05714781

IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 15, NO. 1, FEBRUARY 2011 67

Improving Classical and Decentralized DifferentialEvolution with New Mutation Operator and

Population TopologiesBernabe Dorronsoro and Pascal Bouvry

Abstract—Differential evolution (DE) algorithms compose anefficient type of evolutionary algorithm (EA) for the globaloptimization domain. Although it is well known that the pop-ulation structure has a major influence on the behavior of EAs,there are few works studying its effect in DE algorithms. Inthis paper, we propose and analyze several DE variants usingdifferent panmictic and decentralized population schemes. Asit happens for other EAs, we demonstrate that the populationscheme has a marked influence on the behavior of DE algorithmstoo. Additionally, a new operator for generating the mutantvector is proposed and compared versus a classical one on allthe proposed population models. After that, a new heterogeneousdecentralized DE algorithm combining the two studied operatorsin the best performing studied population structure has beendesigned and evaluated. In total, 13 new DE algorithms arepresented and evaluated in this paper. Summarizing our results,all the studied algorithms are highly competitive compared tothe state-of-the-art DE algorithms taken from the literaturefor most considered problems, and the best ones implementa decentralized population. With respect to the populationstructure, the proposed decentralized versions clearly providea better performance compared to the panmictic ones. The newmutation operator demonstrates a faster convergence on mostof the studied problems versus a classical operator taken fromthe DE literature. Finally, the new heterogeneous decentralizedDE is shown to improve the previously obtained results, andoutperform the compared state-of-the-art DEs.

Index Terms—Differential evolution (DE), heterogeneous algo-rithms, self-adaptation, structured population.

I. Introduction

EVOLUTIONARY algorithms (EAs) [1]–[3] are a familyof population-based optimization metaheuristics designed

for searching optimal values in complex spaces. Individuals inthe population represent tentative solutions to the problem athands, and the algorithm iteratively applies some stochasticvariation operators to them in order to make the populationevolving toward better solutions in the search space.

It is well accepted in the literature that the organizationof individuals in the population has a major influence on the

Manuscript received November 29, 2009; revised April 7, 2010, July 17,2010, and September 6, 2010; accepted September 15, 2010. Date of currentversion February 25, 2011. This work was supported by the Luxembourg FNRGreenIT Project (C09/IS/05).

The authors are with the Faculty of Sciences, Technology and Communi-cation, University of Luxembourg, Luxembourg City L-1359, Luxembourg(e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEVC.2010.2081369

search EAs perform [4]–[7]. The simplest population modelwe can consider is a set of individuals with no structure at allin it. In this case, all individuals can interact with any otherone in the population during the evolutionary process. Thispopulation model is known as panmictic, and it correspondsto the one shown in Fig. 1(a).

However, it is possible to define some structure into the pop-ulation to set some kind of (usually partial) isolation betweenindividuals. The effect is that the scattering of good solutionsthroughout the population is usually slowed down, thus pro-viding the algorithm with higher exploration capabilities ofthe search space with respect to panmictic populations [4]–[6]. These population models are usually called structured ordecentralized. We can distinguish two main canonical kinds ofdecentralized populations, namely distributed [6] (also calledcoarse-grained) and cellular [4] (or fine-grained).

In distributed EAs (dEAs), the population is partitioned intoseveral smaller subpopulations (islands). Then, independentEAs are evolving all the islands, exchanging some informationwith a given frequency. This model corresponds to the onedepicted in Fig. 1(b). It could happen that the algorithmsrunning in the islands have different configurations, like dis-tinct variation operators, their probabilities, the populationmodel, and so on, or even the algorithm itself (i.e., differentmetaheuristics could be implemented in the islands). This isthe case of heterogeneous distributed algorithms [8].

In the case of cellular EAs (cEAs), the population iscomposed of a large number of very small subpopulations(typically composed by only one individual, as in this paper)that are arranged in a toroidal mesh. This way, only thoseindividuals that are close each other in this mesh are allowedto interact during the evolution. In Fig. 1(c), we show anexample neighborhood (the shadowed cross in the center)we could consider for the central individual. It is called L5,Von Neumann, or NEWS (for North, East, West, and South)neighborhood. Then, during the evolutionary process, thecentral individual is only allowed to interact with its neighbors.The neighborhood of the top left-hand individual in the meshis also displayed to show the toroidal effect in the population.

Therefore, dEAs and cEAs define two boundaries in thepopulation structure in terms of the number of subpopulations,their size, and the amount of information they exchange [5].dEAs are composed by several large subpopulations that areloosely connected, meanwhile cEAs are made of tiny tightly

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68 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 15, NO. 1, FEBRUARY 2011

Fig. 1. (a) Panmictic EA has all its individuals—circles—in the samepopulation. Structuring the population usually leads to distinguish between(b) distributed and (c) cellular EAs.

connected subpopulations. Any model between these twoboundaries is, of course, valid and worth to study. Some ofthem are discussed in this paper.

Differential evolution (DE) [9] is a kind of EA that ishighly efficient for global optimization problems [10]–[13].Besides its good performance, they are very attractive due totheir simple design. Basically, they iteratively apply simplemathematical operations to evolve the population.

However, despite their efficiency and popularity, most of theexisting works in the literature implement DEs with panmicticpopulation [14]. A short survey on the main existing worksproposing decentralized population DEs is given in Section II.

The first contribution of this paper is the design, study, andcomparison of DE algorithms with panmictic and decentral-ized populations. Specifically, we consider two panmictic DEs(with synchronous and asynchronous population update), twocellular DEs (the canonical model and a hierarchical one),a distributed DE, and two more DEs with random and small-world topologies. As a second contribution, we designed a newmutation operator that efficiently improves the convergencespeed of the algorithm. We have studied the influence of thisoperator in seven different population models. Finally, ourthird important contribution is the design of a new distributedheterogeneous DE algorithm, which combines two DEs withthe two mutation operators studied in this paper. To the bestof our knowledge, this is the first time that synchronouspanmictic, cellular, random, and small-world populations areproposed for DE.

After this introduction, the paper continues with an overviewof the main existing works proposing either parallel DEapproaches or the use of decentralized populations for DEalgorithms. Section III gathers some relevant works presentingnew population topologies for other EAs different than DEs.

TABLE I

Main Existing Parallel and Decentralized DEs

Reference Topology Description[15] Coarse-grained Distributed islands in a randomly

generated topology[16] Coarse-grained Distributed islands with unidirec-

tional ring topology[17] Fine-grained Population arranged in a ring topol-

ogy[18]–[20] Coarse-grained Parameters influence study in ring

topology[21], [22] Coarse-grained Distributed islands in a toroidal

mesh topology[23], [24] Coarse-grained Co-evolutive dDE for constrained

optimization[25] Coarse-grained Cooperative co-evolutive dDE[26] Fine-grained Tree structured hierarchical popu-

lation[27] Coarse-grained Heterogeneous dDE with different

DE strategies in the islands[28] Coarse-grained Heterogeneous dDE with different

population families for explorationand exploitation

[29] Fine-grained Population arranged in a ring topol-ogy. Use of global and local knowl-edge

[30] Fine-grained Bare bones DE with ring and VonNeumann topologies

[31]–[36] Panmictic Master/slave model parallelized atthe level of fitness function evalu-ation

Then, Section IV describes the functioning of the canonicalDE algorithm, as well as the different population managementtechniques proposed in this paper. Section V introduces thenew mutation operator that we propose to accelerate theconvergence of the algorithm. After that, Section VI presentsthe benchmark of functions used during our experiments, andanalyze and evaluate the results, comparing them to otheralgorithms in the literature. Finally, we conclude and givesome future research lines in Section VII.

