A set of new compact firefly algorithms

Accepted Manuscript A set of new compact firefly algorithms Lyes Tighzert, Cyril Fonlupt, Boubekeur Mendil PII:

S2210-6502(17)30250-X

DOI:

10.1016/j.swevo.2017.12.006

Reference:

SWEVO 337

To appear in:

Swarm and Evolutionary Computation BASE DATA

Received Date: 29 March 2017 Revised Date:

10 November 2017

Accepted Date: 14 December 2017

Please cite this article as: L. Tighzert, C. Fonlupt, B. Mendil, A set of new compact firefly algorithms, Swarm and Evolutionary Computation BASE DATA (2018), doi: 10.1016/j.swevo.2017.12.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A Set of New Compact Firefly Algorithms Lyes TIGHZERT1,*, Cyril FONLUPT2, Boubekeur MENDIL1 1

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Laboratoire de Technologie Industrielle et de l’Information (LTII), Faculté de Technologie, Université de Bejaia, 06000 Bejaia, Algérie (e-mail: [email protected] , [email protected] ). 2 Laboratoire d'Informatique Signal et Image de la Côte d'Opale (LISIC) Université du Littoral - Côte d'Opale BP 719 62228 CALAIS Cedex – France (email : [email protected] )

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Abstract– Population-based algorithms are among the most successful approaches to optimization. These algorithms require high computational capacities such as memory storage. During the last decade, an alternative approach, called compact optimization, was developed. In real-valued compact algorithms, the population is represented by a probabilistic distribution function. Real-valued compact genetic algorithm was first developed and then the idea was extended to other population-based algorithms, like differential evolution, particle swarm optimization and teaching-learning-based optimization. In this paper, we introduce a set of new compact firefly algorithms (cFAs) with minimal computational costs. Our primary aim is to reduce the computational capacity and storage required by the classical variants of FA. The proposed approaches lead to the reduction of the complexity of the attraction model used in FA. The proposed cFAs achieve the optimization with a minimal number of attractions. Several propositions are investigated and reported; such as: elitism strategies, Lévy movements, and opposition-based learning. The proposed algorithms consist of: permanent elitism-based compact firefly algorithm (pe-cFA), non-permanent elitism-based compact firefly algorithms (ne-cFA), permanent elitism-based compact Lévy-flight firefly algorithms (pe-cLFA), non-permanent elitism-based compact Lévy-flight firefly algorithms (ne-cLFA), opposition-based compact firefly algorithms (OBcFA) and opposition-based compact Lévy- flight firefly algorithms (OBcLFA). All the known compact algorithms use normal probability of density function (NPDF) to represent the population. In this paper, a new way is investigated. The alternative solution proposed here is based on uniform PDF (UPDF). Thus, two categories of cFAs are presented: NPDF-based cFAs and UPDF-based cFAs. Hence, for each proposed algorithm, two versions are presented and analyzed. The proposed set of twelve algorithms are tested on the IEEE CEC2014 benchmark functions and compared to the state-of-art of compact evolutionary algorithms (cEAs), swarm intelligent algorithms (SIAs), and the most advanced evolutionary algorithms (EAs). The obtained results show that the proposed cFAs are very competitive and that the uniform distribution is very efficient. The case study of this paper concerns the optimal swing-up control of a gymnastic humanoid robot hanging on a bar. Keys words: compact swarm intelligence; compact representation; optimization; firefly algorithm; humanoid robot; control.

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Computational intelligence (CI) is a subfield of computer science that uses models, methods and techniques inspired from the behaviors [1, 2] of intelligent biological systems. As the natural evolution has endowed breeds with very processed and sophisticated skills, engineers and scientists take inspiration from nature to develop several powerful and robust algorithms to solve difficult engineering problems. The swarm intelligence (SI), introduced by Gerardo Beni and Jing Wang in 1989 [1], is one of the bionic-based computing domains that consists of the exploitation of some mathematical models inspired from the collective behaviors of species [3]. Each swarm agent, also called particle [4-6], is capable of interacting with its environment and locally with the other particles in its sides. The collective comportment of the decentralized swarm makes it a selforganizing system. The proposed algorithms, inspired by swarms, consist of population-based strategies. The particles abilities, such as extracting information from their environment and sharing it in the group, leads the whole population toward the optimal sub-region of the search space. In the literature, very effective optimization techniques based on swarm intelligent systems have been proposed and this includes ant colonies [7, 8], honey bee colonies [9-11], fish schooling [12], firefly algorithm (FA) [13–15], the cuckoo search [16-18], Physarum Polycephalum algorithm [19], and bat algorithm [20-22]. The firefly algorithm (FA) was introduced by Xin-She Yang at Cambridge University, based on the modeling of the brightness and attractiveness characteristics of the fireflies [14-15]. Like other population-based algorithms, it uses the firefly’s positions to represent the temporally found solutions. Each firefly is capable of radiating signals to attract the others towards its own position. The fireflies are assumed bisexual, and the attractiveness is not for sexuality. If the firefly
 is better than
 , the
 will be moved toward
 . Hence, the best fireflies will attract the others toward them. Although FA has shown good performance [13, 23], it can be subject to premature convergence [24-26]. Therefore, improving FA’s performance was the topic of several contributions. We can cite Lévy-flight FA [27], chaos-based FA [28],

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Introduction

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variable step size FA [29], and hybrid FA variants. The attraction model of many FAs is known as full attraction model [2426]. The latter is based on comparing each firefly with all others. In fact, at each generation, each firefly is moved several times and this may induce oscillations [24]. The full attraction model presents a high computational complexity [24-26]. It affects the processing time of the algorithm and leads to difficulty in digital hardware implementation [24-26]. In order to deal with this problem, Hui Wang et al. proposed random attraction FA (RaFA) [25], in which each firefly is attracted, or attracts, only one firefly randomly selected from the swarm. But their proposal can be subject to premature convergence [24, 26]. The complexity of FA’s attraction model is not its first drawback. Like other population-based algorithms, where each individual is stored, the FA variants present the drawback of high information storage. Hence, it requires high computational capacities to achieve the given optimization task. In the field of robotics and many other areas of control, these capacities are not always available. The control system must take into account several other pieces of information, e.g. the navigation map, the position of obstacles, etc. In this study, we investigate the reduction of the computational capacities, the complexity of the attraction model and the memory storage required by FA. During the last decade, several authors have proposed compact representation [30] to reduce the computation cost of several advanced population-based optimization techniques. These algorithms are called compact evolutionary algorithms (cEAs). We find the compact version of GA (cGA) [31], the compact version DE (cDE) [32], the compact version of PSO (cPSO) [33], the compact version of TLBO (cTLBO) [34], the compact version of harmony search algorithms (cHSA) [35], etc. The success of these compact optimization algorithms has motivated us to introduce the compact FA (cFA). Our work has seven main objectives. The first is to reduce the complexity of the attraction model of FA. The second is to reduce the computational capacity required by classical FA. The third is to reduce the memory storage required to store the swarm. The Fourth is to propose a new compact optimization technique, based on FA, which surpasses the known cEAs and the original FA. The fifth is to integrate opposition-based learning into compact optimization. The sixth is to take a step toward compact swarm intelligence. The seventh is to demonstrate that uniform distributions are also suitable to implement cEAs. All the algorithms we propose in this study are implemented both with uniform and normal distributions. To achieve all these goals, this advanced study incorporates to cFAs Lévy-Flight movements, opposition-based learning (OBL) and two elitism strategies. We also study the uses of uniform distributions to implement cEAs. These distributions are not yet used in the field of cEAs. A general approach to use uniform distribution is herein carried out and then applied to propose new cFAs. Thus, this study leads to a set of twelve new compact firefly algorithms (cFAs) with minimal computational costs. The proposed approaches are assessed by means of an extensive comparative study, which includes numerical tests on IEEE CEC2014 benchmark functions, comparison with the state-of-art of compact optimization algorithms, swarm intelligent algorithms and the most advanced evolutionary algorithms. Each proposed variant is studied in a very fine and detailed way. The obtained results show the effectiveness of our proposals and the set of goals are all achieved. The case study of this paper concerns the optimal swing-up control of a gymnast humanoid robot. The remaining of this paper is organized as follows. In the next section, we present the main basics of FA. In section 3 we present a background on compact optimization. In section 4, we present two categories of new compact firefly algorithms. The first one uses normal PDF to represent the population, whereas the second one uses uniform PDF. Thus, for each proposed algorithm, two versions are presented and analyzed. The proposed algorithms consist of: elitism-based compact firefly algorithms (ne-cFA and pe-cFA), compact Lévyflight FA (ne-cLFA and pe-cLFA), opposition-based compact FA (OB-cFA), and opposition-based Lévy-flight FA (OBcLFA). In order to evaluate the effects of the probabilistic function used to represent the population, these six proposed algorithms are implemented by using, in the first time, a normal PDF and, in a second time, a uniform PDF. In section 5, the proposed algorithms are benchmarked against the IEEE CEC2014 functions. The obtained results are compared to the state-ofart of compact evolution algorithms, swarm intelligent algorithms, and the state-of-art of evolutionary algorithms. Section 6 concerns the case of study of this work, which consists of the optimal swing-up control of a gymnast humanoid robot hanging on a high bar. Section 7 concludes this paper.

The Basics of Firefly Algorithms

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The firefly algorithm (FA) was introduced by Xin-She Yang at Cambridge University [14]. To sum up the algorithm steps with more simplicity, we can present it through three steps: 1) Brightness: depending on the distance between two fireflies and the atmospheric absorption coefficient, a lightning signal is associated to each one. This is formulated as:  ( ) =   



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ß( ) = ß   Where ß is the attractiveness of the firefly when  = 0.



(2)

Where  is the intensity of the light source, ɣ is the absorption coefficient,  is the distance between the two considered fireflies and  is the intensity of the light source when  = 0. 2) Attractiveness: it can be expressed by :

3) The moving step : For the entire population and for each pair of fireflies, the less fit firefly is moved toward the costefficient ones, using the following model :

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  +  (−0.5,0.5) (3) Where  is the mutation coefficient which is generally a self-adaptive parameter decreasing through iterations and (−0.5, 0.5) is a normal randomized number between [−0.5, 0.5]. The pseudo code of the FA is shown in Fig. 1.

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Fig. 1 Pseudo-code of FA.

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Algorithm 1: Firefly Algorithm Problem Definition : Objective function f(x) Generate an initial population of n fireflies Define light absorption coefficient γ, I0 and β0 while termination condition do for i = 1 : N do /* all fireflies */ for j = 1 : N (all n fireflies) (inner loop) do if (#$%&'((()$ )> #$%&'((()* )) then /* “>” means better */ Move firefly xj towards firefly xi according to Eqs. (3) end if Vary attractiveness with distance r via using Eqs. (2). Evaluate new solutions and update light intensity. end for j end for i Rank the fireflies and find the current global best g. end while Post process results and visualization

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The Firefly algorithm is known to be very effective, and it is widely used in several real-world applications. We can cite: clustering in wireless networks [36], power systems [37], database [38], etc. The association of swarm intelligence with robotics is baptized "swarm robotics" [39]. FA-based swarm intelligent robotics systems and approaches are also proposed [40]. Fister et al. have given an important comprehensive review on FA and have reported many applications, criticisms and variants [13]. One of the popular variants of FA is the Lévy-flight firefly algorithm (LFA) [27]. This variant uses the Lévy distribution of infinite variance to improve the performance of FA. The basic difference between FA and LFA stands on the formulation model [27, 41] of the moving step. In LFA, the attraction movement is expressed as Eqs. (4).


 =
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 −
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Where levy is an arbitrary vector randomized from the levy distribution (see [27]). This scheme leads, in some cases, to performance improvement. However, the authors have outlined that this depends on the application [27].

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In order to solve a high dimension problem with stochastic population-based algorithms, a large population is required. However, the required computational capacity increases with respect to population size. The compact optimization [51] gives an alternative and promises a small amount of memory and computational requirements. This field started in 1999 [52] when Harik et al. proposed the first compact genetic algorithm (cGA). As estimation of distribution algorithms (EDS) [53, 54], cGA generates offspring population according to the estimated probabilistic model instead of using recombination and mutation operators. The population is represented by the probabilistic vector (PV). The latter contains the probability of each bit to be equal to zero or one. Hence, the whole population is compacted and represented by PV. At each generation, two individuals are sampled using PV. The competition between these two individuals will determine the way in which the PV will be corrected. For example, if the bit j of the winner is set to one and that of the loser is set to zero, the jth component of PV will be increased

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Enhancing the performance of FA was the topic of numerous contributions. In just eight years, the development of FA has been quite spectacular, and it is impossible to present all the related works. Nevertheless, we will briefly review here some hybridization variants and recent publications. A. Rahmani et al. proposed the hybridization of FA with genetic algorithms [42]. P. Kora et al. proposed a hybrid PSO-FA [43]. M. Tuba and N. Bacanin proposed the hybrid FA seeker optimization algorithm [44]. The hybridization of FA with last square algorithms [45] and K-means algorithms [46] were also investigated. Some extensive studies have achieved the combining of fuzzy logic [47] and probabilistic neural network [48] with FA. A. Gálvez and A. Iglesias have proposed a new mimetic self-adaptive FA for continuous optimization [49]. A new history-driven FA was proposed by B. Nasiri and M. R. Meybodi [50]. The works of H. Wang et al., consisting of RaFA [25] and NaFA [24], aim at reducing the complexity of the attraction model used in FA. In this paper, we introduce a set of new compact FAs with minimal computational costs.

Compact optimization

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In 2003 Chang W. Ahn et al. [55] proposed elitism-based compact genetic Algorithms. The authors have reconciled cGA with elitism. They have proposed two strategies. The first is called permanent elitism-based compact genetic algorithm (pecGA) and the second is called non-permanent elitism-based compact genetic algorithm (ne-cGA). The pe-cGA maintains the best temporary solution (the elite) as long as other solutions (i.e., competitors) generated from probability vectors are not better. Hence, it makes a selection pressure that is high enough to offset the disruptive effect of uniform crossover. In ne-cGA, the elite is not maintained longer than a pre-defined inherence length. Thus, it avoids strong elitism that may lead to premature convergence and maintains genetic diversity as a bonus.

