An Algorithm for Global Optimization Inspired by Collective Animal Behavior

A metaheuristic algorithm for global optimization called the collective animal behavior CAB is introduced. Animal groups, such as schools of fish, flocks of birds, swarms of locusts, and herds of wildebeest, exhibit a variety of behaviors including swarming about a food source, milling around a central locations, or migrating over large distances in aligned groups. These collective behaviors are often advantageous to groups, allowing them to increase their harvesting efficiency, to follow better migration routes, to improve their aerodynamic, and to avoid predation. In the proposed algorithm, the searcher agents emulate a group of animals which interact with each other based on the biological laws of collective motion. The proposed method has been compared to other well-known optimization algorithms. The results show good performance of the proposedmethod when searching for a global optimum of several benchmark functions.


Introduction
Global optimization GO is a field with applications in many areas of science, engineering, economics, and others, where mathematical modelling is used 1 .In general, the goal is to find a global optimum of an objective function defined in a given search space.Global optimization algorithms are usually broadly divided into deterministic and metaheuristic 2 .Since deterministic methods only provide a theoretical guarantee of locating a local minimum of the objective function, they often face great difficulties in solving global optimization problems 3 .On the other hand, metaheuristic methods are usually faster in locating a global optimum than deterministic ones 4 .Moreover, metaheuristic methods adapt better to black-box formulations and extremely ill-behaved functions whereas deterministic methods usually rest on at least some theoretical assumptions about the problem formulation and its analytical properties such as Lipschitz continuity 5 .
Several metaheuristic algorithms have been developed by a combination of rules and randomness mimicking several phenomena.Such phenomena include evolutionary processes, for example, the evolutionary algorithm proposed by Fogel et al. 6 , De Jong 7 , and Koza 8 , the genetic algorithm GA proposed by Holland 9 and Goldberg 10 and the artificial immune systems proposed by de Castro and Von Zuben 11 .On the other hand, physical processes consider the simulated annealing proposed by Kirkpatrick et al. 12 , the electromagnetism-like algorithm proposed by İlker et al. 13 , the gravitational search algorithm proposed by Rashedi et al. 14 , and the musical process of searching for a perfect state of harmony, which has been proposed by Geem et al. 15 , Lee and Geem 16 , and Geem 17 .Many studies have been inspired by animal behavior phenomena for developing optimization techniques.For instance, the particle swarm optimization PSO algorithm which models the social behavior of bird flocking or fish schooling 18 .PSO consists of a swarm of particles which move towards best positions, seen so far, within a searchable space of possible solutions.Another behavior-inspired approach is the ant colony optimization ACO algorithm proposed by Dorigo et al. 19 , which simulates the behavior of real ant colonies.Main features of the ACO algorithm are the distributed computation, the positive feedback, and the constructive greedy search.Recently, a new metaheuristic approach which is based on the animal behavior while hunting has been proposed in 20 .Such algorithm considers hunters as search positions and preys as potential solutions.
Just recently, the concept of individual-organization 21, 22 has been widely referenced to understand collective behavior of animals.The central principle of individualorganization is that simple repeating interactions between individuals can produce complex behavioral patterns at group level 21, 23, 24 .Such inspiration comes from behavioral patterns previously seen in several animal groups.Examples include ant pheromone trail networks, aggregation of cockroaches, and the migration of fish schools, all of which can be accurately described in terms of individuals following simple sets of rules 25 .Some examples of these rules 24, 26 are keeping the current position or location for best individuals, local attraction or repulsion, random movements, and competition for the space within a determined distance.
On the other hand, new studies 27-29 have also shown the existence of collective memory in animal groups.The presence of such memory establishes that the previous history of the group structure influences the collective behavior exhibited in future stages.According to such principle, it is possible to model complex collective behaviors by using simple individual rules and configuring a general memory.
In this paper, a new optimization algorithm inspired by the collective animal behavior is proposed.In this algorithm, the searcher agents emulate a group of animals that interact with each other based on simple behavioral rules which are modeled as mathematical operators.Such operations are applied to each agent considering that the complete group has a memory storing their own best positions seen so far, by using a competition principle.The proposed approach has been compared to other well-known optimization methods.The results confirm a high performance of the proposed method for solving various benchmark functions.
This paper is organized as follows.In Section 2, we introduce basic biological aspects of the algorithm.In Section 3, the novel CAB algorithm and its characteristics are both described.Section 4 presents the experimental results and the comparative study.Finally, in Section 5, conclusions are given.