II. Parallel and Decentralized Differential

Evolution Algorithms: State of the Art

This section presents a brief survey on the main existingpapers dealing with decentralized and/or parallel populationDE algorithms in the literature. They are summarized inTable I. As the reader can see, almost all of them are dealingwith distributed (or coarse-grained) populations, and there areonly a few approaches studying the fine-grained model.

Probably, the first paper proposing the use of decentralizedpopulations for DE algorithms was the one by Zaharie andPetcu [15] (there are several earlier papers proposing parallelDE algorithms, but using panmictic populations [31], [32],[37]). This paper presents a parallel distributed self-adaptiveDE algorithm in which islands are connected in a randomstructure. The algorithm was compared versus a canonicalDE, and the results showed a better convergence and fasterexecutions for the new parallel model. The authors used up to16 processors (with one island per processor).

One year later, Tasoulis et al. [16] presented a paralleldistributed DE with the islands arranged in a more classical

DORRONSORO AND BOUVRY: IMPROVING DE WITH NEW MUTATION OPERATOR AND POPULATION TOPOLOGIES 69

topology: an unidirectional ring. They did several experimentswith different migration policies and DE algorithmic models,and they noticed that, even when the amount of informa-tion exchanged has an important impact on the behavior ofthe algorithm, not all the different mutation strategies wereequally sensitive.

After [16], there are a number of papers proposing andstudying different configurations for the unidirectional ringtopology (e.g., migration schemes, migration frequencies,number and size of subpopulations, and so on). Specifically,Kozlov and Samsonov proposed in [18] a simple new migra-tion scheme in which the oldest solution in the populationis replaced by the received one, instead of the worst one.As in the canonical model, every population sends its bestsolution to the neighboring one. Shing et al. [19] presented adetailed study on the parameterization of a parallel distributedDE for solving the learning scheme in asymmetric subsethoodproduct fuzzy neural inference system, studying the influenceof different migration interval and sizes, as well as differentnumber of islands. Finally, Apolloni et al. [20] proposed aparallel distributed DE algorithm for the benchmark problemspresented in CEC’05 special session on continuous optimiza-tion [38]. They studied dDEs with two and four subpopula-tions, comparing among them as well as with the classicalpanmictic DE. As a result, the two islands dDE was betterthan the four islands one, but none of them could significantlyoutperform the canonical panmictic DE.

As an alternative to the random and ring topologies, Falcoet al. [21], [22] proposed a distributed DE in which every sub-population is connected to other µ nodes (in their experiments,they set µ = 4). The subpopulations are arranged in a toroidalmesh, and they exchange solutions with all the other neighborswith some given frequency (after a number of generations).Therefore, although the topology of subpopulations is similarto the one used in the cellular model, it does not have themain features of such model (indeed, the authors claim it is acoarse-grained model), like the tight communications (usuallyevery fitness function evaluation), and the tiny subpopulationssize (typically, composed by only one individual).

Omran et al. studied in [17] the use of a fine-grained popula-tion in which individuals are arranged in a ring topology, and aneighborhood is defined in the population such that only indi-viduals that are located next to each others are able to interact(the proximity is defined by the neighborhood). Notice that thedifference with the previously commented ring topologies isthat, in this case, islands are composed by only one individual,and they are more tightly connected. In this algorithm, theinformation of the best local solution (i.e., the best solution inthe neighborhood of the considered one) is used in the processof generating the new ones. More recently, Das et al. proposedDEGL [29], another fine-grained DE with ring topology and aneighborhood defined on it. The main difference with respectto the one by Omran et al. is that in this case, a combinationof two different strategies using the global and local bestsolutions (in the whole population or at the neighborhoodlevel, respectively) is used in the evolutionary process.

Another fine-grained population DE is the bare bones DErecently presented by Omran et al. [30]. It is based on

the bare bones PSO, and two variants of the algorithm areanalyzed using both the ring and Von Neumann topologies.As a result, the new bare bones DE was demonstrated to behighly competitive with the compared DEs and PSO, speciallyfor the high dimension problems tested.

A DE with hierarchical population was proposed by Shiet al. in [26]. This algorithm organizes the individuals inthe population in a tree hierarchy in such a way that betterindividuals are located in the upper levels (close to the root),as it was previously proposed in [39] for PSO algorithms. Inthe evolution process, the base mutation individual is selectedto be the parent of the current individual, according to thehierarchy. Then, two more random individuals are selectedfrom the whole population to participate in the generationof the offspring. The algorithm outperformed the comparedPSO and standard DE algorithms on the kinetic parameterestimation of two different chemical reactions.

There exist three main works presenting co-evolutionaryapproaches for DE algorithms in the literature. The first coop-erative co-evolutionary DE algorithm was presented in [25]. Inthis paper, the authors followed the classical approach [40] ofsplitting the solution vector of high dimensional problems intosmaller vectors that are separately solved by DE algorithms toco-evolve for the resolution of the big problem. In 2007, twomore co-evolutionary DE algorithms appeared for solving con-strained optimization problems [23], [24]. The two papers pro-pose similar approaches to deal with the problem. Their algo-rithms are composed of two populations: the first one in chargeof optimizing the problem regardless of the constraints, whilethe second population is focussed on minimizing the constraintviolations without considering the fitness of the solutions.

Recently, Izzo et al. designed in [27] a heterogeneousasynchronous island model for DE. They considered fiveislands and five DE strategies (DE/best/1/exp, DE/rand/1/exp,DE/rand-to-best/1/exp, DE/best/2/exp, and DE/rand/2/exp),and studied five distributed DEs using the same DE strategy inall the islands, and a heterogeneous model with one differentDE strategy in every island. As a result, the heterogeneousmodel is not outstanding, but performs as well as the others.Additionally, Weber et al. [28] proposed another heteroge-neous distributed DE which was composed of two familiesof subpopulations. The first one is composed of a set ofsubpopulations with fixed size arranged in a ring topology,and in charge of exploring the search space. The other oneis composed of dynamic size subpopulations that aim to behighly exploitative in the neighborhoods of the best solutionsfound by the first kind of subpopulations. The two subpopula-tion families exchange some solutions with a given frequency.The model was compared versus three other distributed DEs,those presented in [16], [20], and [21] (and already commentedhereinbefore), outperforming all of them.

There exist also several works proposing parallel DE al-gorithms, but implementing panmictic populations. This ap-proach is usually adopted by the authors to solve very complexproblems (generally in the engineering and bioinformaticsfields) by parallelizing the fitness function evaluation followinga master/slave model. Lampinen proposed in [31] a specialparallel implementation of DE algorithm using a cluster of


workstations connected via local area network in which asyn-chronous communications among the different master/slaveprocesses are made through shared disk files. Other panmicticDEs parallelized with the master/slave model have been pro-posed for the learning scheme in asymmetric subsethood prod-uct fuzzy neural inference system [33], for 3-D medical imageregistration [34], for the training of neural networks [35], andfor benchmarks of theoretical functions [36].