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The variants, presented above, use a binary code. In the rest of this paper, they will be referred by bcGA for: binary compact genetic algorithms (bcGA), e.g. pe-bcGA for: permanent elitism-based bcGA. In 2008, Ernesto Mininno et al. [31] proposed the first cGA that works with real values. Their variant is called real-valued compact genetic algorithm (rcGA). It gives better results than bcGAs [31] while significantly reducing the computational cost and avoids the additional computation cost related to the binary-to-float conversions. The rcGA uses a probability density function (PDF), e.g. normal PDF, to represent the solutions. The PV of the binary coded cGAs contains the probability of each allele to be equal to zero or one. On the other hand, the PV of rcGA contains the mean and the standard deviation of each allele. At each generation, individuals are sampled using the PDF function. Fig. 3 illustrates the sampling mechanism.

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Algorithm 2: Binary Compact Genetic Algorithm

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Fig. 2 Pseudo-code of binary compact genetic algorithm.

Gaussian PDF

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N : the population size L : chromosome length Initialize the probability vector for i=1 : L p[i]=0.5 end for while (0
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PDF (µ[i ],σ [i]) =

σ [i ](erf (

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µ[i ] + 1 µ[i ] − 1 ) − erf ( )) 2σ [i ] 2σ [i ]

For the sampling mechanism, first the corresponding cumulative distribution function (CDF) of the truncated Gaussian PDF is calculated. As the codomain of CDF is [0, 1], in order to sample a given gene x from PV, uniform random number rand (0 , 1) is generated, the inverse function of CDF, corresponding to rand(0, 1), is then computed. This value, denoted by xr[i], must be converted into the interval [a, b] by

x[i] = (b − a)

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As suggested in [31-33], the means of the PDF are set to zeros and the standard deviations are set to 10. The rcGA uses a specific updating rule. For example, in permanent elitism-based real-valued compact genetic algorithm (pe-rcGA), at each generation, an individual g is generated from PV, and then a tournament is performed between the best and g. Hence, a winner and a loser are obtained. According to their respective alleles, an updating rule is then performed using Eqs. (7) and Eqs. (8). See [31] for the proof.

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µ t +1 [ i ] = µ t [ i ] +

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(7)

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Where wi and li are, respectively, the alleles of the winner and the loser; n is the size of the compacted population.

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Since the apparition of rcGA, several authors have proposed the compact version of other real-valued optimization algorithms. In 2011, compact differential evolution (cDE) was proposed [32]. In 2012, compact particle swarm optimization (cPSO) was also proposed [33]. In 2013, Yang et al. proposed compact teaching-learning-based optimization (cTLBO) [34]. In 2014, A.S. Soares et al. have proposed mutation-based compact genetic algorithm [56]. In 2016, B. V. Ha et al. [57] have proposed a new modified compact genetic algorithm that works with more than one probability vector. The authors have proposed two updating rules: a local update and a global update. The wide interest of compact optimization is due to three facts. First, it gives satisfying results. Second, it does not compromise on memory and computation costs. Third, it is very easy for implementation. Consequently, compact optimization algorithms are very suitable for hardware implementation because of their reduced memory requirements [58-60]. The success of compact algorithms has led us to study the case of FA. Hence, this paper addresses the problem of the reduction of the memory and the computational cost required by FA. The proposed solution is based on the compact representation. Six new compact FAs are proposed. For each one, two versions are proposed and analyzed in all their fine points. For the first time, a normal PDF is used. For the second time, for more simplicity and also to reduce more the required computational capacity, uniform PDF is used. As a whole, twelve new variants of cFA are proposed in this paper. Their complete description is given in the next section.

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In our last work, we have introduced compact swarm intelligence [61]. The idea was illustrated by proposing compact firefly optimization (cFO) [61]. In this paper, we will deepen this idea at best. Thus, in this section, we will present twelve new real-valued compact firefly algorithms with minimal computational capacity. Six of them are based on normal probability of density function (NPDF) and the six others are based on the uniform probability of density function (UPDF). First, we start with NPDF-based cFAs.

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All the known compact evolutionary algorithms, e.g. rcGA [31], cDE [32], cPSO [33], etc., are implemented with NPDF. We present here six new variants of cFA that are also based on NPDF.

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The proposed compact firefly algorithms

Compact firefly algorithms with normal PDF

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a) Permanent elitism-based compact firefly algorithm

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The permanent elitism strategy was first proposed in [55] for bcGA. This strategy was then used in rcGA [31] and in cDE [32]. The permanent elitism creates pressure selection that is good enough to enhance convergence [55]. The good results obtained by this strategy [31-32] have motivated us to use it. We here introduce the permanent elitism-based compact firefly algorithm (pe-cFA).

Algorithm 3 : permanent elitism-based compact Firefly Algorithm

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N : Virtual population size λ : Initial standard deviation m : problem dimension t=1 /* iteration index */ for j=1:m do /* Initialization of PV */ µjt = 0; σj t = λ; end elite=generate( PV) /* Assume an elite */ while Termination condition do /* Main loop, the termination condition can be fixed by the user */ G1=generate (PV(t)) [ winner, loser ] = compete(G1, elite) G1=loser; elite=winner Move G1 toward the elite using Eqs. (3) [ winner, loser ] = compete(G1, elite) elite = winner ; /* Update elite */ for i=1 : m do /* updating PV */ µi t+1= µi t +1/N × (wit –li t ); 2 [σi t+1] 2 = [σi t ] 2 +[ui t ]2 -[ui t+1 ]2+ × ( [wi t ]2–[ l i t ]2) ; 3 /* where li and wi are the genes of the loser and winner*/ endfor end while return the best individual

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Fig. 4 Pseudo-code of pe-cFA.

The proposed algorithm consists of the following. At each generation, a firefly G1 is generated from PV. We compare (competition) G1 to the elite (the best ever found). If G1 is better than the elite, we update the elite. The loser of the competition is moved according to Eqs. (3) toward the winner, i.e. the fittest one. We evaluate the fitness of the new position, and we compare it to the elite. The elite is updated once the moved firefly surpasses it. According to the alleles of the winner and the loser of the last competition, the means and the standard deviation of the NPDF are updated with the same mechanism proposed in [31-33], see equations Eqs. (7) and Eqs. (8). Fig. 4 gives the pseudo-code of pe-cFA. The termination condition of the proposed algorithms can be fixed by the user. In our work, it is important to note that each fitness evolution is counted and the maximum number of fitness evolution is used as a termination condition (see section 5 for more details) for the entire proposed algorithms.

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b) Permanent elitism-based Lévy-flight firefly algorithm

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We present here permanent elitism-based compact levy-flight firefly algorithms (pe-cLFA). Lévy firefly algorithms (LFA) uses the Lévy distribution [27] to describe the movement of a given firefly attracted by a fitter one. Lévy distribution are used in several algorithms to enhance their performance [27, 62-64]. This has motivated us to use the same idea to implement a first compact version of LFA. The proposed pe-cLFA uses permanent elitism strategy and its pseudo-code is the same as pe-cFA presented in Fig. 4. The only difference consists in replacing Eqs. (3) in line 13 by Eqs. (4). Hence, it is not necessary to give a complete pseudo-code.

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c) Non-permanent elitism-based compact firefly algorithm

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The pe-cFA and pe-cLFA, presented above, keep the current best solution until hopefully a better solution is found. In this subsection, we present another cFA variant that uses non-permanent elitism strategy. The algorithm is called non-permanent elitism-based compact firefly algorithm (ne-cFA). The idea in non-permanent elitism [31-32, 55] consists of relaxing selection pressure, i.e. elitism, by restricting the length of elite chromosome’s inheritance. This means that the elite is updated when other fireflies do not surpass it after a given length of inheritance denoted by η. This scheme diminishes the possibility of premature convergence. The pseudo-code of ne-cFA can be simply derived from pe-cFA by adding or restricting the length of

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elite chromosomes’ inheritance. The allowable length of inheritance was shown to be bounded by the simulated population size [31, 32, 55]. The full pseudo-code is shown in Fig. 5.

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d) Nonpermanent elitism-based compact Lévy-flight firefly algorithm

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The nonpermanent Lévy-flight compact firefly algorithm (ne-cLFA) uses Lévy distribution to realize Lévy-flight movements. It has the same pseudo-code as ne-cFA presented in Fig. 5, except that in line 15 we replace Eqs. (3) by Eqs. (4). Hence, it is not necessary to provide a complete pseudo-code.

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N : Virtual population size λ : Initial standard deviation η= the allowed length of inheritance m : problem dimension t=1 /* iteration index */ for j=1:m do/* Initialization of PV */ t t µj = 0; σj = λ; end elite=generate( PV) /* Assume an elite */ θ=0; /* represent the actual length of inheritance*/ while Termination condition do /* Main loop, the termination condition can be fixed by the user */ G1=generate (PV(t)) [ winner, loser ] = compete(G1, elite) G1=loser; elite=winner Move G1 toward the elite using Eqs. (3) [ winner, loser ] = compete(G1, elite) if winner==G1 OR θ==η then elite = G1; θ=0; end if for i=1 : m do /* updating PV */ µi t+1= µi t +1/N × (wit –li t ); 2 [σi t+1] 2 = [σi t ] 2 +[ui t ]2 -[ui t+1 ]2+ × ( [wi t ]2–[ l i t ]2) ; 3 /* where li and wi are the genes of the loser and winner*/ end for θ=θ+1; end while return the best individual

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Algorithm 4 : Nonpermanent elitism-based compact Firefly Algorithm

Fig. 5 Pseudo-code of ne-cFA.

e) Opposition-based compact firefly algorithm

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Opposition-based optimization (OBO) is founded on opposition-based learning (OBL) [65, 66]. The idea is to define an opposite candidate for each candidate, called the opposite number [65]. The opposite candidates are usually generated in the initialization step. As the distance of the initial guess from the optimal solution determines the computation time, we can speed up the convergence by starting with a fitter solution compared to their opposites. This scheme is widely used in optimization, and several authors have proposed to improve the performances of FA using OBL [67-69].We can also cite: opposition-based differential evolution [70], opposition-based shuffled bidirectional DE [71], opposition-based PSO [72], opposition-based magnetic optimization [73], etc. The performance shown by such algorithms have motivated us to integrate it into compact optimization and especially into cFA. Thus, we propose opposition-based compact firefly algorithm (OBcFA). However, we start by analyzing the drawbacks of the traditional opposite candidate generation described in [65]. The opposite candidate of X is denoted OX. According to [65], the relation between X and OX is given by 45 =  + 6 − 5 (9) Where a, b are respectively the lower and upper search domain bounds. Traditionally [65], opposite candidates generation follows Eqs. (9). The main drawback of this method is the following. If we want to extend it for all generations, i.e. not only in the initialization setup, in such a way that after generating a candidate we also generate its opposite, when the algorithm draws near the optimal solution, the opposite numbers (opposite candidates), generated according to Eqs. (9), are very less fit than the candidates. Thus, the opposite candidates are not beneficial during the last generations. This phenomenon is represented in the right part of Fig. 6. As we can see, when the algorithm approaches towards the optimal solution, the opposite fireflies present a high fitness function. In the context of compact optimization, according to updating rules presented in Eqs. (7) and Eqs. (8), the standard deviation of the used PDF will increase as long as

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ACCEPTED MANUSCRIPT the winner and the loser are more distant. Hence, the algorithm cannot converge as long as the standard deviation is higher and this induces divergence. The opposite candidates can be generated differently. A new way is introduced herein. Let consider a PDF defined by Eqs. (5). Its mathematical shape and the shape of its CDF are represented in Fig. 7. In order to generate a new firefly, we generate a uniform number in the interval [0, 1] denoted Frand. The inverse of CDF at Frand is then calculated. We denote it Fr. Then, the firefly position is returned to the interval [a, b] by

X [i] = (b − a)

244 245

Fr[i] + 1 +a 2

(10)

We denote by OX the opposite candidate of X. To obtain its position, we start by evaluating OFRAND given by

OFRAND[i] = 1 − FRAND[i]

246

RI PT

238 239 240 241 242 243

(11)

247 248

Once OFrand is calculated, we estimate the inverse of CDF at OFrand, the obtained number is denoted OFr. Then, the opposite firefly position is returned to the interval [a, b] using Eqs. (12). The proposed method is illustrated in Fig. 7.

249

OX [i] = (b − a )

Fig. 6 Opposite candidate generation; left: the proposed method; right: the classical method.

