lowest of the three colors; in the case of a tie, the heuristic selects a vertex with the maximal weight; |
in the case of a tie, the heuristic selects a vertex randomly. The main difference between this heuristic and the original DSatur algorithm is in the second step where the heuristic selects the vertices according to the weights instead of degrees. |
Local Search Heuristics The current solution is improved by a sense of local search heuristics. At each evaluation of solution the best neighbor is obtained by acting of the following original local search heuristics: |
inverse, ordering by saturation, ordering by weights, and swap. The evaluation of solution is presented in Algorithm from which it can be seen that the local search procedure ( [MATH] ) is iterated until improvements are perceived. However, this procedure implements all four mentioned local search heuristics. The best ... |
Algorithm 4 Evaluate and improve. [MATH] : solution. 1: [MATH] 2: repeat 3: [MATH] 4: [MATH] 5: [MATH] 6: if [MATH] then 7: [MATH] |
8: [MATH] 9: [MATH] 10: end if 11: until [MATH] In the rest of the subsection, an operation of the local search heuristics is illustrated in Fig. by samples, where a graph with nine vertices is presented. The graph is composed of a permutation of vertices [MATH] , corresponding coloring [MATH] , weights [MATH] and satu... |
The inverse local search heuristic finds all uncolored vertices in a solution and inverts their order. As can be shown in Fig. , the uncolored vertices 4, 6 and 8 are shadowed. The best neighbor is obtained by inverting of their order as is presented on right-hand side of this figure. The number of vertex exchanged is ... |
The ordering by saturation local search heuristic acts as follows. The first uncolored vertex is taken at the first. To this vertex a set of adjacent vertices are selected. Then, these vertices are ordered descending with regard to the values of saturation degree. Finally, the adjacent vertex with the highest value of ... |
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