The works by Takahama [41] and Lai [42] deserve also tobe mentioned here because, even when they are implementingpanmictic populations too, they consider some kind of struc-ture in the representation of the solutions. The two papers arebased on the concept that variables in the solution may havedifferent levels of importance. Therefore, the less importantvariables will have less influence (they could even be disabled)during the evolution process.

To finish this section, it is also worth mentioning the workby Das et al. [43], [44] that proposes a mathematical model ofthe underlying evolutionary dynamics of a simple DE with 1-D fitness landscape (i.e., when solving one variable problems).The model shows that the fundamental dynamics of each agent(parameter vector) in DE employs the gradient-descent typesearch strategy, with a learning rate parameter that dependson the typical DE control parameters.

III. Population Topologies for Evolutionary

Algorithms

After reviewing in Section II the main existing decentralizedpopulation models used in the DE literature and their parallelversions, we now focus on some other relevant topologies thathave appeared for other families of EAs. Then, we borrowsome of these new population topologies to implement theminto DE algorithms in this paper. There are a large number ofpapers studying and proposing different population topologiesfor EAs. It is not the scope of this section to revise all ofthem, but only some of the most outstanding ones.

The influence of using fine-grained and coarse-grainedpopulations in EAs has been deeply investigated in the litera-ture [4]–[7], [45]–[48]. As we already mentioned, they are onlytwo boundary cases of decentralized populations. Recently,several works appeared studying new population topologiesthat share properties of both models. They are discussed below.

Janson et al. proposed in [39] a hierarchical PSO (H-PSO) in which individuals are arranged in a tree hierarchy.Therefore, better solutions move toward the highest levels ofthe hierarchy, exchanging their position with worse ones. Inthis hierarchy, particles are influenced by their personal bestsolution and by its parent in the hierarchy. Later, in [49]the authors proposed a new cellular genetic algorithm (cGA)with a pyramidal hierarchy into the population, such thatthe best individuals will be placed in the highest levels ofthe hierarchy. Therefore, individuals interact in this case withmore than one individual from the hierarchy, as determinedby the neighborhood defined in the cellular population. Theeffect is that the exploitation of the most promising solutionsis enhanced, since they are located next to each other in thepopulation thanks to the hierarchy, and the cellular population

promotes the interaction of neighboring individuals. At thesame time, the diversity of solutions in the population ismaintained due to the evolution of worse individuals at thelower levels of the hierarchy, promoting the exploration ofother regions of the search space different than the ones wherethe most promising current solutions are.

There exist some works analyzing new fine-grained topolo-gies that are not as connected as the panmictic population(which is fully connected), but have shorter characteristic pathlength (i.e., the maximum distance between any two solutions)than the cellular model. In particular, it is worth mentioning thestudies made by Giacobini et al. to both theoretically [50] andempirically [51] analyze the behavior of different GAs usingrandom, scale-free, and small-world topologies. Additionally,Payne et al. addressed in [52] another theoretical study on thebehavior of GAs with scale-free and small-world topologies,and they later extended it in [53] to analyze the effects ofsome characteristics of these kinds of networks, like the scaleand assortativity (the degree of connections of nodes). Theyarrived at the conclusion that increasing the assortativity leadsto shorter convergence time, while high scale networks providelonger convergence rates. However, the main conclusion ofthese studies by Giacobini et al. and Payne et al. is that small-world populations are competitive, but the potential utility ofscale-free population structures is still unclear.

Despite random graphs were not the best performing onesin the studies made by Giacobini et al. for GAs [50], [51],they have become popular for PSO algorithms. Indeed, this isthe population model used in the Standard PSO 2007 (SPSO07) [54]. Kennedy and Mendes [55], [56] deeply investigatedon the use of fine-grained population topologies (e.g., ring,Von Neumann, random, pyramid, or star graphs, to name afew) for PSO algorithms. They concluded by recommendingthe use of Von Neumann topologies. In [57], the same authorspresented a new fully informed PSO algorithm in which theinformation of all the neighbors is taken into account in thegeneration of new solutions.

There exist also several approaches using dynamic topolo-gies. In this sense, Sunganthan presented [58] probably the firstEA with a dynamic population topology: a PSO algorithm withvariable neighborhood sizes. The idea is that every particlestarts with a neighborhood of size 1 (the particle itself), andthen the size of the neighborhood is increasing during the run.One year later, Kennedy [59] proposed splitting the populationinto clusters (groups of swarms), and the centroid of everycluster is the one influencing the other particles in its cluster.A similar concept is the clubs-based PSO [60], but in thiscase particles can belong to different clubs (or clusters), andthey can also join to or quit from a club. A third algorithm wewould like to mention here is the species-based PSO, proposedby Li in 2004 [61]. In this case, the population is split intoswarms of the same species, i.e., containing particles withsome similarities. Only those particles of the same speciesinteract in the creation of the next generation of solutions.

In 2008, Whitacre et al. proposed in [62] a dynamicpopulation structure for GAs that automatically evolves byfollowing a set of simple rules to mimic the interactionnetworks of complex systems. Individuals are dynamically


Fig. 2. Taxonomy of the studied population schemes.

created or removed from the population in terms of theirquality by following some rules. The population is modeledas a graph, and therefore its structure is self-evolving whenadding or removing solutions. As a result, this kind of popu-lation performs longer convergence times compared to cGAs.However, as a consequence, they can provide better resultsafter a high number of evaluations. Similar topologies wereproposed for PSO by Godoy et al. [63] and Clerc [64].

IV. Differential Evolution

This section presents the seven different populationmodels studied for DE algorithms. Among them, onlythe classical DE—presented in Section IV—and thedistributed (Section IV-C) populations were used beforein the literature for DE algorithms. Additionally, someheterogeneous distributed DEs (Section IV-H) were proposedbefore in the literature. However, the classical DE withsynchronous population update (Section IV-B), the cellularDE (Section IV-D), the hierarchical cellular DE (Section IV-E),the random topology DE (Section IV-F), and the small-worldDE (Section IV-G) are applied for the first time to a DEalgorithm, to the best of our knowledge. All these studiedpopulation schemes are classified in Fig. 2.

A. Classical Differential Evolution Algorithm

We describe in this section the functioning of a typicalDE algorithm. The main difference between DE and otherEAs is the way the population of solutions is evolved. Theclassical DE algorithm implements a panmictic population,and the new generated solutions are immediately inserted intothe population, so they can interact with solutions from theirparents’ generation.

The pseudocode for the DE algorithm is given in Fig. 3. Letus suppose the algorithm works on a population P composedby NP tentative solutions (P = (xi), 0 ≤ i < NP) andthe goal is to minimize function f (). The initial populationis generated by assigning random values to the variables ofevery solution. For evolving every individual i, three parentsare selected from P , we call them xr0 , xr1 , and xr2 . Then, thetwo next steps are the main ones characterizing the differentDE variants existing in the literature. They are the mutantvector vi generation and its recombination with the referencevector. We follow in this paper the DE/rand/1/bin scheme, sothese operations are defined by (1) and (2), respectively

vi = xr0 + F · (xr1 − xr2 ) (1)

Fig. 3. DE pseudocode.

ui,j =

{vi,j , if rand(0, 1) ≤ CR or j = jrand

xi,j , otherwise(2)

finally, the fittest solution between the current and the newone will remain in the population, as (3) defines (without lossof generality, we are assuming the minimization of f )

xi =

{ui, if f (ui) ≤ f (xi)xi, otherwise.