AC C

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OFr[i ] + 1 +a 2

(12)

ACCEPTED MANUSCRIPT Gaussian PDF 4

PDF

3 2 1 0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

interval [-1,1] Gaussian CDF 1

Frand

0.6 0.4

OFrand=1-Frand

0.2

OFr [i]

0 -1

-0.8

-0.6

-0.4

-0.2

0

Fr [i] 0.2

0.4

interval [-1,1]

Fig. 7 Opposite candidate generation

0.6

0.8

1

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CDF

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Algorithm 5 : Opposition-based compact firefly algorithm (OBcFA)

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11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

N : Virtual population size λ : Initial standard deviation m : problem dimension t=1 /* iteration index */ for j=1:m do /* Initialization of PV */ µjt = 0; σj t = λ; end elite=generate( PV) /* Assume an elite */ θ=0; /* represent the actual length of inheritance*/ while Termination condition do /* Main loop, the termination condition can be fixed by the user */ G1=generate (PV(t)) OG1=generate opposite candidate of G1 [ winner1, loser1 ] = compete(G1, OG1) Move the loser1towards winner1 using Eqs. (3) for i=1 : m do /* updating PV */ µi t+1= µi t +1/N × (wit –li t ); 2 [σi t+1] 2 = [σi t ] 2 +[ui t ]2 -[ui t+1 ]2+ × ( [wi t ]2–[ l i t ]2) ; 3 /* where li and wi are the genes of the loser1 and winner1*/ end for (winner2, loser2)=compete (elite, winner1) elite=winner2 /* update the elite */ Move loser1 and loser2 towards the elite using Eqs. (3) evaluate the new positions loiser1 and loiser2 [winner, loser]=compete (loiser1, elite) elite=winner for i=1 : m do /* updating PV */ µi t+1= µi t +1/N × (wit –li t ); 2 [σi t+1] 2 = [σi t ] 2 +[ui t ]2 -[ui t+1 ]2+ × ( [wi t ]2–[ l i t ]2) ; 3 /* where li and wi are the genes of the loser and winner*/ end for [winner, loser]=compete (loiser2, elite) elite=winner for i=1 : m do /* updating PV */ 2 µi t+1= µi t + × (wit –li t );

M AN U

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

3

2

35. [σi t+1] 2 = [σi t ] 2 + [ui t ]2 - [ui t+1 ]2 + × ( [wi t ]2–[ l i t ]2) ; 3 36. /* where li and wi are the genes of the loser and winner*/ 37. end for 38. end while 39. return the best individual

257 258 259 260 261 262

Fig. 8 Pseudo-code of OBcFA In Fig. 6, we give a simple experiment, which consists of an optimization process, to show the difference between the proposed approaches for opposite candidates’ generation and the traditional way presented in [65]. The traditional opposite candidates’ generation can only be useful in the initial steps. However, the proposed scheme handles candidates and even their

ACCEPTED MANUSCRIPT 263 264 265 266 267 268 269 270 271 272 273 274 275 276

opposites over all iterations. The proposal is very suitable for compact optimization. Now that the procedure of generating opposite candidates is presented, we integrate it to compact firefly algorithm (cFA). The obtained algorithm is called opposition-based compact firefly algorithm (OBcFA). The idea is to generate, according to the procedure presented above, at each iteration a firefly and its opposite candidate. It allows each firefly to interact with its opposite candidate. Therefore, each firefly can move towards or attract its opposite candidate. The firefly is moved towards its opposite when this latter is fitter. Otherwise, the opposite candidate is moved towards the firefly (which is also its opposite). If the candidate and its opposite are not better than the best, they will be both moved towards it. The proposed strategy does not compromise on the computational cost. The complete OBcFA pseudo-code is shown in Fig. 8.

277

f) Opposition-based compact Lévy-flight firefly algorithm

278 279 280 281 282 283

This subsection introduces the Lévy-flight version of OBcFA. The algorithm is called Opposition-based compact Lévyflight firefly algorithm (OBcLFA). The aim is to investigate the influence of Lévy distribution [16, 62-63], recommended for FA [27], on the performance of OBcFA. The OBcLFA herein introduced is similar to the OBcFA presented above (see pseudo-code shown in Fig. 8). The only difference is that the moving step is based on the Lévy-flight distribution. Hence, the pseudo-code of OBcLFA is the same as that of OBcFA, except that we replace Eqs. (3) by Eqs. (4) in lines 14 and 22 of the pseudo-code presented Fig. 8.

284

4.2.

285

a) Uniform PDFs and their interest

286 287 288

All the known compact algorithms use the normal PDF (NPDF) to represent the population, e.g. cGA [31], cDE [32], and cPSO [33]. In this paper, we investigate the influence of the chosen PDF on the performance of the proposed compact firefly algorithms (cFAs).

289 290 291

One of the drawbacks of normal PDF is the computational complexity of its inverse CDF. It is constructed by means of Chebyshev polynomials according to the procedure described in [74] which is used by [31-33]. For the simplicity, we propose a simpler way and we will analyze its performance in the next section. The proposal consists of uniform PDF (UPDF).

294 295 296

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SC

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∈ [, 6] 789: (
) = ;< = 0 ?@ℎB,+ 

We integrate (13) and we obtain 7C9: (
)

EP

293

Compact firefly algorithms with uniform PDF

AC C

292

In OBcFA, the PV (PDF’s means and standard deviation) is updated three times per cycle. First, in line 15 of Fig. 8, the PV is updated using the alleles of the generated firefly and its opposite candidate. In the second and the third time of updating PV, we take into consideration the elite and the moved fireflies. The PV is also updated, in a second and a third time, line 26 and 33 respectively, using the alleles of the elite and the moved fireflies. After each competition (comparison) between the elite and a given firefly, the elite is updated if it is surpassed, i.e. in lines 21 and 32 of Fig. 8.

7C9: (
) = D



< =


−

=

< =


∈ [, 6]

1
> 6 0 ?@ℎB,+

(13)

(14)

The inverse function of UCDF, denoted U-1CDF, in the interval [0, 1] is given by  () 7C9: = (6 − )  + 

(15)

297 298 299

Each allele, i.e. optimization parameter, in the interval [a, b], is represented by a UPDF. Unlike NPDF presented in section 2, the bounds a and b of a UPDF vary if its mean and its standard deviation vary. The relationships between the mean (µ), standard deviation (σ), a and b are expressed by Eqs. (16) and Eqs. (17)

300

(16)

301 302 303 304

6 = √3 I + J

 = J − √3 I

(17)

We note that equations Eqs. (16) and Eqs. (17) can be easily found by computing the mean (µ) and the standard deviation (σ) of a given UPDF defined in an interval [a, b]. Now, if we want to sample (generate) an individual from PV defined by a UPDF, we start by calculating a and b given in Eqs. (16) and Eqs. (17). Then we generate a uniform rand number in the

ACCEPTED MANUSCRIPT 305 306 307

interval [0, 1], and then we calculate the position using Eqs. (15). The graphical representation of UPDF, its CDF and the sampling mechanism are shown in Fig. 9. Uniform PDF in interval [-0.5, 0.15] 2

PDF

1.5 1

0 -1

-0.5 a

0

b

0.5

search space Uniform CDF in interval [-0.5, 0.15] 2

opposite candidate

1

r=rand(0,1)

0.5

or=1-r

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0 -1

-0.5 OFr

Fr 0 search space

0.5

1

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Fig. 9 Generation of a candidate and its opposite from uniform PDF Fig. 9 also illustrates the opposite candidate generation from a UPDF. The candidate (firefly) is generated from the random number r, its opposite is generated using or. Even though PDF functions are different, the mechanism shown in Fig. 9 is similar to that of Fig. 7. b) Uniform compact Firefly algorithms In the previous subsections we have proposed six new variants of compact firefly algorithms: i.e. pe-cFA, ne-cFA, pecLFA, ne-cLFA, OBcFA and OBcLFA. All these algorithms use NPDF to represent the population. We note that, in the field of compact optimization, we have not found works that analyze and propose other distributions functions than NPDF. This has motivated us to investigate another way using UPDF. Thus, here we propose six new variants consisting of the introduced pecFA, ne-cFA, pe-cLFA, ne-cLFA, OBcFA and OBcLFA that use uniform PDF. A UPDF is characterized by the simplicity of its mathematical representation. Its CDF and the inverse of its CDF are very simple to calculate. The use of UPDF will consequently reduce the computational cost and will simplify more compact optimization algorithms. This has motivated us to propose UPDF-based cFAs. The idea is to make use of uniform distributions to represent the population rather than Gaussian distributions. It is not necessary to give the pseudo-code of the proposed UPDF-based cFAs given that they follow the same scheme of NPDF-based cFAs; the only difference is in the representation of the variables. So, before sampling a firefly from UPDF, the bounds must be calculated using Eqs. (16) and Eqs. (17). Another advantage of UPDF consists of the fact that it gives the same probability for the entire virtual population in the interval [a, b], i.e. there is no region which is privileged. This means that a UPDF leads to diversity enhancements. Furthermore, as the limits of the interval a and b vary according to equations Eqs. (16) and Eqs. (17), the UPDF’s limits will retract towards the optimal subdomain when this latter is in [a, b]. When the solution is out of the interval [a, b], according to Eqs. (16) and Eqs. (17), UPDF will undergo an expansion. Therefore, the algorithm can find a solution even if the latter is out of the initial interval [a, b]. One can see that UPDFs are very suitable. The performance of the twelve herein proposed algorithms will be tested and analyzed in detail in the next section. 4.3. On the complexity of the attraction model of the proposed algorithms The computational complexity of the standard FA is O(Gmax ×N2×f ) [24], where Gmax is the maximum number of generations, N is the population size, and f stands for the fitness function. However, as N is usually not very large, the number of attractions during Gmax generations could be more important. The proposed algorithms use the same attraction model as classical FA. But the number of attractions per fitness evaluation is not the same. Hence, we will just compare the number of attractions performed during an optimization task by classical FA, RaFA, NaFA and the proposed cFAs during a FES fitness evaluations. Wang et al. have analyzed the complexity of the attraction model of FA [24]. FA’s attractions model is known as full attraction model (FAM) [24-26]. FA requires the comparison of each firefly with all the N agents in the swarm. At each comparison, one of the fireflies is moved. This means that each one is moved with an average of (N-1)/2 times per generation [24], where N is the number of agents in the swarm. Thus, as there are N fireflies in the swarm, at each generation, N×(N-1)/2

AC C

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candidate

1

SC

CDF

1.5

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0.5

ACCEPTED MANUSCRIPT attractions are performed by classical FA. The number of attractions during a determined FES fitness evaluation is FES*(N1)/2. In order to reduce the number of attractions pe r generation in FA, i.e. to reduce the complexity of the attraction step, Wang et al. have proposed a random FA (RaFA) in which each firefly is attracted only by a randomly selected one in the population [25]. The authors have reduced the number of attractions to N attraction per generation [25]. Hence, for a determined FES fitness evaluations, FES attractions are performed. This method simplifies the attraction model and limits the oscillations induced by the FAM, but RaFA can result in a premature convergence in multimodal problems because of the poverty of its attraction model [24]. Another attempt, called neighborhood FA (NaFA), was proposed in [24]. In NaFA [24], each firefly can interact with pre-defined k fireflies, i.e. k-neighborhoods. Hence, the number of attractions per generation is k×N. Therefore, k×FES attractions are performed to solve a given problem. The authors have shown that k is an important factor in the performance of NaFA. In contrast, our approaches lead to less attractions per fitness evaluation. In the case of pe-cFA and pe-cLFA, we have FES/2 attractions to achieve the optimization. In the case of ne-cFA and ne-cLFA, we have an average of FES/2 attractions to achieve the optimization. In the case, OBcFA and OBcLFA, ¾ FES attractions are done to achieve an optimization task. So, the complexity of attraction model is systematically reduced. The advantage of our proposal is that not only the complexity of attraction model is reduced but also the memory storage. The population is replaced by the virtual population which is statistically represented. Furthermore, as a bonus, each firefly can affect the whole population using the model described in Eqs. (7) and Eqs. (8). In addition, the introduced UPDF more simplifies the sampling mechanism. Thus, our approach uses a strict minimal computation cost. In conclusion, during FES fitness evaluations, the number of attractions in classical FA is FES*(N-1)/2, in RaFA is FES, in NaFA is k×FES. But in the proposed cFAs the number of attractions is reduced to FES/2 for pe-cFA, pe-cLFA, ne-cFA and ne-cLFA, and reduced to ¾ FES in the case of OBcFA and OBcLFA. These results are available for both UPDF-Based cFA and NPDF-based cFAs. Hence, the proposed approaches present the most minimal computational complexity.

5.

Experimental study

368

5.1.

Benchmark functions

369 370 371 372 373 374 375 376 377 378 379 380

This section presents and analyzes the numerical results. The proposed algorithms are benchmarked on 30 test functions, with 30 dimensions (D=30), taken from IEEE CEC2014 [75]. All the test functions are defined in the search space [-100, 100] D . Each function is tested 25 times. The maximum number of fitness evaluation is set to 10000*D as it is suggested in [75]. Despite the fact that the number of fitness evaluations per generation differs from an algorithm to another, each fitness evaluation is counted. The termination condition of the algorithms used in this comparative study is based on the maximum number of fitness which is set to 10000*D [75]. The function error value (f (xbest)−f (x*)) of each run is recorded, where x* is the optimal solution and xbest is the best solution found at the end of a run. The average and standard deviations of the function error values in all runs (denoted as “Mean” and “SdD.”) are considered as two-performance metrics to assess the performance of the algorithms. Moreover, Wilcoxon’s rank sum test at a 0.05 significance level is used to test the statistical significance between pairwise algorithms [76]. In addition, the proposed algorithms are ranked using the Friedman test [76, 77].The algorithms are implemented in Matlab 2014a code source. The computer used for our simulations is Intel® Core™ i52350M CPU 2.30 GHz and 8 GB of RAM.

381

5.2.

382 383 384 385 386 387 388 389 390 391

In order to assess the performance of the twelve proposed algorithms, they are compared here to three sets of well-known effective algorithms. The first set consists of compact evolutionary algorithms (cEAs). The second set consists of swarm intelligent algorithms (SIAs). We note that the proposed algorithms are inspired from swarm intelligence but the information of swarm is compacted and represented by mathematical PDF. Hence, we consider the proposed algorithms as compact swarm intelligent algorithms (cSIAs). This concept is not yet explored in the literature [61]. Thus, this paper trend to take a step towards this concept. In cSIAs, the approaches are the same as those of swarm intelligence but populations, positions, food, etc. are compacted, i.e. represented by probability functions. Therefore, in order to state the performance of proposed cSIAs, we must compare them to the same swarm intelligent algorithms. The third set consists of effective modern variants of evolutionary algorithms (EAs). The comparison of our algorithms to these advanced variants will give us an overview of the limits of the proposed approaches.

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5.3.

393

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343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367

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Algorithms used for comparison

Algorithms’ parameters

A) The of the proposed cFAs’ parameters.

ACCEPTED MANUSCRIPT 394 395 396 397 398

We report in Fig. 10 the performance of NPDF-based pe-cFA on two benchmarks F2 and F13 with D=30. Where F2 is a unimodal problem and F13 is a multimodal problem. Both are taken from the benchmark tests of IEEE CEC2014 [75]. In this experiment, we make N vary form N=30 to N=600 and K from K = 2 to K = 35. For each value of N and K, the test is repeated 10 times and then we compute the average. The results are presented in Fig. 10. From the contour of the 3D plot, we can see that the algorithm gives a better result when λ=10 and N=10×D=300.