Biologic Fundamentals
The remarkable collective behavior of organisms such as swarming ants, schooling fish, and flocking birds has long captivated the attention of naturalists and scientists.Despite a long history of scientific research, the relationship between individuals and group-level properties has just recently begun to be deciphered 30 .
Grouping individuals often have to make rapid decisions about where to move or what behavior to perform in uncertain and dangerous environments.However, each individual typically has only a relatively local sensing ability 31 .Groups are, therefore, often composed of individuals that differ with respect to their informational status and individuals are usually not aware of the informational state of others 32 , such as whether they are knowledgeable about a pertinent resource or about a threat.
Animal groups are based on a hierarchic structure 33 which considers different individuals according to a fitness principle called dominance 34 which is the domain of some individuals within a group that occurs when competition for resources leads to confrontation.Several studies 35, 36 have found that such animal behavior lead to more stable groups with better cohesion properties among individuals.
Recent studies have begun to elucidate how repeated interactions among grouping animals scale to collective behavior.They have remarkably revealed that collective decisionmaking mechanisms across a wide range of animal group types, from insects to birds and even among humans in certain circumstances seem to share similar functional characteristics 21, 25, 37 .Furthermore, at a certain level of description, collective decisionmaking by organisms shares essential common features such as a general memory.Although some differences may arise, there are good reasons to increase communication between researchers working in collective animal behavior and those involved in cognitive science 24 .
Despite the variety of behaviors and motions of animal groups, it is possible that many of the different collective behavioral patterns are generated by simple rules followed by individual group members.Some authors have developed different models, one of them, known as the self-propelled particle SPP model, attempts to capture the collective behavior of animal groups in terms of interactions between group members which follow a diffusion process 38-41 .On the other hand, following a biological approach, Couzin and krauze 24, 25 have proposed a model in which individual animals follow simple rules of thumb: 1 keep the current position or location for best individuals, 2 move from or to nearby neighbors local attraction or repulsion , 3 move randomly, and 4 compete for the space within of a determined distance.Each individual thus admits three different movements: attraction, repulsi ón, or random and holds two kinds of states: preserve the position or compete for a determined position.In the model, the movement, which is executed by each individual, is decided randomly according to an internal motivation .On the other hand, the states follow a fixed criteria set.
The dynamical spatial structure of an animal group can be explained in terms of its history 36 .Despite such a fact, the majority of studies have failed in considering the existence of memory in behavioral models.However, recent research 27, 42 have also shown the existence of collective memory in animal groups.The presence of such memory establishes that the previous history of the group structure influences the collective behavior which is exhibited in future stages.Such memory can contain the location of special group members the dominant individuals or the averaged movements produced by the group.
According to these new developments, it is possible to model complex collective behaviors by using simple individual rules and setting a general memory.In this work, the behavioral model of animal groups inspires the definition of novel evolutionary operators which outline the CAB algorithm.A memory is incorporated to store best animal positions best solutions considering a competition-dominance mechanism.

Collective Animal Behavior Algorithm (CAB)
The CAB algorithm assumes the existence of a set of operations that resembles the interaction rules that model the collective animal behavior.In the approach, each solution within the search space represents an animal position.The "fitness value" refers to the animal dominance with respect to the group.The complete process mimics the collective animal behavior.
The approach in this paper implements a memory for storing best solutions animal positions mimicking the aforementioned biologic process.Such memory is divided into two different elements, one for maintaining the best locations at each generation M g and the other for storing the best historical positions during the complete evolutionary process M h .

Description of the CAB Algorithm
Following other metaheuristic approaches, the CAB algorithm is an iterative process that starts by initializing the population randomly generated random solutions or animal positions .Then, the following four operations are applied until a termination criterion is met i.e., the iteration number NI .
1 Keep the position of the best individuals.4 Compete for the space within a determined distance update the memory .