(3)

As it can be seen in the description given before, DE algo-rithms require two control parameters, F and CR, to generatethe mutant vector and the new solution, respectively. Typically,their recommended values in the literature are F = (0, 2] andCR = [0, 1]. However, finding the correct value for themis a critical issue, since they have a major impact on thebehavior of the algorithm [12], [65]. Additionally, the bestcontrol parameters setting can be different depending on theproblem to optimize [20].

We are interested in avoiding a strong dependency on thecontrol parameters in our results. Therefore, we adopt a self-adaptive strategy that automatically chooses good values forthe two parameters. It lies in assigning different CR andF values to every solution. Then, when a new solution iscreated, it inherits these values from the reference vector, andthey are modified with probability 0.1 by assigning randomvalues in the intervals [0.1, 1.0] and [0.0, 1.0] for F andCR, respectively—as shown in (4), where rand1 to rand4 arerandom values in the interval [0, 1]

F =

{0.1 + 0.9 ∗ rand1, if rand2 ≤ 0.1

F, otherwise

C =

{rand3, if rand4 ≤ 0.1

CR, otherwise. (4)

This strategy was proposed by Brest et al. in [66], wherethis self-adaptive DE was highly competitive with respect to


Fig. 4. Synchronous DE (sDE) pseudocode.

the equivalent standard DE (with fixed values for the controlparameters) and other EAs.

B. Synchronous Differential Evolution

The synchronous DE (sDE) is another kind of panmicticalgorithm. The difference with respect to DE is that the popu-lation is updated with all the new solutions at the same time,instead of updating the solutions just after being generated.We give its pseudocode in Fig. 4. As it can be seen, thebest solution between the current one xi and the new oneui is stored in an auxiliary population. Once we have doneit for all the individuals (i.e., a generation is complete), thecurrent population is replaced by the auxiliary one, and thenthe algorithm is ready to perform the next generation.

The effects of using either a synchronous or an asyn-chronous population update have been studied for other EAs inthe literature, both theoretically and experimentally in severaldomains [2], [4], [67], [68]. Basically, we can say that the con-vergence speed of the population is slower in the case of syn-chronous populations. The reason is that for the asynchronousmodel, new solutions are inserted into the population just afterbeing generated, replacing usually worse ones, and they cantake part in the evolution process of their parents’ generationsolutions. In the case of synchronous panmictic populations,these new individuals are stored in an auxiliary structure, andtherefore, they cannot interact in the evolution process until thewhole generation is finished. Consequently, the convergenceof the population is slowed down, while at the same time theexploration capabilities of the algorithm are enhanced.

C. Distributed Differential Evolution

In distributed DE (a pseudocode is provided in Fig. 5), thepopulation is partitioned into a number of islands, which are

Fig. 5. Distributed DE (dDE) pseudocode.

evolved by independent DEs. We may also refer to the popula-tions in the islands as subpopulations of the algorithm, as it iscommonly done in the literature. In our study, we have chosenthe regular DE algorithm to evolve these subpopulations. Wecould use any other population model for the islands of thedDE, and it is a matter of study for future work.

The subpopulations are typically arranged in a unidirec-tional ring topology. This means that every subpopulation cansend information to only one other subpopulation, and in thesame way it can only receive information from one singlesubpopulation. With a given frequency, every subpopulationsends its best current solution to the neighboring subpopula-tion. This process is called migration. When an island receivesthe best solution from its neighboring island, it is inserted intoits subpopulation replacing the worst one. This replacement isalways carried out, even when the received solution is lessaccurate than the worst one in the local subpopulation. Thiswill allow the algorithm to better benefit from the diversity ofsolutions in the different islands.

D. Cellular Differential Evolution

In the case of the cellular topology [4], solutions are spreadin a 2-D toroidal mesh, and they are allowed to interact onlywith their neighbors. Therefore, when using this topology weare introducing some kind of isolation by distance among thesolutions in the population. The reason is that multiple gener-ations are required by the algorithm before the information ina good solution will affect a distant solution in the populationdue to the reduced size of the neighborhood.

It has been demonstrated for several EA families (see [4]for some examples) that the performance of the algorithm isgenerally improved after implementing the cellular topologyinto the population. Consequently, we are highly motivated toanalyze if this is the case of DE algorithms too.

The cellular DE (cDE) algorithm we are introducing in thispaper is similar to the canonical cEA model, and it followsthe pseudo-code included in Fig. 6. In this basic cDE, thepopulation is structured in a regular grid of two dimensions,and a neighborhood is defined on it. The algorithm iteratively


Fig. 6. Cellular DE (cDE) pseudocode.

considers as current each solution in the grid, and this solutioncan only interact with its neighboring solutions in the evolutionprocess. Therefore, parents are in this case chosen among theneighbors. Then, both the mutant vector and the new solutionare generated, the fitness value for the latter is computed, andit is inserted instead of the current solution in the populationfollowing a given replacement policy. This loop is repeateduntil the termination condition is met.

E. Hierarchical Cellular Differential Evolution

The hierarchical cDE (HcDE) algorithm is a cDE in whichsolutions are arranged in the population according to theirfitness. Specifically, good solutions will be moved towardthe central positions of the population, while worse ones arelocated far from it, as it is displayed in Fig. 7. The pursuedeffect is to enhance the exploitation of promising regions of thesearch space, since all the good solutions are placed togetherin the center of the population, allowing high quality solutionsto interact among them during the evolution process. Thiswill allow to reach better solutions faster. However, prematureconvergence is avoided at the same time since low qualitysolutions are generating diversity in regions far from the centerof the population.

We present a pseudocode for HcDE in Fig. 8. As it can beseen, after every generation all the solutions are re-arrangedby moving them one step toward the center of the population.This movement is done by swapping it with its neighbor closerto the center in the case the former has better fitness value

Fig. 7. Hierarchy used in the population in HcDEs.

Fig. 8. Hierarchical cellular DE (HcDE) pseudocode.

than the latter. This hierarchical model was designed for thefirst time in [49] for cellular GAs, and we adopt it in thispaper for the cellular DE algorithm. At the very best of ourknowledge, the only precedent DE with hierarchical popula-tion was proposed in [26] for a tree-structured population.Additionally, we have improved the model in order to maintainthe hierarchical structure every time a new solution is insertedby allowing swap movements. In the H-cGA presented in [49]this was done at the same time for all the individuals afterevery generation. Therefore, the proposed model fits betterwith the cellular DE implementation.

F. Random Topology Differential Evolution

The DE algorithm presented in this section, called randDE,borrows from other EAs [50], [54], [55] the topology basedon pseudo-random graphs. In randDE, solutions are arrangedat the vertices of a randomly generated unidirectional graph,and the neighborhood of every solution is determined by theother solutions to which it is connected in the graph.