399 F2

F13 6

650

600 1000

500 0

VP's size (N)

40

20 Standard deviation

0

4

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fitnees

fitnees

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2

0 1000

500 VP's size (N)

0

500 VP's size

500 400

100

400 300

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F13

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40 30 20 Standard deviation

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VP's size

F2 600

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0

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5

10 15 20 25 Standard deviation

400

30

35

5

10 15 20 25 Standard deviation

30

35

401

Fig. 10 Average of the fitness value obtained by pe-cFA on benchmarks F2 (right) and F13 (left) of IEEE CEC2014.

402 403 404 405

The parameters of UPDF-based cFAs are calculated as follow: UPDF is centered, we have initially µ=0 and b=100 (the = 

upper bound of the search space), then we calculate K using equation Eqs. (13). Hence, K = = = 57.7350. This

TE D

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√M

B) The parameters of the algorithms used for comparison

The parameters of each algorithm used in this experimentation are taken from the literature. They are reported in Table 1.

EP

406

√M

parameter represents the initial value of I. The tuning of other parameters, i.e. β0, γ and α, is inspired from the work of XinShe Yang [79]. They are taken as follow: α 0=0.25, β0=0.20 and γ=1.

409 410

pe-cGA [31]

ne-cGA[31]

AC C

Algorithms

n=200

K = 10

Parameters

n=200 η=100

K = 10

Table 1 Algorithms’ parameters pe-cDE [32]

ne-cDE [32]

cPSO [33]

cTLBO [34]

Np=60

Np=60

Np=300

Np=300

K = 10

K = 10

η=100

K = 10

K = 10

Φ1 =0.2, Φ2 = 0.07, Φ3 = 3.74, c1 = 1, and c2 = 1.

Algorithms

FA & LFA [78] N=30 α0=0.25

SLPSO[80] N=30 M=100

CoBiDE[81]

OXDE[82]

CODE[83]

SaDE[84]

N=30

NP = 30

N=30

N=50

-Strategy:

F = 0.9

1- Strategies 1) “rand/1/bin”;

F, CR are self adaptive

ACCEPTED MANUSCRIPT Parameters

β0=0.20 γ=1

α=0.5

rand/1/bin

CR = 0.9

2) “rand/2/bin”; 3) “current-to-rand/1”. 2-Parammeters

pb = 0.4,

-Strategy:

ps = 0.5.

DE/rand/1/bin

parameters

1) [F = 1.0, Cr = 0.1]; 2) [F = 1.0, Cr = 0.9]; 3) [F = 0.8, Cr = 0.2].

411 412

Table 2

Experimental results of pe-rcGA, ne-rcGA, pe-cDE, ne-cDE, cPSO, cTLBO, normal pe-cFA, normal ne-CFA over 25 independent runs on 30 test functions of 30 variables with 300,000 FES. “Means” and “StD” indicate the average and standard deviation of the function error values obtained in 25 runs, respectively. ne-cDE 2.5121E7 1.6949E7 7.7172E3 7501.345 32554.34 19099.83 143.2137 31.58930 20.24295 0.172997 9.046724 2.918259 0.013430 0.015942 31.55111 6.181573 32.49893 9.750440 1368.826 459.4891 2580.560 645.6110 0.382341 0.225341 0.300150 0.077900 0.398677 0.080043 6.453839 2.826148 11.60754 0.642657 1.9150E6 1.6670E6 2660.337 3326.768 22.14020 22.82207 3.4580E4 1.8899E4 6.9171E5 5.7411E5 421.2512 203.7960 315.6271 0.537860 234.3337 5.599290 214.4956 3.938666 127.3166 49.98393

cPSO 2.4995E7 2.3088E7 1.1774E4 4277.744 50997.92 32101.85 73.82909 9.832563 20.90826 0.068868 2.458911 1.697425 4.910816 7.379351 119.0778 16.84161 123.8396 33.10723 1207.340 650.2743 4155.123 779.8695 2.290121 0.403891 0.095830 0.016103 0.430972 0.035018 3.807756 0.951385 12.88967 0.119158 8.8945E6 4.6290E6 587.2613 672.4460 7.475836 2.250322 70856.98 19551.45 2.3389E6 1.5945E6 84.17412 64.07763 301.4148 37.44990 200.8364 0.068885 200.0000 2.273736 200.0072 0.002644

cTLBO 2.4743E8 3.0602E7 4.2314E9 9.4725E8 43060.65 5318.255 501.8909 54.47159 20.94141 0.050145 17.31376 1.857338 41.82243 5.551956 102.4903 8.767605 185.4046 29.16267 2947.128 533.8589 5029.304 812.1966 2.076128 0.673974 0.961914 0.651418 10.80495 4.589137 52.58306 22.86195 12.58763 0.232524 8.3652E6 3.0008E6 551.4883 470.9240 76.25553 9.195112 22579.20 5143.285 1.3123E6 6.3929E5 598.1669 174.5048 358.5587 3.832806 200.0000 1.261024 200.0000 1.327022 108.1470 27.08753

pe-cFA 9.8360E6 1.2537E7 1379.036 6755.752 608.7337 1705.655 102.4210 40.19495 20.20928 0.269940 15.35241 4.199814 0.009160 0.008307 83.68709 25.03764 89.10027 18.75452 2883.510 538.3599 3678.531 590.3333 0.464030 0.459535 0.439784 0.090154 0.267330 0.053561 6.445791 2.074163 12.39033 0.429704 1.7984E5 3.6170E5 1705.945 2125.453 16.06878 5.164228 3240.405 2606.432 66729.45 87781.17 515.0630 217.4455 317.5292 3.428078 228.2157 3.874118 212.9412 2.146389 124.3316 42.54021

SC

pe-cDE 2.7007E7 1.5939E7 1.2016E4 9038.292 53786.57 40353.72 168.4092 44.41905 20.04539 0.077099 30.56642 3.979747 0.492646 0.847709 112.5440 30.35302 145.1294 52.78022 3535.975 924.8130 4781.577 557.3375 1.680707 0.7641085 0.466005 0.116271 0.324923 0.175519 1706.552 1690.456 13.07618 0.364157 3.4903E6 3.5951E6 4.8114E3 4884.317 103.9494 76.19329 5.0312E4 2.8130E4 8.1783E5 7.3492E5 870.7301 326.0758 317.3702 3.551247 250.9901 6.407442 227.7318 8.063287 159.9093 66.50269

M AN U

ne-rcGA 1.5847E7 1.1543E7 1.1235E4 8885.174 4.1563E4 2.4154E4 147.3855 33.81822 20.01729 0.044276 10.55761 2.492981 0.026770 0.033933 43.76262 12.85336 52.09911 11.02813 2067.936 565.3012 2909.981 612.1179 0.172516 0.090099 0.338289 0.078828 0.343401 0.052622 9.486343 4.789198 12.11649 0.560284 1.5140E6 9.6466E5 4218.988 5390.635 25.93472 22.22380 3.0323E4 1.2639E4 7.3166E5 6.1142E5 529.1435 218.3069 315.5452 1.150287 237.8945 4.757349 216.7693 4.070577 124.4377 42.82045

TE D

pe-rcGA 6.1775E7 7.510E7 1.4989E6 7.2524E6 6.8264E4 4.8778E4 235.5145 93.06517 20.27571 0.273812 28.26883 4.033062 0.559368 0.768745 148.3930 42.40161 140.9210 48.92422 3859.884 881.9020 4964.486 929.4055 1.561998 0.713748 0.492678 0.145036 0.326922 0.122057 1057.490 1226.943 13.13839 0.558361 4.2846E6 3.5474E6 5572.433 5713.241 77.68631 30.56548 5.8076E4 4.3299E4 1.5501E6 2.3585E6 804.9517 256.8120 317.5246 2.322643 252.2659 7.833692 228.4817 9.448906 167.4849 62.60428

EP

Function F 1 Means StD F 2 Means StD F 3 Means StD F 4 Means StD F 5 Means StD F 6 Means StD F 7 Means StD F 8 Means StD F 9 Means StD F 10 Means StD F 11 Means StD F 12 Means StD F 13 Means StD F 14 Means StD F 15 Means StD F 16 Means StD F 17 Means StD F 18 Means StD F 19 Means StD F 20 Means StD F 21 Means StD F 22 Means StD F 23 Means StD F 24 Means StD F 25 Means StD F 26 Means StD

AC C

414 415 416

RI PT

413

ne-cFA 1.0815E7 1.3484E7 1563.859 7661.183 81.82413 171.5210 115.0932 38.45409 20.15960 0.258002 11.36745 3.422150 0.006304 0.009134 51.06152 12.15603 71.92963 20.74118 2395.067 811.1477 3032.817 665.7516 0.389244 0.325460 0.414828 0.118204 0.270432 0.041381 5.969381 2.103608 11.53813 0.715996 8.7094E5 9.3891E5 1.2901E3 1250.1014 15.021021 4.7444735 5585.9101 3413.4052 93482.272 130116.97 458.52299 165.25604 316.72467 2.5668653 231.26377 6.4009961 212.92984 2.8328264 128.32895 44.733156