Initializing the Population
The algorithm begins by initializing a set A of N p animal positions A {a 1 , a 2 , . . ., a N p } .Each animal position a i is a D-dimensional vector containing parameter values to be optimized.Such values are randomly and uniformly distributed between the prespecified lower initial parameter bound a low with j and i being the parameter and individual indexes, respectively.Hence, a j,i is the jth parameter of the ith individual.All the initial positions A are sorted according to the fitness function dominance to form a new individual set X {x 1 , x 2 , . . ., x N p }, so that we can choose the best B positions and store them in the memory M g and M h .The fact that both memories share the same information is only allowed at this initial stage.

Keep the Position of the Best Individuals
Analogous to the biological metaphor, this behavioral rule, typical from animal groups, is implemented as an evolutionary operation in our approach.In this operation, the first B elements {a 1 , a 2 , . . ., a B } , of the new animal position set A, are generated.Such positions are computed by the values contained inside the historical memory M h , considering a slight random perturbation around them.This operation can be modeled as follows: where l ∈ {1, 2, . . ., B} while m l h represents the l-element of the historical memory M h .v is a random vector with a small enough length.

Move from or to Nearby Neighbors
From the biological inspiration, animals experiment a random local attraction or repulsion according to an internal motivation.Therefore, we have implemented new evolutionary operators that mimic such biological pattern.For this operation, a uniform random number r m is generated within the range 0, 1 .If r m is less than a threshold H, a determined individual position is attracted/repelled considering the nearest best historical position within the group i.e., the nearest position in M h ; otherwise, it is attracted/repelled to/from the nearest best location within the group for the current generation i.e., the nearest position in M g .Therefore such operation can be modeled as follows: with probability H with probability 1 − H ,

3.3
where i ∈ {B 1, B 2, . . ., N p }, m nearest h and m nearest g represent the nearest elements of M h and M g to x i , while r is a random number between −1, 1 .Therefore, if r > 0, the individual position x i is attracted to the position m nearest h or m nearest g , otherwise such movement is considered as a repulsion.

Move Randomly
Following the biological model, under some probability P , one animal randomly changes its position.Such behavioral rule is implemented considering the next expression:

3.4
with i ∈ {B 1, B 2, . . ., N p }and r a random vector defined in the search space.This operator is similar to reinitializing the particle in a random position, as it is done by 3.1 .

Compete for the Space within a Determined Distance (Update the Memory)
Once the operations to keep the position of the best individuals, such as moving from or to nearby neighbors and moving randomly, have been applied to all N p animal positions, generating N p new positions, it is necessary to update the memory M h .
In order to update memory M h , the concept of dominance is used.Animals that interact within the group maintain a minimum distance among them.Such distance, which is defined as ρ in the context of the CAB algorithm, depends on how aggressive the animal behaves 34, 42 .Hence, when two animals confront each other inside such distance, the most dominant individual prevails meanwhile other withdraw.Figure 1 depicts the process.
In the proposed algorithm, the historical memory M h is updated considering the following procedure.
1 The elements of M h and M g are merged into If the distance between both elements is less than ρ, the element getting a better performance in the fitness function prevails meanwhile the other is removed.
3 From the resulting elements of M U from Step 2 , it is selected the B best value to build the new M h .
The use of the dominance principle in CAB allows considering as memory elements those solutions that hold the best fitness value within the region which has been defined by the ρ distance.
The procedure improves the exploration ability by incorporating information regarding previously found potential solutions during the algorithm's evolution.In general, the value of ρ depends on the size of the search space.A big value of ρ improves the exploration ability of the algorithm although it yields a lower convergence rate.
In order to calculate the ρ value, an empirical model has been developed after considering several conducted experiments.Such model is defined by following equation: where a low j and a high j represent the prespecified lower and upper bound of the j-parameter respectively, within an D-dimensional space.