The main idea behind the random topology is to have ashort characteristic path length, but still in a decentralizedpopulation. This means that the characteristic path length(i.e., the maximum distance between any two solutions inthe population) is shorter than in the cellular topology, butmuch longer than in the case of panmictic populations (withcharacteristic path length of one link).

A pseudocode of randDE is provided in Fig. 9. The first stepis to generate the initial population with random solutions, andto evaluate them. After that, the algorithm enters the loop forevolving the population until the termination condition is met(lines 3 to 22). The first operation to perform in this loop is thecreation of the quasi-random population topology [54] that willbe considered in the current evolution step (notice that this willbe done only when no solution was improved in the populationin the previous generation, as it is described in [54]). For that,we define the maximum allowed neighborhood size, i.e., themaximum number of edges every vertex can have, as [54]

max neighb size = 10.0 + 2.0 · √num vars (5)

where num vars is the number of variables to opti-mize for the considered problem. As suggested in [54],max neighb size is not allowed to be greater than 910, soin the case that (5) exceeds that value (this will happen forproblems having more than 202 500 variables), the variable isset to this upper bound. Then, the quasi-random topology isbuilt by adding every possible solution to the neighborhood ofthe considered one with probability pconnect , defined in (6) [54]

pconnect = 1.0 −(

1.0 − 1.0

max neighb size

)3

. (6)

Once the topology is generated, the algorithm iterates toevolve every solution just as in the case of the cellular model,but taking into account the neighborhoods defined by thequasi-random topology.

G. Small-World Differential Evolution

Small-world topologies were adopted for EAs for the firsttime by Giacobini et al. in [50] as an attempt to slowdown the convergence speed of similar EAs with randomtopologies. Small-world graphs [69], [70] share propertiesof both random graphs (short characteristic path length) andregular lattices (high clustering coefficient). On the one hand,the short characteristic path length allows a faster diffusionof the information contained within good solutions throughthe population with respect to an equivalent algorithm witha regular lattice having a similar connectivity degree. Thiseffect enhances the exploitation performed by the algorithmon the search space. On the other hand, the high clusteringcoefficient (i.e., high connectivity degree among the vertices inthe graph) allows the formation of different niches, containingsimilar individuals each. This provides the algorithm withbetter exploration capabilities with respect to an equivalentone with random or fully-connected topologies.

We present in this section swDE, a new DE algorithm withsmall-world topology population. The small-world topology is

Fig. 9. DE with random topology (randDE) pseudocode.

generated in swDE as proposed by Watts and Strogatz [71].The process starts by creating a regular ring topology in whicheach node has k neighbors (with k � the number of vertices).Then, for every node, we change the links to its neighbors bya link to a randomly chosen solution in the population withprobability β = 0.05. Examples of the starting and finishingnetworks for this process are shown in Fig. 10.

The pseudocode for the proposed swDE algorithm is pre-sented in Fig. 11. It starts by randomly generating the popula-tion of new individuals, which are evaluated afterward. Then,the small-world topology is created as explained above todefine the relationship among the solutions in the population.After that, the algorithm iteratively evolves every individuallike the cellular DE does, but taking into account the neigh-borhoods defined by the small-world topology.

H. Heterogeneous Distributed Differential Evolution

Heterogeneous EAs are distributed EAs that implementdifferent configurations in the different subpopulations [8].The differences of the configurations of the islands can be,for instance, on some parameters in the variation operators,on some of the variation operators, on the encoding scheme,or on the algorithm running in every island [72], [73]. Thesealgorithms are useful to explore the search space in a differentway in all the subpopulations, obtaining a better overallperformance with respect to the algorithms running in theislands themselves.

In this paper, we designed a simple heterogeneous DEmodel composed by two islands, and a different mechanismis used in every island for generating the mutant vector. One


Fig. 10. Small-world graph (right-hand side) is generated by randomlyrewiring some edges from a regular ring-shape graph (left-hand side).

Fig. 11. DE with small-world topology (swDE) pseudocode.

of them is likely the most commonly used method in DE,already described in Section IV-A, while the other one is anew operator introduced in the next section.

V. GPBX-α: A New Mutation Operator for DE

We propose in this section a new operator for generating themutant vector in DE algorithms. We call it Gaussian PBX-α(or GPBX-α). It is based on the PBX-α [74] recombinationoperator, designed for genetic algorithms to solve continuousoptimization problems.

PBX-α is a parent centric recombination operator that cre-ates an offspring solution from two parents. It is called parentcentric because the offspring is created around the region ofthe dominant parent (called the female parent in the literature),which is slightly modified with the information of the otherparent (the male one). This idea is similar to the one usedin DE for generating the mutant vector: in the classical DE

Fig. 12. PBX-α operator.

approach, for instance, it is computed by adding to a referencevector (the equivalent to the female parent in PBX-α) thedifference between two other vectors (two male parents).We find some analogy between the mutant vector generationoperator in DE and parent centric recombination operatorsfrom other kinds of EAs. Therefore, we find that it is verysuitable to adapt this type of recombination operators for DE.

PBX-α is a parent centric version of the well-known BLX-αrecombination operator [75], [76]. PBX-α works on two real-coded chromosomes x = (x1, . . . , xn) and y = (y1, . . . , yn),where n is the number of variables of the problem to solve.It is considered x to be a dominant parent (called female par-ent by the authors) and y a dominated one (the male parent),and then an offspring z = (z1, . . . , zn) is generated such thatevery zi in z is a randomly (uniformly) chosen number fromthe interval

[lowi, upi

], where:

1) lowi = max {mini, xi − I · α}, where mini is the mini-mum allowed value for variable i;

2) upi = min {maxi, xi + I · α}, where maxi is the maxi-mum allowed value for variable i;

3) I = |xi − yi|.We graphically show in Fig. 12 how new solutions are

generated with PBX-α. As it can be seen, every variableof the new solution belongs to an interval centered in thevalue the dominant parent has for this variable, and its rangedepends on the value of the dominated parent for this variable.Even though the PBX-α operator introduces an additionalparameter, this can be considered acceptable because addi-tional functionality with respect to managing the exploration/exploitation tradeoff is obtained.

Gaussian PBX-α (GPBX-α) is a new operator for generatingthe mutant vector in DE algorithms that is inspired by someideas coming from PBX-α. We show in Fig. 13 how thisoperator generates new solutions, and we give its definitionin (7)—the same nomenclature introduced in Section IV isadopted here, where subindex i refers to a position in thepopulation, and j is the variable index of the solution

vi,j = xr0,j + Gaussian (0.0, 1.0) · F · (upi,j − lowi,j

);

upi,j = min(maxj, xr0,j − Ij · α) ;lowi,j = max(minj, xr0,j + Ij · α) ;Ij = xr1,j − xr2,j

(7)where Gaussian (0.0, 1.0) returns a random number from aGaussian distribution centered in 0.0 with deviation 1.0, and


Fig. 13. New GPBX-α operator.

Fig. 14. Search performed by the regular mutation (left-hand side) andGPBX-α (right-hand side).

maxj and minj are the upper and lower allowed bounds forvariable j, respectively.

For variable α, we propose a new adaptive approach similarto the one adopted for F . For the new offspring solutions, the α

value is inherited from the current solution in 90% of the cases,and with probability 0.1 its value is modified to a randomlychosen one from the interval [0.2, 0.8], as it is defined in

α =

{0.2 + 0.6 ∗ rand1, if rand2 ≤ 0.1α, otherwise.