ACCEPTED MANUSCRIPT F 27 Means StD F 28 Means StD F 29 Means StD F 30 Means StD

913.8140 262.9263 2732.278 900.7618 7.9588E5 2.6842E6 5.9500E4 6.3306E4

620.2015 85.08544 1274.096 281.4757 1.163372 1.968922 23949.60 14488.62

988.1042 176.6211 2510.430 751.5803 7.1584E5 2.5450E6 3.5222E4 2.1479E4

564.1921 117.8953 1165.485 273.1807 2.3872E5 4.1511E5 2.3343E4 2.2656E4

428.7588 162.4599 1003.407 100.8980 1.3523E6 6.4924E6 12628.08 6946.630

618.7316 104.2638 1793.580 789.4957 3.7500E6 7.3456E6 3.3951E5 6.1856E4

506.7060 148.0933 1763.530 625.2309 3131.102 3619.982 21862.05 13789.71

632.75418 121.69199 1164.2752 435.28396 3605.3544 4366.8473 5900.6496 14928.138

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ACCEPTED MANUSCRIPT Table 3 Experimental results of FA, SLPSO, LFA, SaDE, OXDE, normal pe-cLFA, normal ne-cLFA, normal OBcFA and normal OBcLFA over 25 independent runs on 30 test functions of 30 variables with 300,000 FES. “Means” and “StD” indicate the average and standard deviation of the function error values obtained in 25 runs, respectively. Function FA SLPSO LFA SaDE OXDE pe-cLFA ne-cLFA OBcFA OBcLFA 4.9518E6 3.7710E5 1.3238E7 1.6107E7 F 1 Means 2.8895E6 4.7519E6 1.50964E5 3.0351E7 2.6514E7 2.0965E6 2.8256E5 3.2861E7 3.0371E7 StD 1.8735E6 3.7585E6 2.1147E5 2.8443E7 3.1533E7 3.6076E6 6.252E-14 0.02454 0.0209 F 2 Means 8.8006E3 1.2927E4 0.0 0.240988 7.2128E7 4.9933E5 1.942E-13 0.04649 0.0354 StD 8.2315E3 8.0119E3 0.0 0.260297 8.2607E7 1.0347E4 16.59538 21.8999 12.3079 F 3 Means 8.8758E3 7936.416 1.364E-14 2.0877E2 1.6474E6 4.1385E3 32.45895 25.5723 18.5938 StD 3.6859E3 4767.299 3.694E-14 5.0636E2 2.5602E4 1.0448E2 35.41133 127.5052 121.7514 F 4 Means 9.2590E1 32.0001 2.553034 7.9607E1 1.2786E4 31.4687 8.850978 33.05580 56.27990 48.9217 StD 11.1487 12.42053 3.9677E1 3.4076E3 20.9261 2.0942E1 20.5468 20.8937 F 5 Means 20.0022 19.99910 19.9989 2.0811E1 2.0932E1 0.06350 0.042743 0.0546 0.1043 StD 0.0020 4.6083E-4 5.9939E-4 0.243710 0.106419 0.8358 2.694082 5.0729 1.4695 21.2528 18.5242 F 6 Means 0.7208 3.9506E1 3.9897E1 0.764700 0.430728 2.2569 0.7702 3.2715 3.7557 StD 0.5568 4.217365 1.905767 1.028710 0.0072 0.0028 0.0104 0.0060 F 7 Means 0.0050 1.45E-13 0.093467 7.7411E2 0.009905 0.0113 0.0049 0.0092 0.0081 StD 0.0040 5.10E-14 0.109572 1.1181E2 17.854 3.076432 13.5314 142.5659 142.5173 F 8 Means 28.2571 0.0795 6.4251E 3.7205E2 6.3046 9.392264 4.3958 18.0877 16.1275 StD 9.6053 0.2699 2.3396E1 2.1471E1 18.7322 2.9701E1 37.8083 25.7097 170.3361 F 9 Means 23.1234 167.1523 3.0351E6 2.6514E7 4.6106 6.991817 9.9694 7.5179 19.9049 StD 7.0031 22.5886 2.8443E6 3.1533T7 1.6163E3 0.2670 125.8422 2.5876E3 2.5725E4 F 10 Means 1.3836E3 380.2653 0.240988 7.2128E7 7.1505E2 0.3983 109.3647 453.2316 399.4921 StD 5.5404E2 236.4876 0.260297 8.2607E7 4.5875E3 3.5581E3 3.3887E3 F 11 Means 1.5955E3 8.9215E3 1.9740E3 3.1772E3 2.0877E2 1.6474E7 2.4190E3 540.0940 660.5649 StD 7.4229E2 2.9807E2 1.0514E3 530.1398 5.0636E2 2.5602E4 2.2885 0.299427 0.7803 2.0521 0.2122 0.1356 F 12 Means 0.0094 7.9607E1 1.2786E4 0.4931 0.045565 0.1058 0.49170 0.1398 0.0892 StD 0.0031 3.9677E1 3.4076E3 0.1668 0.2628 0.2882 0.5160 0.4803 F 13 Means 0.1767 0.193608 2.0811E1 2.0932E1 0.0324 0.0395 0.0473 0.1057 0.1099 StD 0.0383 0.043060 0.2437E1 0.106419 0.3992 0.313670 0.2898 0.2845 0.2815 F 14 Means 0.2991 0.2334 3.9506E1 3.9897E1 0.0775 0.045378 0.0429 0.1120 0.0410 StD 0.0350 0.0328 4.217365 1.905763 5.4012 9.387063 4.6341 9.0842 8.3790 4.7011 F 15 Means 3.2826 0.093467 7.7411E2 4.2750 1.338882 2.0624 5.1162 3.8982 1.1452 StD 0.6391 0.109578 1.1181E2 12.0562 1.2117E1 10.9932 11.8086 12.5210 12.0689 F 16 Means 11.5315 6.4251E1 3.7205E2 0.3460 0.536276 0.3765 0.3639 0.4926 StD 0.4722 0.3739 2.3396E1 2.1471E1 1.0007E6 1.2355E6 F 17 Means 1.0823E5 8.4737 E4 2.7311E5 9.9866E3 2.6762E3 3.0351E6 2.6514E7 1.4042E5 7812.176 2.0271E6 1.9448E6 StD 8.4909E5 4.2118E4 3.2676E3 2.8443E6 3.1533E7 1.0280E5 279.5385 1415.797 1044.897 F 18 Means 2005.282 774.3159 13.4962 0.240988 7.2128E7 2.3735E4 236.8633 655.5064 656.7539 StD 2285.302 1031.105 8.1842 0.260297 8.2607E7 6.9148 9.407984 4.1949 25.9596 25.2339 F 19 Means 9.5086 3.0163 2.0877E2 1.6474E5 1.0974 2.020704 1.2754 3.5056 5.5867 StD 1.7563 1.1627 5.0636E2 2.5602E4 7.0573E2 129.4737 5461.806 4236.490 F 20 Means 592.0731 2.0878E4 12.0762 7.9607E1 1.2786R4 8.8844E2 148.8812 2874.599 2733.764 StD 512.6437 9.8820E3 4.0357 3.9677E1 3.4076E3 7.0572E4 3297.063 31026.99 47188.23 F 21 Means 71358.32 1.0165E5 342.7553 2.0811E1 2.0932E1 5.2488E4 3901.271 38245.52 54518.62 StD 50080.64 9.5516E4 222.6483 0.243710 0.106419 2.3417E2 142.5669 788.3575 741.9728 F 22 Means 198.5338 147.8259 135.0107 3.9506E1 3.989784 9.7300E1 65.96675 231.6572 205.9960 StD 89.67583 97.98714 122.4289 4.217365 1.905767 3.1561E2 315.2441 315.2441 280.284 267.8035 F 23 Means 315.3579 315.2441 0.093467 7.7411E2 1.272912 0.125175 2.27E-13 1.25E-13 65.9873 65.6201 StD 0.1317 0.109578 1.1181E2 2.1387E2 226.7343 226.7058 209.8227 202.2906 F 24 Means 210.5534 229.0262 6.4251E1 3.7205E2 5.803405 1.0194E1 2.939845 4.9450 6.6993 3.51027 StD 11.5412 2.3396E1 2.1471E1 2.0571E2 208.6687 200.0000 200.2410 F 25 Means 205.6842 205.1102 203.0015 3.0351E6 2.6514E7 1.1473 0.850339 2.4316 5.268E-13 1.18092 StD 1.2774 0.3624 2.8443E6 3.1533E7 1.0019E2 108.2566 100.2804 200.0113 200.0071 F 26 Means 100.1535 124.1478 0.240988 7.2128E7 42.6505 0.038979 27.0611 0.05281 0.0048 0.0059 StD 0.0355 0.260297 8.2607R7 3.9479E2 416.8225 355.9377 2034.811 1896.600 F 27 Means 328.6778 380.8583 2.0877E2 1.6474E5 43.4091 4.2512E1 33.7438 40.2606 1063.595 967.0348 StD 34.4151 5.0636E2 2.5602E4 9.7031E2 888.6709 844.7182 2557.480 2222.932 F 28 Means 843.5430 885.8657 7.9607E1 1.2786E4 34.48397 2.0738E2 33.5962 32.3567 1237.776 932.1751 StD 31.6400 3.9677E1 3.4076E3 1.2389E7 1.0171E3 6.2608E6 1.9609E4 F 29 Means 2.0862E4 1.7535E3 845.1462 2.0811E1 2.0932E1 9.7563E6 157.3320 3.0971E6 8.3051E4 StD 1.6578E3 485.3746 156.9603 0.243710 0.106419 7.2661E3 1613.099 1.0367E4 1.0800E4 F 30 Means 5460.613 3283.203 828.0993 3.9069E1 3.9897E1 2.3256E3 494.3105 1.6062E4 1.4664E4 StD 1290.831 1294.858 418.7854 4.217365 1.905767

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5.4.

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The proposed algorithms, consisting of pe-cFA, ne-cFA, pe-cLFA, ne-cLFA, OBcFA and OBcLFA, are compared here to six state-of-art of compact evolutionary algorithms (cEAs). The comparison includes: permanent elitism-based real-valued compact genetic algorithms (pe-rcGA) [31], non-permanent elitism-based real-valued compact genetic algorithm (ne-rcGA) [31], permanent elitism-based compact differential evolution (pe-cDE) with “/rand/1/” strategy [32], non-permanent elitismbased compact differential evolution (ne-cDE) with “/rand/1/” strategy [32], compact particle swarm optimization (cPSO)[33] and compact teaching-learning-based optimization (cTLBO) [34]. Their parameters are taken from their original literature (see Table 1). The results given by cEAs listed above are shown in Table 2. A. NPDF-based cFAs

432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450

We start by comparing the six proposed cFAs that use NPDF. The results given by pe-cFA and ne-cFA are presented in Table 2. The results given by pe-cLFA, ne-cLFA, OBcFA and OBcLFA are presented in Table 3. In order to compare the proposed algorithms to the cited cEAs, Wilcoxon’s rank sum test at 0.05 significance level is used to test the statistical significance of the difference between pairwise algorithms. As we have much comparison to do, the results of Wilcoxon’s rank sum test are presented in a compact format in Table 4. Each proposed algorithm is compared to the cited cEAs. When the proposed algorithm is better than a given cEA for a given benchmark function, the sign ‘+’ appears. If the algorithms are equivalent, i.e. no significant difference, the sign “=” appears. If one of the proposed algorithm is less effective than one of the cEAs, the sign “-”appears. In order to clarify and to allow a precise reading of the tables that present the results of Wilcoxon’s rank sum test, we give an example. In Table 4, the intersection between the first columns and the first line illustrates the comparison between pe-cFA and pe-rcGA. The obtained result means that the proposed pe-cFA with NPDF is not more effective than pe-rcGA for problems: F9, F27 and F29; and it is equivalent to pe-rcGA for problems F8, F13, F22, F23, F26, F28 and F30. For the remaining twelve functions, the proposed pe-cFA is better than pe-rcGA. If we count the number of successes, i.e. “+”, equivalences, i.e. “=”, and failures, i.e. “-”, of each pairwise comparison we obtain the results shown in Table 5. From the latter, one can see that the proposed algorithms are very competitive except for OBcLFA that is outperformed by all cEAs used here for comparison. We also note that cPSO is only surpassed by ne-cLFA. The comparison between cPSO and OBcFA gives 13 “+”s and 13 “-”s, we can conclude that they are equivalent for the considered IEEE CEC2014 benchmarks. The ne-cDE also outperforms the proposed pe-cFA, but ne-cFA, pe-cLFA, ne-cLFA and OBcFA surpass it. Table 5 presents more details.

451 452 453 454

Table 4 Wilcoxon’s rank sum test at a 0.05 significance level is performed between each one of the proposed normal cFAs with each pe-rcGA, ne-rcGA, pe-cDE, ne-cDE, cPSO, cTLBO. For each benchmark fi, “+” means our approaches is better, “-” means our approaches is worst, “=” means equivalence between the pairwise algorithms.

Normal

Distribution

Normal

455 456

Algorithm pe-cFA ne-cFA pe-cLFA ne-cLFA OBcFA OBcLFA Compared to Algorithm pe-cFA ne-cFA pe-cLFA ne-cLFA OBcFA OBcLFA

ne-rcGA Functions : From 1 to 30

cPSO Functions : From 1 to 30

+++++++=-+++=++++++++==++=-=-=

+++=+-+----=-+=-+=-++-=++---+=

+++-+-----++-+-++--++-=-------

+++++++=-+++==+++++++==++=-=++

-++++-+----=-++=+++++-=++---++

+++-+-=---++-+-++--++-=--=--++

++++--+++++-==+=++++++++++++=+

++++--=------=--+=+++=+==+=+=+

+++---=+=-=--+--+--++-=--+-=++

++++--+++++++-+-++++++++++++++

++++--+++++-+-+-++++++++++++++

+++----+++++--+-+-=++----+=+++

++++--+++++-==++++++++++++++++

++++-----=---==-+=+++=+=+==-=+

+++-=--++-+=-+-++--++-=--+--+=

---- - -- - - - - - - - - + - --=== - - - = - + + +

-------------------------+--==

----=----------=+--==----+--+-

pe-cDE Functions : From 1 to 30

+++++++--+++==+++=+++==++=-=-=

ne-cDE Functions : From 1 to 30

cTLBO Functions : From 1 to 30

+++=+-=----+-+=-+=-++-=++-----

+++++-+-++++=++=+-+++-+------+

+++++++--+++==+++++++==++=-=++

+++++-+----+-+=-+=-++-=++---++

+++++=+-++++=++++-+++-+----=++

++++--+=+++-==+=+=+++++++=++=+

++++--=------+--+==++=+-=+===+

++++=-+=+=+-=++-+-+++++--+=+++

++++--+++++++-+-++++++++++++++

++++--+++++++-+-+=+++++++=++++

++++--+++++++++-+-+++++--+++++

++++--=++++-==+++=++++++++++++

++++---------+--+==++=+==+=-==

+++++-+++++==++=+-+++++--+==++

---------------=---=-=---=-+==

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----=-----------+---=----+-=++

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Comparison to compact evolutionary algorithms (cEAs)

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B. UPDF-based cFAs

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The performance of compact optimization algorithms also depends on the probabilistic function that is used to represent the population. Hence, in this paper we investigate its effect. We compare the proposed cFAs, i.e. pe-cFA, ne-cFA, pe-cLFA, necLFA, OBcFA and OBcLFA, that use UPDF to represent the population, to the set of cEAs presented above. The results of each proposed UPDF-based cFAs are shown in Table 12. Table 6 gives the results of the pairwise comparisons between UPDF-based cFAs and cEAs. Table 7 summarizes the results presented in Table 6. The proposed algorithms, that use UPDF to store the population, surpass the cEAs used here except in

ACCEPTED MANUSCRIPT four cases, which are: ne-rcGA, ne-cDE and cPSO surpass pe-cLFA. The ne-cDE surpasses OBcFA. Unlike the OBcLFA that uses NPDF, the OBcLFA implemented using UPDF is only surpassed by pe-cDE and cTLBO. Table 5 Summarized results of Wilcoxon’s rank sum test at a 0.05 significance level between each one of the proposed normal cFAs with each pe-rcGA, ne-rcGA, pe-cDE, ne-cDE, cPSO, cTLBO Comparison results between the proposed cFAs and a set of cEAs

Normal

pe-cFA ne-cFA pe-cLFA ne-cLFA OBcFA OBcLFA

pe-rcGA + = 20 7 3 21 7 2 23 4 3 26 0 4 25 2 3 4 4 22

pe-cDE + = 18 8 4 21 6 3 20 7 3 26 0 4 23 4 3 1 6 23

ne-cDE + = 11 5 14 15 3 12 11 8 11 24 2 4 10 8 12 1 3 26

+ 11 13 11 16 13 3

cPSO = 1 18 3 14 5 14 2 12 4 13 4 23

cTLBO + = 18 2 10 20 3 7 19 5 6 24 0 6 21 5 4 4 3 23

Table 6 Wilcoxon’s rank sum test at a 0.05 significance level is performed between each one of the proposed UPDF-based cFAs with each pe-rcGA, ne-rcGA, pe-cDE, ne-cDE, cPSO, cTLBO. For each benchmark fi, “+” means our approaches are better, “-” means our Compared to

ne-rcGA Functions : From 1 to 30

cPSO Functions : From 1 to 30

++++=+++++++=+++++++++=+++++++

++++--+------++=++=++==++=+-+=

+++-+--++-++-+-++=-++----+--+-

++++++++++++++++++++++=+++++++

++++==+=-==--+++++=++==++==+++

+++-+-=++-++-+-++--++-=--+-=++

++++--+++++-==+-+=++++++++++++

++++--+------=--+==++=====-==+

+++---=++-=--+--+--++-=--+--+=

++++--+++++++-+-+=++++++++++++

++++--+++++++-+-+=++++++++++++

++++---+++++--+-+-=++----+++++

++++--+++++-=-+++=++++++++++++

++++--+-------+-+=+++=+====+=+

+++-=-=++-+=-=-=+--++----+-=++

++=+--+++++-+-+++-++++++++-+++

++-+--+++=--=-+-+-+++++++=-+=+

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pe-rcGA Functions : From 1 to 30

pe-cDE Functions : From 1 to 30

ne-cDE Functions : From 1 to 30

cTLBO Functions : From 1 to 30

++++=+++++++==++++++++=+++++++

++++=-=----=-+=-+==++==+=++-+=

+++++++++=++=++++=+++=+---+=++

++++=+++++++==++++++++=+++++++

+++++-+----=-+==+==++==+==-=++

++++++++++++=++++-+++++---=+++

++++--+=+=+-=-+-+=+++++++=++=+

++++--=------=--+=+++=+-=+-===

++++--+=+=+-=++-+-+++=+--+-+++

++++--+++++++-+-+=++++++++++++

++++--+++++++-+-+=++++++++++++

++++--+++++++++-+-+++++--+++++

++++--+++=+-=-+++=++++++++++=+

++++--+------==-+-=++=+-==-==+

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approaches are worse, “=” means equivalence between the pairwise algorithms.