Computational Procedure
The computational procedure for the proposed algorithm can be summarized as follows: Step 1. Set the parameters N p , B, H, P , and NI.
Step 3. Sort A according to the objective function dominance to build X {x 1 , x 2 , . . ., x N p }.
Step 4. Choose the first B positions of X and store them into the memory M g .
Step 5. Update M h according to Section 3.1.5during the first iteration: M h M g .
Step 6. Generate the first B positions of the new solution set A {a 1 , a 2 , . . ., a B } .Such positions correspond to the elements of M h making a slight random perturbation around them, being v a random vector of a small enough length.
Step 7. Generate the rest of the A elements using the attraction, repulsion, and random movements.
for i B 1 : N p {if r 1 < P then attraction and repulsion movement if r 2 < H then where r 1 , r 2 ∈ r and 0, 1 and r ∈ −1, 1 Step 8.If NI is completed, the process is finished; otherwise, go back to Step 3.
The best value in M h represents the global solution for the optimization problem.

Test Suite and Experimental Setup
A comprehensive set of 31 functions that have been collected from 43-54 , they are used to test the performance of the proposed approach.Tables 12-17 in the appendix present the benchmark functions used in our experimental study.Such functions are classified into four different categories: unimodal test functions Table 12 , multimodal test functions Table 13 , multimodal test functions with fixed dimensions Tables 14 and 15 , and GKLS test functions Tables 16 and 17 .In such tables, n is the dimension of function, f opt is the minimum value of the function, and S is a subset of R n .The optimum location x opt for functions in Tables 12  and 13 fall into 0 n , except for f 5 , f 12 and f 13 with x opt falling into 1 n and f 8 in 420.96 n .A detailed description of all functions is given in the appendix.
To study the impact of parameters P and H described in Sections 3.1.3and 3.1.4over the performance of CAB, different values have been tested on 5 typical functions.The maximum number of iterations is set to 1000.N p and B are fixed to 50 and 10, respectively.The mean best function values μ and the standard deviations σ 2 of CAB, averaged over 30 runs, for the different values of P and H are listed in Tables 1 and 2, respectively.The results suggest that a proper combination of different parameter values can improve the performance of CAB and the quality of solutions.Table 1 shows the results of an experiment which consist in fixing H 0.8 and varying P from 0.5 to 0.9.On a second test, the experimental setup is swapped, that is, P 0.8 and H varies from 0.5 to 0.9.The best results in the experiments are highlighted in both tables.After the best value in parameters P and H has been experimentally determined with a value of 0.8 , it is kept for all tests throughout the paper.
In order to demonstrate that the CAB algorithm provides a better performance, it has been compared to other optimization approaches such as metaheuristic algorithms Section 4.2 and continuous methods Section 4.3 .The results of such comparisons are explained in the following sections.

Performance Comparison with Other Metaheuristic Approaches
We have applied CAB to 31 test functions in order to compare its performance to other wellknown metaheuristic algorithms such as the real genetic algorithm RGA 55 , the PSO 18 , the gravitational search algorithm GSA 56 , and the differential evolution method DE 57 .In all cases, population size is set to 50.The maximum iteration number is 1000 for functions in Tables 12 and 13, and 500 for functions in Table 14 and 16.Such stop criteria have been chosen as to keep compatibility to similar works which are reported in 14 and 58 .
Parameter settings for each algorithm in the comparison are described as follows.
1 RGA: according to 55 , the approach uses arithmetic crossover, Gaussian mutation, and roulette wheel selection.The crossover and mutation probabilities have been set to 0.3 and 0.1, respectively.
2 PSO: In the algorithm, c 1 c 2 2 while the inertia factor ω is decreasing linearly from 0.9 to 0.2.
3 In GSA, G 0 is set to 100 and α is set to 20; T is the total number of iterations set to 1000 for functions f 1 -f 13 and to 500 for functions f 14 -f 31 .Besides, K 0 is set to 50 total number of agents and is decreased linearly to 1.Such values have been found as the best configuration set according to 56 .
4 DE: the DE/Rand/1 scheme is employed.The parameter settings follow the instructions in 57 .The crossover probability is CR 0.9 and the weighting factor is F 0.8.
Several experimental tests have been developed for comparing the performance of the CAB algorithm against other metaheuristic algorithms.The experiments have been developed considering the following function types.
3 Multimodal test functions with fixed dimensions Tables 14 and 15 .