(8)

Every variable j of the mutant vector vi is a random numbertaken from a gaussian probability distribution. This gaussiandistribution is centered in xr0,j (the value of the referencevector for variable j), and the deviation is set to upi,j − lowi,j

multiplied by the control parameter F , typically used in DEfor the mutant vector generation. The gaussian probability wasused in order to better promote small changes with higherprobability than big ones in the variables of the referencevector. Therefore, new solutions will be allocated in the searchspace close to either the mutant vector (which is, at the sametime, in the neighborhood of the reference vector) or thecurrent solution, as it is shown in Fig. 14.

VI. Experimentation

We present in this section all the experiments we haveperformed for this paper. First, we describe in Section VI-A theparameters used in the configuration of our algorithms. Then,

we summarize the results for all the population models withthe two mutation schemes, the regular one (Section VI-B) andGPBX-α (Section VI-C). After that, the behavior of the studiedDEs with the two mutations is compared in Section VI-D.Section VI-E presents the results obtained with HdDE, andfinally our best algorithm is compared versus other state-of-the-art DE algorithms in Section VI-F.

In order to analyze and compare all the studied algorithms,we have selected the set of problems proposed for CEC com-petition on real-parameter optimization problems in 2005 [38].This benchmark, composed by 25 minimization problems, iswidely used in the literature. The mathematical description ofthe 25 functions is provided in that work. These functions haveseveral different features, such as epistasis, multimodality,noise, deceptiveness, and so on, that make all of them difficultto be solved. In addition, they all have been displaced in orderto move the optimum away from the center of the searchspace. Functions F1 to F5 are unimodal, while the other onesare multimodal: F6 to F12 are basic multimodal functions,F13 and F14 are expanded functions, and F15 to F25 arehybrid functions. The number of variables of all the functionsis scalable (up to 50), and we have considered in this paperdimensions D = 30 and D = 50 for all the problems.

The authors provide the implementation of all the functionsin several programming languages.1 The Java implementationwas used in the experiments presented in this paper.

All the results shown in this section represent the error of thesolutions found by the algorithms with respect to the optimalsolution to the problem, which is known for all of them. Weconsider the most accurate precision we can get with Javaversion 1.5 in the hardware used for the experiments (IntelXeon 2.0 GHz processor): 4.9E−324. All the presented resultsare obtained after performing 100 independent runs, and theyare compared using the Iman-Davenport (to check differencesbetween all the algorithms for all the problems), Holm (foranalyzing the differences between pairs of algorithms for allthe problems), and Wilcoxon unpaired signed-ranks (to lookfor significant differences between pairs of algorithms forevery single problem). After analyzing several statistical testsin [77], these ones were considered to be highly appropriateto compare algorithms on the benchmark of functions usedin this paper. Therefore, all the differences commented in thispaper are always statistically significant according to the tests,unless the opposite is said.

A. Parameterization

The parameters we have used in our algorithms are summa-rized in Table II. We have studied the behavior of the classicalDE/rand/1/bin algorithm with seven different population man-agement schemes, namely panmictic with synchronous andasynchronous population update, distributed, cellular, hierar-chical cellular, random, and small-world; all of them explainedin Section IV. The algorithms have been tested with tworecombination operators, the classical one in DE (describedin Section IV-A), and the new GPBX-α operator (presented inSection V).

1Available at http://www3.ntu.edu.sg/home/EPNSugan.


TABLE II

Summary of the Configurations Used in Our Algorithms

DE model DE/rand/1/binPopulation schemes Panmictic (synchronous and asynchronous), distributed,

cellular, hierarchical cellular, random, and small-worldPopulation size 50 solutions

distributed algorithms: 100 sols.Pop. arrangement 4 islands (in a ring) of 25 sols. for dDE

7 × 7 mesh in cDE and HcDE2 islands of 25 and 75 solutions in HdDE

Selection scheme RandomMutation strategy Regular: vi = xr0 + F · (xr1 − xr2)

GPBX-αReplacement Replace if non worseControl parameters F and CR are self-adaptive [66]Stopping criterion Find optimum or max. number of evaluations

Fig. 15. C9 is the neighborhood used in the cellular populations.

The population size was set to 50 individuals for the pan-mictic algorithms, as well as for the random and small-worldtopologies, to 49 individuals (the square population with theclosest size to 50) for the two cellular ones (arranged in a meshof 7 × 7), and to 100 individuals in the distributed algorithms.In the case of dDE, individuals are distributed in four islandsof 25 individuals each, arranged in a ring topology. For HdDE,we have two populations with different sizes: 75 individualsfor the population with the regular mutation strategy, and 25individuals for the population with GPBX-α. The populationsizes for dDE and HdDE were experimentally chosen fora good performance. In the two distributed algorithms, theregular population model is used in the islands. The migrationof solutions between the populations is done every 1000fitness function evaluations for dDE. In the case of HdDE,this number was increased to 5000 in order to allow thedifferent mutation strategies in the subpopulations to evolveindependently for longer, enhancing this way the explorationof different regions of the search space in the islands. Theneighborhood used in the cellular models is C9, composed bythe central solution and the closest eight ones (see Fig. 15).

The solutions involved in the evolution process are ran-domly selected, as it is commonly adopted in the DE literature.We would like to mention that this selection is different tothe one typically used in the literature when DEs implementsome neighborhood structure defined in the population, sincein those cases the best neighbor is always used in the evolution

TABLE III

Pairwise Comparison of the Algorithms with Standard

Mutation Operator by Means of Holm Test. Adjusted p-Values

30 VariablesDE sDE dDE cDE HcDE randDE

sDE 1.0 – – – – –dDE 0.7055 1.0 – – – –cDE 1.0 1.0 1.0 – – –HcDE 1.0 1.0 1.0 1.0 – –randDE 1.0 1.0 0.0157 0.1318 1.0 –swDE 1.0 1.0 1.0 1.0 1.0 0.2677

50 VariablesDE sDE dDE cDE HcDE randDE

sDE 1.0 – – – – –dDE 0.1758 4.2002E − 2 – – – –cDE 1.0 0.3644 1.0 – – –HcDE 1.0 1.0 1.0 1.0 – –randDE 1.0 1.0 6.7700E − 4 1.5991E − 2 0.1236 –swDE 0.3997 0.1236 1.0 1.0 1.0 2.9293E − 3

process. All the selected parents are forced to be differentsolutions of the population.

The values for the CR and F control parameters are self-assigned by the individuals using the self-adaptive criterionproposed in [66], and explained in Section IV-A in orderto make this paper self-contained. Regarding the other self-adaptive parameter, α, its value is initialized to 0.5 for everyindividual. Then, it automatically evolves as described inSection V. New solutions replace the current ones in thepopulation if they have the same fitness value or if the newsolution is better. This policy is followed in order to increasethe diversity in the population along the run. Finally, thestopping condition is either to find the optimal solution tothe problem (with an error given by the maximum accuracyof our system, i.e., 4.9E − 324), or to perform a maximumnumber of evaluations.