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Table 7 Summarized results of Wilcoxon’s rank sum test at a 0.05 significance level between each one of the proposed uniform cFAs with each pe-rcGA, ne-rcGA, pe-cDE, ne-cDE, cPSO, cTLBO Comparison results between the proposed cFAs and a set of cEAs.

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Comparison to swarm intelligence algorithms

All the proposed algorithms can be considered as swarm intelligence algorithms (cSIAs). The only difference between our Compared to Distribution Algorithm pe-cFA ne-cFA pe-cLFA Normal ne-cLFA OBcFA OBcLFA

FA Functions : From 1 to 30

LFA Functions : From 1 to 30

SLPSO Functions : From 1 to 30

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=++-+-=------+--==--+-==+-----

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-++=+-=------=---=--+-==+---+=++=--=------=--==--+-+-----+=

-++=+-=------=---=--+-==+---+=++=--=------=--==--+-+-----+=

-++-+------+-++=---++-=++----= -++-=------=-+--=--++-=--+----

=++=--++++=-=-=-==+-+=+---=+++

=++=--++++=-=-=-==+-+=+---=+++

-++----===-+=-=-=-=++=+=-==+=+

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proposal and SIAs is in the representation of swarm’s information. Hence, we consider them as compact swam intelligence algorithms (cSIAs). Therefore, we will compare our approaches to some SIAs. The selected algorithms are: the original firefly algorithm (FA) described in [78], the original Lévy-flight firefly algorithm (LFA) [27], and the social learning particle swarm optimization (SLPSO) introduced in [80]. The results given by FA, LFA and SLPSO are presented in Table 3.

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A. NPDF-based cFAs

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We here compare the proposed cFAs, i.e. pe-cFA, ne-cFA, pe-cLFA, ne-cLFA, OBcFA and OBcLFA, based on NPDF with FA, LFA, and SLPSO. The results of Wilcoxon’s rank sum test are presented in Table 8. Table 9 summarizes the results given in Table 8. One of the important results is that the proposed ne-cLFA surpasses FA, LFA, and surpass SLPSO in several cases. The other proposed algorithms are also competitive compared to this advanced swarm intelligent algorithms. For example, ne-cFA outperforms SLPSO in 10 cases; the OBcFA surpasses SLPSO in seven cases. See Table 9 for more results.

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Table 8 Wilcoxon’s rank sum test at a 0.05 significance level is performed between each one of the proposed NPDF-based cFAs with each one of FA, LFA and SLPSO. For each benchmark fi, “+” means our approaches are better, “-” means our approaches are worse, “=” means equivalence between the pairwise algorithms.

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Table 9 Summarized results of Wilcoxon’s rank sum test at a 0.05 significance level between each one of the proposed NPDF-based cFAs with each one of FA, LFA,SLPSO

ACCEPTED MANUSCRIPT

Compared to Distribution Algorithm pe-cFA ne-cFA Uniform pe-cLFA ne-cLFA OBcFA OBcLFA

FA Functions : From 1 to 30

LFA Functions : From 1 to 30

SLPSO Functions : From 1 to 30

=++=--=------+--==--=-=-----+=++-=-=------+-=-=--=-=-----+-

=++=--=------+--==--=-=-----+=++-=-=------+-=-=--=-=-----+-

-++-+------+-+--==-++--=----=-++-+------+-+-+---+=-==----==

=++=--=---------==--+-=-----+++++--++++=-=-=--=+-+=+---=+++ =+++--=---------=---+-+-----==

=++=--=---------==--+-=-----+++++--++++=-=-=--=+-+=+---=+++ =+++--=---------=---+-+-----==

-++----------=--=--++-=--+---=++----==--+=-=---=++-+=-=++=+ -++-=------=-=--=--++-+--+-=--

=+-+--+=------=-=-=-=-+----++-

=+-+--+=------=-=-=-=-+----++-

-+--+------=--=-=--+=-+--+-+--

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B. UPDF-based cFAs

565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602

In Table 10, we present the comparison between the proposed UPDF-based cFAs with the swarm intelligent algorithms presented above, i.e. FA, LFA and SLPSO. The comparison results are shown in Table 10 and summarized in Table 11. The proposed frameworks surpass FA, LFA and FA LFA SLPSO SLPSO in several cases. distribution + = + = - + = 7 19 4 7 19 7 4 19 pe-cFA 4 7 19 4 7 19 7 5 18 ne-cFA 4 4 6 4 6 20 5 3 22 pe-cLFA 20 Uniform 6 10 ne-cLFA 14 14 6 10 9 9 12 5 20 5 5 20 6 5 19 OBcFA 5

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Table 10

FA + = 6 6 6 6 5 7 9 12 5 5 1 0

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Wilcoxon’s rank sum test level is performed proposed uniform cFAs SLPSO. For each our approaches are better, are worse, “=” means pairwise algorithms.

18 18 18 9 20 29

+ 6 6 5 12 5 1

LFA = 6 6 7 9 5 0

18 18 18 9 20 29

SLPSO + = 9 2 19 10 3 17 6 4 20 8 12 10 7 3 20 1 1 28

at a 0.05 significance between each one of the with each FA, LFA and benchmark fi, “+” means “-” means our approaches equivalence between the

Table 11 Summarized results of Wilcoxon’s rank sum test at a 0.05 between each one of the proposed uniform cFAs with each FA, LFA, and SLPSO.

ACCEPTED MANUSCRIPT OBcLFA

6

6

18

6

6

6

18

4

20

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603 604 605 606 607 608 609 610 611 612 613

5.6.

614 615 616 617 618 619 620

In this subsection, we propose to compare our algorithms with the most efficient variants of differential evolution (DE). The set of DE includes self-adaptive differential evolution (SaDE) [84]; OX-based DE (OXDE) [82]; Differential evolution with composite trial vector generation strategies (CODE) [83] and differential evolution based on covariance matrix learning and bimodal distribution parameter setting (CoBiDE) [81].These algorithms are very advanced and enhanced with selfadaptive strategies, e.g. (SaDE), advanced reproduction operators, e.g. OXDE, covariance matrix, e.g. CoBiDE. This comparison will record the limits of the proposed approaches and the limits of compact optimization. The results given by OXDE and SaDE are presented in Table 3. The results given by CODE and CoBiDE are presented in Table 12.

621

A. Comparing NPDF-based cFAs to EAs

622 623 624 625 626

The results of the comparison between the proposed NPDF-based cFAs with modern DE variants are presented in Table 13 and are summarized in Table 14. Despite the high effectiveness of advanced DE variants, the very simple proposed algorithms, with very less memory requirement, have surpassed these modern DE variants in several cases. For example, ne-cLFA surpasses CoBiDE in 9 cases and they give similar results in 6 cases.

627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644

The comparison between UPDF-based cFAs and advanced DE variants is presented here. The results are shown in Table 15 and summarized in Table 16. Here also, the proposed algorithms, with minimal memory storage, surpass the most efficient DE variants in several cases. We demonstrate through this work that simple algorithms, i.e. the proposed cFA, with reduced memory requirements, can be competitive and can surpass the most powerful modern evolutionary algorithm in some problems. The modern improved DE variants require high computational capacities compared to our approaches. For example, the CoBiDE uses the covariance matrix. Hence, at each generation, the eigenvalues and eigenvectors must be computed, the crossover is done in a rotated space, and then we must turn back the results to the research space (original coordinates). The covariance matrix and the multi-strategies used by CoBiDE [81] make its very efficient, but this algorithm requires high computational costs. We can conclude that compact optimization present the advantages of simplicity and minimal computational cost requirements and the drawback of relative weakness compared to advanced optimization algorithms. It does not imply that we have to stop at this level with regard to the compact optimization. It is very interesting to push, in the future, the field of compact optimization further so that it reaches the efficiency level of the most advanced algorithms.

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B. Comparing UPDF-based cFAs with EAs

SC

Comparison to the state-of-art of evolutionary algorithm (EAs)

AC C

Table 12 Experimental results of CoBiDE, CODE, uniform pe-cFA, uniform ne-cFA, uniform pe-cLFA, uniform ne-cLFA, uniform OBcFA and uniform OBcLFA over 25 independent runs on 30 test functions of 30 variables with 300,000 FES. “Means” and “StD” indicate the average and standard deviation of the function error values obtained in 25 runs, respectively. Function F 1 Means StD F 2 Means StD F 3 Means StD F 4 Means StD F 5 Means StD F 6 Means StD

CoBiDE 1.6428E5 1.3488E5 0.0 0.0 0.0 0.0 0.1594 0.7812 20.2401 0.26215 1.4189 1.5285

CODE 3.0849E4 2.4466E4 0.0 0.0 4.547E-15 1.542E-14 6.0567E-4 0.001263 20.0323 0.0411 1.7153 1.4203

pe-cFA 9.8360E6 1.2537E6 1.3790E3 6.7557E3 6.0873E2 1.7056E3 1.0242E2 4.0194E1 2.0209E1 0.269944 1.5352E1 4.199814

ne-cFA 1.0815E7 1.3484E7 1.5638E3 7.6611E3 8.1824E1 1.7152E2 1.1509E2 3.8454E1 2.0159E1 0.258011 1.1367E1 3.422150

pe-cLFA 3.2006E6 4.2382E6 0.016959 0.038047 4.319097 8.350470 90.36647 31.81074 20.99247 0.079448 42.12292 1.462129

ne-cLFA 1.9261E6 2.6880E6 0.036288 0.063893 141.4758 261.6495 73.72880 27.70708 21.01484 0.049928 42.09243 1.218225

OBcFA 2.8607E6 2.8727E6 0.042822 0.096208 1.7927E3 5.8716E3 7.4766E1 3.5483E1 2.0767E1 0.334122 4.1207E1 2.864179

OBcLFA 3.1443E6 3.7464E6 4.5003E1 1.5417E2 6.0704E4 3.1073E4 7.8509E1 1.6193E1 2.0621E1 0.230719 4.0725E1 3.911800

ACCEPTED MANUSCRIPT

F 12 F 13 F 14 F 15 F 16 F 17 F 18 F 19 F 20 F 21 F 22 F 23 F 24 F 25 F 26 F 27 F 28 F 29 F 30

647 648 649 650

0.013159 0.021218 99.25889 32.35045 3.2006E6 4.2382E6 0.016959 0.038047 4.319097 8.350470 90.36647 31.81074 20.99247 0.079448 42.12292 1.462129 0.013159 0.021218 99.25889 32.35045 3.2006E6 4.2382E6 0.016959 0.038047 4.319097 8.350470 90.36647 31.81074 20.99247 0.079448 42.12292 1.462129 0.013159 0.021218 99.25889 32.35045 3.2006R6 4.2382E6 0.016959 0.038047 4.319097 8.350470 90.36647 31.81074 20.99247 0.079448 42.12292 1.462129

0.000295 0.001449 18.02867 4.618372 1.9261E6 2.6880E6 0.036288 0.063893 141.4758 261.6495 73.72880 27.70708 21.01484 0.049928 42.09243 1.218225 0.000295 0.001449 18.02867 4.618372 1.9261E6 2.6880E6 0.036288 0.063893 141.4758 261.6495 73.72880 27.70708 21.01484 0.049928 42.09243 1.218225 0.000295 0.001449 18.02867 4.618372 1.9261E6 2.6880E6 0.036288 0.063893 141.4758 261.6495 73.72880 27.70708 21.01484 0.049928 42.09243 1.218225

0.010199 0.020316 60.11316 2.0325E1 6.8067E1 2.5915E1 3.0807E3 6.5340E2 3.8063E3 517.5737 2.399836 0.776339 0.466278 0.131771 0.538521 0.264016 6.278689 3.230026 12.71747 0.438809 2.7274E5 2.7478E5 8637.176 9582.334 15.23854 14.51286 1116.185 2067.141 44230.03 30336.09 449.7572 156.7530 314.7622 0.563191 237.7605 5.905352 215.1285 4.831669 100.4794 0.132926 634.1462 130.2443 1080.832 293.6923 19583.30 12020.90 10161.37 12022.75

2.875E-11 5.175E-12 3.1333E1 8.752581 3.7779E1 14.13956 1.9815E3 4.7312E3 4.4597E3 7.8870E2 2.284520 0.587408 0.317960 0.100028 0.792244 0.158672 3.113726 0.787691 12.75615 0.357430 4.2767E5 5.1109E5 1.8727E4 1.1233E4 10.23049 2.149631 784.1576 614.2401 128264.3 154139.3 379.4773 146.2668 314.4768 0.163100 234.5342 4.528812 207.5513 2.068693 100.5425 0.109039 1274.403 152.6050 808.8508 56.90484 10145.18 3651.592 13558.66 12083.40

Table 13 Wilcoxon’s rank sum test at a 0.05 significance level is performed between each one of the proposed normal cFAs with each SaDE, OXDE, CODE and CoBiDE. For each benchmark fi, “+” means our approaches are better, “-” means our approaches are worse, “=” means equivalence between the pairwise algorithms.