Unimodal Test Functions
In this test, the performance of the CAB algorithm is compared to RGA, PSO, GSA and DE, considering functions with only one minimum/maximum.Such function type is represented by functions f 1 to f 7 in Table 12.The results, over 30 runs, are reported in Table 3 considering the following performance indexes: the average best-so-far solution, the average mean fitness   In order to statistically analyze the results in Table 3, a non-parametric significance proof known as the Wilcoxon's rank test has been conducted 59, 60 , which allows assessing result differences among two related methods.The analysis is performed considering a 5% significance level over the "average best-so-far" data.Table 4 reports the P values produced by Wilcoxon's test for the pairwise comparison of the "average best so-far" of four groups.Such groups are formed by CAB versus RGA, CAB versus PSO, CAB versus GSA, and CAB versus DE.As a null hypothesis, it is assumed that there is no significant difference between mean values of the two algorithms.The alternative hypothesis considers a significant difference between the "average best-so-far" values of both approaches.All P values reported in the table are less than 0.05 5% significance level which is a strong evidence against the null hypothesis, indicating that the CAB results are statistically significant and that it has not occurred by coincidence i.e., due to the normal noise contained in the process .

Multimodal Test Functions
Multimodal functions, in contrast to unimodal, have many local minima/maxima which are, in general, more difficult to optimize.In this section the performance of the CAB algorithm is compared to other metaheuristic algorithms considering multimodal functions.Such comparison reflects the algorithm's ability to escape from poor local optima and to locate a near-global optimum.We have done experiments on f 8 to f 13 of Table 13 where the number of local minima increases exponentially as the dimension of the function increases.The dimension of these functions is set to 30.The results are averaged over 30 runs, reporting the performance indexes in Table 5 as follows: the average best-so-far solution, the average mean fitness function and, the median of the best solution in the last iteration the best result for each function is highlighted .Likewise, P values of the Wilcoxon signed-rank test of 30 independent runs are listed in Table 6.
For f 9 , f 10 , f 11 , and f 12 , CAB yields a much better solution than the others.However, for functions f 8 and f 13 , CAB produces similar results to RGA and GSA, respectively.The Wilcoxon rank test results, presented in Table 6, show that CAB performed better than RGA, PSO, GSA, and DE considering the four problems f 9 -f 12 , whereas, from a statistical viewpoint, there is not difference in results between CAB and RGA for f 8 and between CAB and GSA for f 13 .Evolutions of the "average best-so-far" solutions over 30 runs for functions f 10 and f 12 are shown in Figure 3.

Multimodal Test Functions with Fixed Dimensions
In the following experiments the performance of the CAB algorithm is compared to RGA, PSO, GSA, and DE considering functions which are extensively reported in the metaheuristicbased optimization literature 49-54 .Such functions, represented by f 14 to f 23 in Tables 14  and 15, are all multimodal with fixed dimensions.Table 7 shows the outcome of such process.Results, presented in Table 7, show how metaheuristic algorithms maintain a similar average performance when they are applied to low-dimensional functions 58 .The results show that RGA, PSO, and GSA have similar solutions and performances that are nearly the same as it can be seen in Figure 4.

GKLS Test Functions
This section considers GKLS functions which are built using the GKLS-generator described in 54 .In the construction, the generator uses a set of user-defined parameters for building a multimodal function with known local and global minima.For conducting the numerical experiments, eight GKLS functions been employed which are defined by f 24 to f 31 .Details of their characteristics and parameters for their construction are listed in Tables 16 and  17.Results, over 30 runs, are reported in Table 8 the best result for each function test is boldfaced .According to this table, CAB provides better results than RGA, PSO, GSA, and DE for all GKLS functions, in particular for functions holding bigger dimensions f 28 -f 31 .Such performance is directly related to a better tradeoff between exploration and exploitation which is produced by CAB operators.Likewise, as it can be observed from Figure 5, the CAB algorithm possesses better convergence rates in comparison to other metaheuristic algorithms.
In order to statistically validate the results of Table 8, the Wilcoxon's test has been conducted.Table 9 shows the P values obtained after applying such analysis over 30

Comparison to Continuous Optimization Methods
Finally, the CAB algorithm is also compared to continuous optimization methods by considering some functions of the appendix.Since the BFSG algorithm 61 is one of the most effective continuous methods for solving unconstrained optimization problems, it has been considered as a basis for the algorithms used in the comparison.
In order to compare the performance of CAB to continuous optimization approaches, two different tests have been conducted.The first one tests the ability of BFGS and CAB to face unimodal optimization tasks see Section 4.3.1 is evaluated.The second experiment analyzes the performance of CAB and one BFGS-based approach, when they are both applied to multimodal functions review Section 4.3.2 .