We would like to emphasize at this point that the config-urations of the algorithms are exactly the same for all theproblems and dimensions, unlike other papers in the literaturethat tune the algorithms for every problem in order to havebetter performance [13], [20], [78].

Finally, we adopt the same evaluations limit proposedin [38]: 300 000 and 500 000 for the 30 and 50 variables prob-lems, respectively. Additionally, all the compared algorithmsstart from the same initial population (but we used differentinitial populations for every independent run). In the case ofthe populations with more than 50 solutions, 50 of them arecommon to the ones of the other compared algorithms, whilethe rest are randomly generated.

B. Population Models Comparison

We compare in this section the canonical DE model versusthe other proposed DEs. The best and average solutions (withstandard deviation) found by all the algorithms for everyproblem with 30 and 50 dimensions are shown in Tables IVand V, respectively. When the optimal solution was found bythe algorithms, the percentage of runs in which it happened isindicated between parenthesis. Figures in bold font representthe best results we found for every problem. The backgroundcolor of the cells is related to the unpaired Wilcoxon statistical


tests performed to compare DE with the other algorithms forevery problem, this will be addressed later in a few lines.

First, we make a statistical study to compare the behaviorof all the algorithms on the 25 studied problems, for everyproblem dimension, as suggested in [77]. For that, we appliedthe Iman-Davenport test on the average solutions found bythe algorithms on every problem. The p-values we obtainedwere 0.0243 and 3.6730E − 5 for the 30 and 50 dimensionalproblems, respectively. Therefore, the test states that thereare significant differences on the behavior of the consideredalgorithms for all the problems. Consequently, we proceedto perform a pairwise comparison of the algorithms in allthe problems by means of the Holm test. The results areshown in Table III (bolded figures correspond to significantdifferences). As it can be seen, the test detects a significantlydifferent behavior between randDE and dDE for all the 30dimensional problems (the latter being better than the former).In the case of the big problems, we detected more differencesamong the algorithms. In particular, dDE outperforms sDE andrandDE, being this latter one outperformed by cDE and swDEtoo.

After analyzing the performance of the algorithms on all theproblems, we perform now a much more detailed statisticalstudy to compare the behavior of the algorithms for everyproblem with respect to DE. For that, we applied the unpairedWilcoxon test to compare the performance of DE versus theother studied algorithms in terms of the quality of solutionsachieved in every independent run (see Tables XVI and XVII).Those figures in Tables IV and V with light gray backgroundare significantly worse than DE according to our statistical test(with 95% confidence), while the dark gray background standsfor significantly better results. We did not find significantdifferences with DE in the other cases.

If we focus on the 30 dimensional problems shown inTable IV, we can see that DE is never (significantly) betterin terms of the found solutions than all the other comparedmodels. The best algorithms for these problems are dDE,better than all the other ones according to our statistical testfor 9 functions (F2, F3, F4, F6, F10, F11, F13, F16, andF17), and swDE, outperforming the other algorithms in twoproblems (F9, and F15). None of the other algorithms couldoutperform all the other ones for any problem. However, onlydDE (for problem F8) and randDE (for F9, F17, F18, F19,F20, F22, and F23) were outperformed by all the otheralgorithms. All the studied algorithms perform similarly forF1 and F24.

We did not find statistically significant differences betweenDE and the other panmictic DE, sDE. The only exception isF8, for which sDE is significantly better than DE. Regardingthe decentralized algorithms, the number of problems in whichthey outperform DE is higher than those for which DE is better(with the exception of randDE): dDE, cDE, HcDE, and swDEare better than DE for 12, 10, 8, and 11 problems, respectively,being outperformed by DE in seven, four, seven, and sixproblems. Finally, it stands out that randDE outperforms DEfor nine problems, but it was worse for 11 out of the 25studied problems. We are always speaking about statisticallysignificant differences.

We proceed now to analyze the behavior of the samealgorithms for the big problem instances. The results areprovided in Table V. As it happened for the small problems,DE is never outperforming all the compared algorithms fornone of them. Once again, dDE is the algorithm whichsignificantly outperforms all the other ones in more cases (F2,F3, F7, F10, F11, F13, F16, F17, and F25). In secondplace, HcDE and swDE are the best ones in functions F6and F23, respectively. The dDE algorithm was, as for thesmall problems, the worst one for F8, while randDE wasoutperformed by the others for F9, F12, F18, F19, and F20.

Comparing the new proposed algorithms versus DE, allthe decentralized algorithms outperform it in a larger numberof problems (15, 12, 10, 10, and 13 for dDE, cDE, HcDE,randDE, and swDE, respectively) than the cases where theywere worse (nine, six, seven, seven, and six, respectively), asshown in Table XVII. The behavior of the other panmicticalgorithm, sDE, is similar for these problems to DE, sincewe found significant differences only for problems F1 (wheresDE is better) and F3 (for which DE outperforms sDE).

The best algorithms in Table V are dDE, cDE, and swDE,since they outperform DE with higher frequency. Amongthese three algorithms, even when dDE is outperforming DEmore times, both cDE and swDE stand out as those onesbeing outperformed by DE for less problems (i.e., six). Ifwe compute for all the algorithms the difference between thenumber of times they outperform DE and the number of timesthey are worse than it, we find that swDE is the best algorithm,with seven, followed by dDE and cDE, with six, HcDE andrandDE, with three, and sDE, with zero.

We also compare the algorithms in terms of efficiency, i.e.,in terms of the number of evaluations they require to findthe optimal solution. We can only compare them for functionF1, since it is the only problem for which they can findthe optimal solution. We graphically show the comparisonof the algorithms for F1 with 30 (left-hand side) and 50(right-hand side) variables in the boxplots in Fig. 16. Inthese boxplots, the bottom and top of the boxes represent thelower and upper quartiles of the data distribution, respectively,while the line between them is the median. The whiskersare the lowest and highest datum still within 1.5 IQR ofthe lower and higher quartiles, respectively. The crosses aredata not included between the whiskers. The notches in theboxes display the variability of the median between samples.If the notches of two boxes are not overlapped, then itmeans that there is statistical significant difference in thedata with 95% confidence. The box plot diagrams providedin this paper add value in assisting the reader in visualizingthe confidence intervals associated with different algorithmevaluations.

We can see in Fig. 16 that HcDE and swDE are the fastestones converging to the optimal solution for F1, with statisticalsignificance. HcDE is the fastest one for the small problem,and swDE is the best one for the 50 variables case. It is alsointeresting that DE is always among the algorithms requiringa higher number of evaluations to find the optimal solution,together with randDE for the small instances, and with sDE,dDE, and randDE for the large ones.