AC C

645 646

0.006304 0.009134 5.1061E1 1.2156E1 1.0815E7 1.3484E7 1.5638E3 7.6611E3 8.1824E2 1.7152E2 1.1509E2 3.8454E1 2.0159E1 0.258002 1.1367E1 3.422150 0.006304 0.009134 5.1061E1 1.2156E1 1.0815E7 1.3484E7 1.5638E3 7.6611E3 8.1413E1 1.7152R2 1.1509R2 3.8454E1 2.0159E1 0.258002 1.1367E1 3.422150 0.006304 0.009134 5.1061E1 1.2156E1 1.0815E7 1.3484E7 1.5638E3 7.6611E3 8.1824E1 1.7152E2 1.1593E2 3.8454E1 2.0159E1 0.258002 1.1367E1 3.422150

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F 11

0.009160 0.008307 8.3687E1 2.5037E1 9.8360E6 1.2537E6 1.3790E3 6.7557E3 6.0873E2 1.7056E3 1.0242E2 4.0194E1 2.0209E1 0.269940 1.5352E1 4.199814 0.009160 0.008307 8.36870E1 2.5037E1 9.8360E6 1.2537E7 1.3790E3 6.7557E3 6.0873E2 1.7056E3 1.0242E2 4.0194E1 2.0209E1 0.269940 1.5352E1 4.199814 0.009160 0.008307 8.3687E1 2.5037E1 9.8360E6 1.2537E7 1.3790E3 6.7557E3 6.0873E2 1.7056E3 1.0242E2 4.0194R1 2.0209E1 0.269940 1.5352E1 4.199814

SC

F 10

2.958E-4 0.0014 0.0 0.0 39.5993 11.2178 0.4831 0.5518 1.8456E3 445.2318 0.0654 0.0260 0.2321 0.0567 0.2236 0.0344 3.2722 0.6844 8.8806 0.8781 1060.443 823.4652 13.5628 5.5912 2.7656 0.4254 11.7811 8.1628 230.2410 137.2547 141.9830 118.4240 315.2441 1.25E-13 225.1749 2.84644 203.3426 0.4168 100.2362 0.04484 385.4954 33.0277 826.2442 31.3761 779.7216 146.7673 869.4063 401.6876

M AN U

F9

0.0 0.0 0.0 0.0 38.1799 10.4528 62.3792 11.8414 1.6702E3 357.9046 0.2507 0.3303 0.2211 0.0460 0.2343 0.0349 3.2674 0.8772 9.8183 0.7500 250.9556 163.9537 10.8316 3.7500 2.7753 0.4497 7.6625 2.52995 167.5802 110.8002 110.3446 68.04158 315.2441 0.0 220.8600 6.2126 203.0878 0.4218 100.2188 0.04413 373.9293 43.3385 810.0831 25.3899 585.8562 232.2142 681.5464 195.9768

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F8

Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD Means StD

EP

F7

Compared to

Distribution Algorithm pe-cFA ne-cFA Normal pe-cLFA ne-cLFA OBcFA OBcLFA Compared to Distribution Algorithm pe-cFA ne-cFA

SaDE Functions : From 1 to 30

OXDE Functions : From 1 to 30

----+-----=+----------=++------=-+-+---=+--=-------=++--------------------------=--+---------=-+-+++-+------=+=+=++== --=-------------------+--+----------------------------+---CODE Functions : From 1 to 30

----+-----=+-==-------=++--------+-----=+-===------=++--------------=-----------=------------+=+-+++-+-------+=-==+-----=-----==-==-------+----------=-----=------------------CoBiDE Functions : From 1 to 30

----+-----------------=++--------+-----------------=++-----

----+------+----------=++--------+------=----------=++---=-

ACCEPTED MANUSCRIPT

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----------------------=--------------+-+++-=-------+--==+-----------------------+------------------------------------

SC

Table 14 Summarized results of Wilcoxon’s rank sum test at a 0.05 significance level between each one of the proposed normal cFAs with each SaDE, OXDE, CODE, CoBiDE

Distribution

pe-cFA ne-cFA pe-cLFA ne-cLFA OBcFA OBcLFA

+ 4 5 1 9 2 1

SaDE = 2 24 4 21 1 28 6 15 1 27 0 29

+ 3 3 0 6 1 0

OXDE = 1 26 1 26 1 29 5 19 0 29 0 30

+ 4 4 0 8 1 0

CODE = 4 22 5 21 2 28 4 18 5 24 2 28

CoBiDE + = 4 2 24 5 4 21 1 1 28 9 6 15 2 1 27 1 0 29

Normal Table 15 Wilcoxon’s rank sum test at a 0.05 significance level is performed between each one of the proposed uniform cFAs with each SaDE, OXDE, CODE and CoBiDE. For each benchmark fi, “+” means our approaches are better, “-” means our approaches are worse, “=” means equivalence between the pairwise algorithms.

EP

Compared to

AC C

Distribution Algorithm pe-cFA ne-cFA Uniform pe-cLFA ne-cLFA OBcFA OBcLFA Compared to Distribution Algorithm pe-cFA ne-cFA uniform pe-cLFA ne-cLFA OBcFA OBcLFA

692 693 694 695 696 697

----------------------=------------+-+-+-+-=------=+=-==+-----------------------+------------------------------------

M AN U

666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691

pe-cLFA ne-cLFA OBcFA OBcLFA

TE D

651 652 653 654 655 656 657 658 659 660 661 662 663 664 665

Normal

SaDE Functions : From 1 to 30

OXDE Functions : From 1 to 30

--=-+------+-----------=---------+-+---=+----------=--------=-------------------=--+---=-----=-+-+++-+-------+-++++-= --=-------------------+--+-=-----=-=-=---=-+-------+-++-+-CODE Functions : From 1 to 30

----+-----=+-==------------------+-----=+-===------=----------------=-----------=------------+-+-+++-+-------+--+=+-----=-----=---=-------+----------+-+---===-+-------+----+-CoBiDE Functions : From 1 to 30

----=----------------------------=-----------------=----------------------------=--------------+-=++-=-------+--=++-----------------------+------------+-=-----=-------+----+--

----=----------------------------=-----------------=----------------------------=--------------+-=++-=-------+--==+-----------------------+--------------=-----=-------+----=--

Table 16 Summarized results of Wilcoxon’s rank sum test at a 0.05 between each one of the proposed uniform cFAs with each SaDE, OXDE, CODE, CoBiDE

ACCEPTED MANUSCRIPT

distribution

pe-cFA ne-cFA pe-cLFA ne-cLFA OBcFA OBcLFA

Uniform

+ 2 3 1 10 2 5

SaDE = 2 26 2 25 2 27 3 17 2 26 4 21

+ 0 0 0 6 1 3

CODE = 1 29 2 28 1 29 3 21 0 29 2 25

+ 2 2 0 9 1 5

OXDE = 3 25 5 23 2 28 1 20 3 26 3 22

CoBiDE + = 0 1 29 0 2 28 0 1 29 5 4 21 1 0 29 1 3 26

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698 699 700 701 702 703 704 705 706 707 708 709

710

5.7.

711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752

As it is presented above, for each proposed algorithm, e.g. ne-cLFA, two versions are presented. The first is based on the classical NPDF and the second is based on the proposed UPDF. A comparative study is presented here between our algorithms, i.e. UPDF-based cFAs and NPDF-based cFAs. The obtained results are presented in Table 17 and summarized in Table 18. From Table 17, we conclude that UPDF-based pe-cFA, ne-cFA, OBcFA and OBcLFA give better results than their respective NPDF-based cFAs. However, the NPDF-based pe-cLFA and ne-cLFA are more efficient than their respective UPDF-based cFAs. We note that all the known works in the field of compact optimization [51] use NPDF. We think, as it is analyzed here, that UPDF are also suitable. The efficiency of UPDF can be explained by the fact that its bounds vary when its average and standard deviation vary. The adaptation rule used in compact optimization, i.e. Eqs. (7) and Eqs. (8), affects the mean and standard deviation. Consequently, the UPDF’s bounds are retracted towards the optimal subregion. Eqs. (16) and Eqs. (17) give the relationship between the bounds of UPDF and its mean and its standard deviation. The bounds are attracted and will surround the optimal sub region at convergence. Hence, the algorithm’s precision can be enhanced. The UPDF is also easy to implement and requires less calculus to get its CDF and the inverse of its CDF compared to NPDF. Thus, the implementation of cEAs can be simplified using UPDF. In future works, we propose the use of UPDFs in others compact algorithms, e.g. cGA, cPSO, cDE, as well as an evaluation of the performance. Other compact representations can also be evaluated, e.g. triangular PDFs. The field of compact optimization must be improved to get the same performance of advanced population-based algorithms while reducing computational capacities. The concepts of covariance matrix, hybridization and self-adaptive strategies can also be incorporated in the future into compact optimization.

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Comparison between UPDF-based cFAs and NPDF-based cFAs

EP

Table 17 Wilcoxon’s rank sum test at a 0.05 significance level is performed between each one of the proposed uniform cFAs with each one of the proposed normal cFAs. For each benchmark fi, “+” means our approaches are better, “-” means our approaches are worse, “=” means equivalence between the pairwise algorithms.

AC C

Compared to

Distribution Algorithm

pe-cFA ne-cFA Uniform PDF pe-cLFA ne-cLFA OBcFA OBcLFA Compared to Distribution Algorithm

Uniform PDF

pe-cFA ne-cFA pe-cLFA ne-cLFA OBcFA OBcLFA

pe-cFA Functions : From 1 to 30

==-=-+=++==-+===+=++=+=--++++= ====-+-++-=-==--+=+==+=--++=+==-=++==+==+++++======-++==-+=+-=++-------+-++=-==-------==---+++-=-=+=++====-==-+==+-+= +++++++++++++++++++++++++-+=+=

Normal PDF ne-cFA Functions : From 1 to 30

pe-cLFA Functions : From 1 to 30 ==++--=++---=---==++=+=--+++++ ++==---++---=---+-++=+=--++++=++=--===-=-===-============-= =++====------+-==--+=--------=++=--+-=-=-==--=======-=-==+= ++++--+++++-+++-++++++++++++=+

ne-cLFA Functions : From 1 to 30

=-==-+=++=+-+=++==+=-+=--+++++ =--=-+=++==-+=-+==+==+=--++++= ===-++=+++++++++-==-==-++===++ -+=-++=------+-+-=--=-------==-=-+++++-++++++-==-==-+===+++ +++++++++++++++++++++++++-++++ Normal PDF OBcFA Functions : From 1 to 30

+-=+--+++++++-+-==+=-+=--+++++ +-=+--+++++++-+-=-+==+=--+++++ +=-=--+++=++=-+--=+==+=+++++++ =+====-==--===========--====== ++==--+++++++-+-==+-=+++++++++ ++++--+++++++++-++++++++++++++

==-+--=++-=-=-+==-++-+=--+++++ +==+--=++---=-=-+-++=+=--+++++ ==-==-=++====-++--=+==+==+==-= =+-++==------=-+=--+=--------=+====+=+-===-+-=-=========+== ++++=-+++++=+++=++++++++++++-+

=--+--++++--+-+-=-++-+=--+++++ +--+--=+++--+-+-=-++-+=--++++=--=+-=+++==+-++--++-++++=-+-= =+-++=+-------=+=--+----=----=--=+=+++=-=+-+-=-+=-=+++--++= +++++-+++++++++=++++++++++-+=+

OBcLFA Functions : From 1 to 30

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Normal PDF pe-cLFA ne-cLFA + = + = 10 17 5 8 12 8 12 5 8 13 17 5 2 22 6 6 16 8 8 24 5 4 18 1 4 17 9 5 20 5 4 3 25 1 27 0

+ 12 13 7 5 5 25

OBcFA = 8 10 7 10 17 6 6 19 21 4 3 2

OBcLFA + = 15 3 12 14 3 13 14 7 9 6 5 19 12 9 9 26 2 2

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ne-cFA + = 14 10 6 11 12 7 15 11 4 5 5 20 6 15 9 1 29 0

SC

5.8. Algorithms’ ranking

M AN U

To further detect the significant differences between the proposed algorithms and the other competitors, the Friedman’s test was carried out, in which Bonferroni–Dunn’s procedure was used as a post hoc procedure [76, 77]. Veček et al. have suggested in their work on the influence of the number of algorithms [77] on the performance of the Friedman’s Test that the number of algorithm (k) should be inferior to n/2, where n is the number of problems. In our experiment, the number of problems, e.g. the number of benchmarks in IEEE CEC2014 [75], is n=30. Hence, the maximal number of algorithms in each comparison is 14. Therefore, we will compare the proposed algorithms with the most known cEAs first and then with the advanced EAs and SIAs.

a) Comparing cFAs to cEAs

One of the purposes of this work is to propose a new compact optimization technique that surpasses the known cEAs, such as pe-cGA, ne-cGA, pe-cDE, ne-cDE, cPSO and cTLBO. Thus, we will compare the proposed NPDF-based cFA to cEAs using the Friedman’s test [76-77]. The results are presented in Table 19. The Friedman’s Test is used to obtain the global ranking of each algorithm. The best algorithm of this comparison is the proposed ne-cLFA. This algorithm gives the best rank=3.82. We find in the second place ne-cFA with rank =5.12. The proposed ne-cLFA presents the best results because of two facts: first, it uses a non-permanent elitism strategy, second, it is enhanced by Lévy flight movements. The rank of the other algorithms is presented in Table 19.

TE D

769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808

Uniform PDF

pe-cFA ne-cFA pe-cLFA ne-cLFA OBcFA OBcLFA

pe-cFA + = 12 13 5 9 12 9 11 15 4 6 6 18 13 8 9 1 27 2

Table 19 A global ranking of the proposed normal cFAs and the cEAs according to Friedman’s test at a 0.05 significance level. NPDF- based cFAs Rank cEAs Rank

pe-cFA 5.77 pe-cGA 8.62

EP

768

Table 18 Summarized results of Wilcoxon’s rank sum test at a 0.05 significance level between each one of the proposed uniform cFAs with each one of the proposed normal cFAs.

AC C

753 754 755 756 757 758 759 760 761 762 763 764 765 766 767

ne-cFA 5.12 ne-cGA 5.76

pe-cLFA 5.80 pe-cDE 8.21

ne-cLFA 3.82 ne-cDE 5.39

OBcFA 5.50 cPSO 5.30

OBcLFA 10.56 cTLBO 8.10

In Table 20, we present the results of the comparison between the proposed UPDF-based cFAs and the cEAs according to Friedman’s test. The UPDF-based ne-cLFA gives the best rank=4.36. The proposed ne-cLFA, both with NPDF and UPDF, presents the best results with a minimal computational cost.

A global ranking Friedman’s test at a of the proposed cEAs.