Local Optimization
In the first experiment, the performance of algorithms BFGS and CAB over unimodal functions is compared.In unimodal functions, the global minimum matches the local minimum.Quasi-Newton methods, such as the BFGS, have a fast rate of local convergence although it depends on the problem's dimension 62, 63 .Considering that not all unimodal functions of Table 12 fulfill the requirements imposed by the gradient-based approaches i.e., f 2 and f 4 are not differentiable meanwhile f 7 is nonsmooth , we have chosen the Rosenbrock function f 5 as a benchmark.
In the test, both algorithms BFGS and CAB are employed to minimizef 5 , considering different dimensions.For the BFGS implementation, B 0 I is considered as initial matrix.Likewise, parameters δ and σ are set to 0.1 and 0.9 respectively.Although several performance criteria may define a comparison index, most can be applied to only one method timely such as the number of gradient evaluations .Therefore, this paper considers the elapsed time and the iteration number once the minimum has been reached as performance indexes in the comparison.In the case of BFGS, the termination condition is assumed as g 5 x ≤ 1 × 10 −6 , with g 5 x being the gradient of f 5 x .On the other hand, the stopping criterion of CAB considers when no more changes to the best element in memory M h are registered.Table 10 presents the results of both algorithms considering several dimensions From Table 10, we can observe that the BFGS algorithm produces shorter elapsed times and fewer iterations than the CAB method.However, from n 70, the CAB algorithm contend with similar results.The fact that the BFGS algorithm outperforms the CAB approach cannot be deemed as a negative feature considering the restrictions imposed to the functions by the BFGS method.

Global Optimization
Since the BFGS algorithm exploits only local information, it may easily get trapped into local optima restricting its use for global optimization.Thus, several methods based on continuous optimization approaches have been proposed.One of the most widely used techniques is the so-called multistart 64 MS .In MS a point is randomly chosen from a feasible region as initial solution and subsequently a continuous optimization algorithm local search starts from it.Then, the process is repeated until a near global optimum is reached.The weakness of MS is that the same local minima may be found over and over again, wasting computational resources 65 .
In order to compare the performance of the CAB approach to continuous optimization methods in the context of global optimization, the MS algorithm ADAPT 66 has been chosen.ADAPT uses as local search method the BFGS algorithm, which is iteratively executed.Thus, ADAPT possess two different stop criteria, one for the local procedure BFGS and other for the complete MS approach.For the comparison, the ADAPT algorithm has been implemented as suggested in 66 .
In the second experiment, the performance of the ADAPT and the CAB algorithms is compared over several multimodal functions described in Tables 13 and 14.The study considers the following performance indexes: the elapsed time, the iteration number, and the average best so-far solution.In case of the ADAPT algorithm, the iteration number is computed as the total iteration number produced by all the local search procedures as the MS method operates.The termination condition of the ADAPT local search algorithm BFGS is assumed when g k x ≤ 1 × 10 −5 , g k x being the gradient of f k x .On the other hand, the stopping criterion for the CAB and the ADAPT algorithms is considered when no more changes in the best solution are registered.Table 11 presents results from both algorithms considering several multimodal functions.In order to assure consistency, results ponder 0.00030    the averaged elapsed time AET , the averaged iteration number AIN and the average bestso-far solution ABS over 30 different executions.In Table 11, the averaged number of local searches ALS executed by ADAPT during the optimization is additionally considered.Table 11 provides a summarized performance comparison between the ADAPT and the CAB algorithms.although both algorithms are able to acceptably locate the global minimum for both cases, there exist significant differences in the required time for reaching it.When comparing the averaged elapsed time AET and the averaged iteration number AIN in Table 11, CAB uses significantly less time and fewer iterations to reach the global minimum than the ADAPT algorithm.