UPDF- based cFAs Rank cEAs Rank

Table 20 pe-cFA 6.01 pe-cGA 6.69

ne-cFA 5.720 ne-cGA 6.01

pe-cLFA 7.37 pe-cDE 6.90

ne-cLFA 4.36 ne-cDE 4.53

OBcFA 6.54 cPSO 6.65

OBcLFA 6.31 cTLBO 10.87

according to 0.05 significance level uniform cFAs and the

ACCEPTED MANUSCRIPT

b) Comparing cFAs with EAs and SIAs

Table 21 A global ranking of normal cFAs, SIAs, according to

NPDF- based cFAs Rank cEAs Rank

pe-cFA 8.32 FA 7.34

ne-cFA 7.78 LFA 8.34

pe-cLFA 9.08 SLPSO 6.49

ne-cLFA 6.42 CoBiDE 2.76

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In this section, we rank, according to Friedman’s test, the proposed algorithms and the advanced EAs, such as CoBiDE, CODE, OXDE, and SaDE, the SIAs, such as FA and LFA, and the advanced SLPSO [80]. This ranking will compare the performance of our proposed algorithms, which uses a strict minimal computational cost and storage, with the performance of the most advanced DE variants and SIAs. Table 21 shows the comparison of NPDF-based cFAs with EAs and SIAs. Although the best rank is achieved by CoBiDE, which is very sophisticated by covariance matrix, parameters adaptation, and multiple strategies, the proposed ne-cLFA surpasses the SIAs (FA, LFA, and the advanced variant SLPSO). Hence, our purpose of enhancing the performance of FA by means of compact optimization is reached. Our variant also surpasses SLPSO. OBcFA 8.90 CODE 3.283

SaDE 5.30

the proposed and advanced EAs, Friedman’s test at a

M AN U

SC

0.05 significance level

OBcLFA 12.25 OXDE 4.67

TE D

Table 22 shows the comparison of UPDF-based cFAs with advanced EAs and SIAs. Similar results of those presented in Table 21 are obtained. We conclude from this study that the proposed algorithms, which can be seen as compact swarm intelligence algorithms (cSIAs), are very effective. They give competitive results. Their performance surpasses that of cEAs and the SIAs. The proposed algorithms remain relatively weak compared with the advanced EAs, but our proposed algorithms require a strict UPDF- based cFAs pe-cFA ne-cFA pe-cLFA ne-cLFA OBcFA OBcLFA minimal computational cost Rank and storage. The 8.96 8.64 10.10 6.45 9.34 8.89 field of compact cEAs optimization can be FA LFA SLPSO CoBiDE CODE OXDE SaDE enhanced in the Rank future years to 7.58 8.58 6.51 2.65 3.19 4.66 5.38 performance of the reach the advanced EAs.

EP

Table 22 A global ranking of the proposed uniform cFAs, SIAs, and advanced EAs according to Friedman’s test at a 0.05 significance level.

AC C

809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854

6.

855 856 857 858 859 860 861

In this section we present a real application of the proposed algorithms. The objective is to realize a swing-up movement by a humanoid robot hanging on a high bar [85]. Let us first describe the robot. 6.1.

Case study: swing-up control of a gymnast humanoid robot

Virtual model Virtual humanoid gymnast robot is designed on Solidworks2014. The robot hanging on a high bar is represented in Fig. 11. The dynamic parameters of the robot are given in Table 23.

862 863 864

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Fig. 11 Virtual model of the humanoid designed with Solidworks.

Table 23 Robot’s parameters

Link1

O(kg)

8.0076

Link2

31.8146

Link 3

21.2453

P(QROS )

TU (M)

T (M)

SC

The link

0.505

0.39874

0.50542

1.52895

0.23665

0.57

M AN U

865 866 867 868 869 870 871 872 873 874 875

3.10134

0.48835

0.9767

876

6.2.

877 878 879 880 881 882

The Gymnastic humanoid robot can be seen as being composed of three main links [85]. The first link represents the arms, the second represents the torso and the third represents the legs. The joints of the robot are the hands, the shoulders, and the hips. Both the shoulders and hips are actuated by two servos on each side. The hands are not actuated. Indeed, the system can be approximated by a three link under actuated pendulum. The dynamic behavior of this multi-body robotic system can be derived from the classical Euler-Lagrange equations. The model is represented in Fig. 12.

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Nonlinear model:

896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913

M AN U

V is the angle of joint i in respect to the previous link. W is the mass of link i.  is the inertia of link i. X is the torque actuated on the active joint i. Y is the length between joint i and joint i+1. YZ is the length between joint i and the center of gravity of the mass of the link i. The direct cinematic model of the center of mass of each link is:

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We precise here the used notations:


 = YZ Z?+ (V ) 1 = YZ sin(V )
^ = Y Z?+ (V ) + /Z^ Z?+(V + V^ ) 12 = Y1 +,(V1) + YZ2 +,(V1 + V2)
3 = Y1 cos(V1) + Y2 cos(V1 + V2) + YZ3 cos(V1 + V2 + V3) 13 = Y1 sin(V1) + L2 sin(V1 + V2) + YZ3 sin(V1 + V2 + V3) The Lagrangian of the system is given by: Y = ∑Md(c − 7 ) Where the kinetic energy T and the potential energy U of links are given by: ^  c = [W 
e ^ + 1e ^  +  Ve

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Fig. 12 The gymnast humanoid as a three link under-actuated pendulum.

^

c^ = fW^ 
^e + 1^e  +  (Ve  + V^e )^ g ^  cM = fW^ 
Me ^ + 1Me ^  +  (Ve  + V^e + VMe )^g ^ 71 = W -1 72 = W^ -1^ 73 = WM -1M The nonlinear model is then derived from Euler-Lagrange equations. We found the model given by  ^ M Vk G n 0 h^ ^^ ^M i jVk^ l + hG^ i + hn^ i = hX i X^ GM nM M M^ MM VMk Where the inertial terms are:  = A + 2W^ Y YZ^ Z?+(V^ ) + 2WM Y Y^ cos(V^ ) + 2WM Y^ YZM cos(VM ) + WM Y YZM Z?+(V^ + VM ) ^ = A^ + W^ Y YZ^ Z?+(V^ ) + WM Y Y^ Z?+(V^) + 2WM Y^ YZM Z?+(VM ) + WM Y YZM Z?+(V^ + θM ) M = qM + WM Y^ YZM Z?+(VM ) + WM Y YZM Z?+(V^ + VM )

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^

^

(18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30)

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The gravitational terms are:

r = B Z?+(V) + B^ Z?+(V + V^ ) + Bt Z?+(V + θ^ + VM ) r^ = BM Z?+(V + V^ ) + ut Z?+(V + V^ + VM ) rM = ut Z?+(V + V^ + VM ) B = (W /Z + W^ / + WM / )B^ = (W^ /Z^ + WM /^)BM = (W^ /Z^ + WM /^)ut = WM /ZM -

RI PT

914

^ = ^ ^^ = q^ + 2WM Y^ YZM Z?+(VM ) ^M = qM + WM Y^ YZM Z?+(VM) M = M M^ = ^M MM = qM + WM Y^ YZM cos(VM ) q = W YZ^ + W^ Y^ + W^ YZ^^ + WM Y^ + WM Y^^ + WM YZM^ + 1 + 2 + 3 q^ = W^ YZ^^ + WM Y^^ + WM YZM^ + 2 + 3 qM = WM YZM^ + 3

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921 922 923

We are interested in the achievement of the gymnastic movement. In the literature, we talk about the swing-up control problem. The objective is to realize and to stabilize the humanoid at the vertical unstable equilibrium position. As the model derived from Euler-Lagrange equations is highly nonlinear, we derive the linearized model around the vertical position.

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Linearized model at swing-up position

V = y ^ 0 0{ z

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  h ^ M

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|

And Ve = [0 0 0]|

The linearized model of the humanoid for the vertical unstable equilibrium is given by the following equations:

Where:

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And the Coriolis terms are: n = Ft V^e + VMe 2Ve + V^e + VMe +,(V^ + VM ) + F V^e 2Ve + Ve^ +,(V^) + F^ w^e 2Ve + Ve^ +,(V^) + FM VMe 2Ve + 2V^e + VMe +,(VM) ^ ^ e n^ = −x V +,(V^ ) − F^ Ve +,(w2) + FM VMe (2Ve + 2V^e + VMe ) +,(VM) − Ft we ^ +,(V^ + VM ) ^ ^ nM = − FM Ve + V^e  +,(VM ) − Ft Ve +,(V^ + VM ) x = −W^Y YZ^ , F^ = − WM Y Y^ , xM = − WM Y^ YZM , Ft = − WM Y YZM

^ ^^ M^

}wk + rw = ~X

- M Vk 1 ^M i hVk 2i + h g^ gM MM Vk 3

g^ ^^ g M^

gM V − 0 ^ g ^M i j V l = h1 ^ g MM 0 V €

M

 = A + 2W^ Y YZ^ + 2WM Y Y^ + 2WM Y^ YZM + 2WM Y YZM ^ = A^ + W^ Y YZ^ + WM Y Y^ + 2WM Y^ YZM + WM Y YZM M = qM + WM Y^ YZM + WM Y YZM ^ = ^ ^^ = q^ + 2WM Y^ YZM ^M = qM + WM Y^ YZM M = M M^ = ^M MM = qM + WM Y^ YZM - = u + u^ + ut -^ = uM + ut -^ = Bt -^^ = -^ = -^

0 X  0i yX { ^ 1

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-MM = -M = -M^ = -^M = -M

6.4. The state space model of the humanoid The state vector is constituted by the position vector  and e , hence we have:

(34)


e (@) = q
(@) + uƒ(@)

(35)

€ ^

The state space model is given by

933

1(@) = n
(@) + }ƒ(@)

934 935

‚ = [w − w^ wM we w^e wMe ]

The matrix A, B, C and D are given by q=„

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0…×… }  r

…×… 0 ‡, u = „ …׈ ‡, 0…×… }  c

(36)

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n = …×… and } = 0

937

The numerical values of the state space matrix A and B are:

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5.5

939 940 941 942 943 944 945

The objective is to stabilize the robot on a vertical unstable position. In order to realize this swing-up movement, we propose the use of an optimal linear quadratic controller (LQR). The corresponding optimization criteria is given by n = ”•
– —
+ ƒ– ˜ƒ™ @ (37) The control feedback is given by ƒ = −š ∗
(38)

0 1 0 0 0 0

0 0 ‹ 0’ 0 ‘ Š 1‘ 0 ; u = Š 0‘ Š−0.2169 Š 0.4124 0‘ ‰ 0.1147 0

0 ’ 0 ‘ 0 ‘ 0.0326 ‘ −0.1147 ‘ − 0.1410

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Achievement of the Swing-up movement by cFAs

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Cost

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pecFA necFA pecLFA necLFA OBcFA OBcLFA

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Uniform PDF-based cFAs

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40 Interation × 1000

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0 0 ‹ 0 0 Š 0 0 q=Š Š 18.8797 −21.4186 Š−18.3221 46.7787 ‰ −0.2202 −10.0165

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Fig. 13 Optimization process Torque

Torque 200

100 0 -100

0

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4 6 Time (S) Positions

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4 6 Time (S) Speed

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10 200 velocity (°/s)

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velocity (°/s)

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Fig. 14 The best obtained results, left: without noise, right: with noise.

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The matrix k is obtained using the Riccati equation. The performance of the obtained controller depends on the matrix Q and R. In order to realize a swing-up movement with a minimal cost, i.e. minimization of the criteria given in Eqs. (37), the proposed cFAs using uniform and normal distribution are used. The results of the optimization process are given in Fig. 13. The UPDF-based cFAs give better results than the NPDF-based cFAs. The best solution, i.e. which minimizes more Eqs. (37), is found by the UPD-based pe-cLFA. The joint angles corresponding to the best solution are shown in Fig. 14. The latter also gives the application to the robot without noise (see Fig. 14 at left) and the robot with noised joint actuators (see Fig. 14 at right). Despite the presence of the noise, the LQR controller designed and optimized by the proposed cFAs stabilizes the nonlinear model of the gymnast humanoid robot on the unstable upright position.

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In this paper, a set of twelve new compact firefly algorithms (cFAs), designed for global optimization problems, is introduced. The proposed algorithms can be viewed as compact swarm intelligence algorithms. We subdivided them into two categories, normal cFAs and uniform cFAs. Six of them are based on the classical commonly used normal distribution function. The six others use a proposed uniform distribution function to represent the population. All the proposed algorithms only require a minimal computational cost; the swarm information is compacted and represented by probabilistic functions. All the goals of our study are reached and several proposals are presented and deeply analyzed. This includes population storage, elitism strategies, opposition-based learning and Lévy-flight movements. The proposed algorithms use minimal attractions per fitness evaluation. Furthermore, a new way of generating opposite candidates is also introduced. Unlike the classical opposition-based optimization, which generates opposite candidates only in the initialization step, the proposed method performs this over all iterations. The proposals are assessed by an extensive numerical experimental study. The proposed algorithms are tested on the IEEE CEC2014 benchmark functions and compared to the state-of-art of compact evolutionary algorithms, a set of swarm intelligent algorithms and a set of advanced population-based algorithms. The obtained results show that the proposed algorithms are very competitive compared to the existing compact evolutionary algorithms, the original FA and LFA. This paper also shows the relative weakness of the existing compact optimization algorithms compared to most advanced evolutionary algorithms such as CoBiDE and SaDE but they present the advantage of less memory storage. The efficiency of the proposed uniform distribution function is also demonstrated. Due to their minimal memory and computational cost requirement, the proposed algorithms are very suitable for hardware implementation in systems control where the

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ACCEPTED MANUSCRIPT computational cost is limited such as robotics. The proposed cFAs were applied for the realization of an optimal swing-up movement of a humanoid robot hanging on a bar. Acknowledgments The authors sincerely thank the anonymous reviewers for their constructive and helpful comments and suggestions. With their interesting comments and suggestions, they helped us to enrich the quality of our paper.

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