Conclusions
This article proposes a novel metaheuristic optimization algorithm that is called the collective animal behavior algorithm CAB .In CAB, the searcher agents emulates a group of animals that interact with each other considering simple behavioral rules which are modeled as mathematical operators.Such operations are applied to each agent considering that the complete group has a memory storing the best positions seen so far by using a competition principle.
The CAB algorithm presents two important characteristics: 1 CAB operators allow a better tradeoff between exploration and exploitation of the search space; 2 the use of its embedded memory incorporates information regarding previously found local minima potential solutions during the evolution process.
CAB has been experimentally tested considering a challenging test suite gathering 31 benchmark functions.In order to analyze the performance of the CAB algorithm, it has been compared to other well-known metaheuristic approaches.The experiments, statistically validated, have demonstrated that CAB generally outperforms other metaheuristic algorithms for most of the benchmark functions regarding the solution quality.In this study, the CAB algorithm has also been compared to algorithms based on continuous optimization methods.The results have shown that althogh continuous-based approaches outperform CAB for local optimization tasks, they face great difficulties in solving global optimization problems.

2
Move from or to nearby neighbors local attraction and repulsion .3 Move randomly.

ρFigure 1 :
Figure 1: Dominance concept as it is presented when two animals confront each other inside of a ρ distance.

Figure 2 :
Figure 2: Performance comparison of RGA, PSO, GSA, DE, and CAB for minimization of a f 1 and b f 7 considering n 30.

Figure 3 :
Figure 3: Performance comparison of RGA, PSO, GSA, DE, and CAB for minimization of a f 10 and b f 12 considering n 30.

Figure 4 :
Figure 4: Performance comparison of RGA, PSO, GSA, DE, and CAB for minimization of a f 15 and b f 22 .

Figure 5 :
Figure 5: Performance comparison of RGA, PSO, GSA, DE and CAB for minimization of the GKLSfunctions: a f 28 and b f 31 .

Table 1 :
Results of CAB with variant values of parameter P over 5 typical functions, with H 0.8.

Table 2 :
Results of CAB with variant values of parameter H over 5 typical functions, with P 0.8.

Table 3 :
Minimization result of benchmark functions in Table 12 with n 30.Maximum number of iterations 1000.
algorithm.Moreover, the good convergence rate of CAB can be observed from Figure2.According to this figure, CAB tends to find the global optimum faster than other algorithms and yet offer the highest convergence rate.

Table 4 :
P values produced by Wilcoxon's test comparing CAB versus RGA, PSO, GSA, and DE over the "average best-so-far" values from Table3.

Table 5 :
Minimization of benchmark functions in Table13with n 30.Maximum number of iterations 1000.

Table 6 :
P values produced by Wilcoxon's test comparing CAB versus RGA, PSO, GSA, and DE over the "average best-so-far" values from Table5.

Table 7 :
Minimization result of benchmark functions in Table14with n 30.Maximum number of iterations 500.

Table 8 :
Minimization result of GKLSfunctions in Table16.Maximum number of iterations 500.

Table 9 :
P values produced by Wilcoxon's test comparing CAB versus RGA, PSO, GSA, and DE over the "average best-so-far" values from Table8.Since all P values, presented in Table9, are less than 0.05, it indicates that the CAB results are statistically better.

Table 10 :
Performance comparison between the BFGS and the CAB algorithm, considering different dimensions over the Rosenbrock function.The averaged elapsed time AET is referred in seconds.

Table 11 :
Performance comparison between the ADAPT and the CAB algorithm considering different multimodal functions.The averaged elapsed time AET is referred in the format M's Minute'second .10, 30, 50, 70, 100, 120} of f 5 .In order to assure consistency, such results represent the averaged elapsed time AET and the averaged iteration number AIN over 30 different executions.It is additionally considered that at each execution both methods are initialized in a random point inside the search space .

Table 14 :
Multimodal test functions with fixed dimensions.

Table 17 :
Optimum locations of the used GKLS functions.