TABLE IV

Solutions Found by the Different Studied Algorithms with Standard Mutation Operator (30 Variables Problems)

DE sDE dDE cDE HcDE randDE swDEF1 0.0(100%) 0.0(100%) 0.0(100%) 0.0(100%) 0.0(100%) 0.0(100%) 0.0(100%)

±0.0 ±0.0 ±0.0 ±0.0 ±0.0 ±0.0 ±0.0

F2 4.4866E1 4.6782E1 1.2880E − 6 1.1460E1 1.3478E1 8.1931 2.0334

±5.0874E1 ±7.1499E1 ±2.4278E−6 ±1.5049E1 ±1.6528E1 ±1.2513E1 ±2.8277

F3 1.4079E7 1.3986E7 2.1803E5 9.1138E6 7.8988E6 9.7103E6 6.5822E6

±7.3263E6 ±7.5272E6 ±1.2527E5 ±3.8501E6 ±3.8734E6 ±5.5070E6 ±2.3821E6

F4 6.7348E2 5.8448E2 5.3015E1 4.5263E2 4.3376E2 1.4731E3 7.8223E2

±5.2630E2 ±4.6222E2 ±2.9130E2 ±3.7162E2 ±3.3452E2 ±1.4973E3 ±4.4604E2

F5 6.1987E2 6.3635E2 6.5288E2 1.3017E3 1.2686E3 2.1121E3 2.1483E3

±3.6420E2 ±3.9947E2 ±4.5679E2 ±4.8205E2 ±5.1051E2 ±7.8744E2 ±4.2444E2

F6 3.1782E1 3.1854E1 1.5589E1 3.0274E1 3.0398E1 2.8325E1 2.8460E1

±2.6279E1 ±2.6417E1 ±2.4397E1 ±2.5477E1 ±2.8125E1 ±2.8786E1 ±2.7774E1

F7 4.6963E3 4.6963E3 4.6963E3 4.6963E3 4.6963E3 4.6963E3 4.6963E3±4.6609E−13 ±4.2874E−13 ±5.6347E−13 ±3.9844E−13 ±2.5854E−13 ±6.7288E−13 ±4.1888E−13

F8 2.0952E1 2.0939E1 2.1023E1 2.0949E1 2.0949E1 2.0941E1 2.0945E1

±5.6082E−2 ±5.3660E−2 ±1.0370E−1 ±5.0359E−2 ±4.9712E−2 ±4.9445E−2 ±5.2934E−2

F9 2.6466(14%) 2.2499(15%) 5.0403(1%) 1.5415(42%) 2.0408(7%) 1.2120E1 8.6999E − 1(29%)

±5.3794 ±4.1545 ±2.5733 ±2.9835 ±1.3660 ±6.1852 ±1.2023

F10 1.4783E2 1.4967E2 5.8855E1 1.2228E2 1.3758E2 1.1555E2 1.0548E2

±2.7927E1 ±2.6237E1 ±1.9690E1 ±3.7688E1 ±3.2934E1 ±4.6236E1 ±3.5540E1

F11 3.4573E1 3.4317E1 2.2480E1 3.1782E1 3.4126E1 3.2465E1 2.9828E1

±2.2143 ±2.3643 ±5.7736 ±3.0603 ±2.5706 ±3.6743 ±3.4326

F12 6.5319E3 7.8102E3 5.8824E3 6.2877E3 9.4917E3 1.0489E4 6.1534E3

±7.5344E3 ±1.0716E4 ±9.0684E3 ±5.1038E3 ±9.4230E3 ±1.0704E4 ±5.1221E3

F13 5.4451 5.6387 2.2919 4.1928 5.5304 3.2051 4.4109

±1.6870 ±1.6307 ±6.1155E−1 ±2.0579 ±1.5783 ±1.7750 ±1.6313

F14 1.3269E1 1.3284E1 1.2992E1 1.3194E1 1.3243E1 1.3237E1 1.3054E1

±1.5715E−1 ±1.5527E−1 ±4.5610E−1 ±1.9305E−1 ±1.7126E−1 ±1.8485E−1 ±2.0202E−1

F15 3.2635E2 3.2969E2 3.5602E2 3.1813E2 3.0494E2 3.3406E2 2.2667E2±8.3973E1 ±7.3004E1 ±8.4479E1 ±9.4227E1 ±9.5326E1 ±1.0207E2 ±9.2896E1

F16 1.8656E2 1.8045E2 9.1258E1 1.5449E2 1.5863E2 1.8117E2 1.3576E2

±5.8222E1 ±4.7974E1 ±2.7395E1 ±4.6946E1 ±5.0123E1 ±9.5619E1 ±4.2044E1

F17 2.2258E2 2.1415E2 1.1711E2 2.0708E2 2.0465E2 2.6053E2 2.0256E2

±4.9700E1 ±3.7604E1 ±4.0709E1 ±4.4317E1 ±3.5780E1 ±8.7622E1 ±4.0113E1

F18 9.0418E2 9.0439E2 9.0549E2 9.0441E2 9.0478E2 9.0723E2 9.0573E2

±7.6546E−1 ±1.0691 ±1.3292 ±8.4445E−1 ±9.6665E−1 ±2.7073 ±1.1026

F19 9.0426E2 9.0417E2 9.0519E2 9.0447E2 9.0482E2 9.0710E2 9.0586E2

±9.8032E−1 ±7.3660E−1 ±1.2745 ±8.5046E−1 ±1.0288 ±2.7059 ±1.0885

F20 9.0430E2 9.0415E2 9.0518E2 9.0443E2 9.0480E2 9.0726E2 9.0479E2

±9.3980E−1 ±7.0476E−1 ±1.2987 ±8.0205E−1 ±1.0137 ±2.6933 ±1.0642E1

F21 5.0000E2 5.0000E2 5.0000E2 5.0000E2 5.0000E2 5.1264E2 5.0000E2±6.1531E−14 ±6.7597E−14 ±9.5767E−14 ±9.2119E−14 ±7.4269E−14 ±6.0938E1 ±1.3495E−13

F22 8.9009E2 8.8906E2 8.9574E2 8.9735E2 8.9308E2 9.0423E2 9.0608E2

±1.4542E1 ±1.5392E1 ±1.9343E1 ±1.2657E1 ±1.5426E1 ±1.6007E1 ±1.2247E1

F23 5.3416E2 5.3416E2 5.3416E2 5.3416E2 5.3416E2 5.5163E2 5.3416E2

±3.5222E−4 ±3.4261E−4 ±3.8687E−4 ±3.4176E−4 ±8.8623E−4 ±8.6951E1 ±4.2183E−3

F24 2.0000E2 2.0000E2 2.0000E2 2.0000E2 2.0000E2 2.0000E2 2.0000E2±3.2878E−13 ±0.0 ±0.0 ±0.0 ±0.0 ±0.0 ±0.0

F25 1.2683E3 1.2462E3 1.3809E3 1.2821E3 1.2724E3 1.3839E3 1.3878E3

±1.2195E2 ±1.2202E2 ±2.8788E1 ±1.2015E2 ±1.2361E2 ±4.4611E1 ±3.3378E1


TABLE V

Solutions Found by the Different Studied Algorithms with Standard Mutation Operator (50 Variables Problems)

DE sDE dDE cDE HcDE randDE swDE

F1 3.6380E − 14(36%) 2.7285E − 14(52%) 1.2506E − 14(78%) 1.7053E − 15(97%) 2.8422E − 15(95%) 2.2488E − 14(60%) 1.1369E − 15(98%)

±2.7422E−14 ±2.8542E−14 ±2.3666E−14 ±9.7456E−15 ±1.2451E−14 ±2.7949E−14 ±7.9982E−15

F2 4.3597E3 4.7612E3 2.7466E − 3 1.8598E3 1.4892E3 2.0568E3 4.8499E2

±2.8872E3 ±3.0451E3 ±3.3206E−3 ±1.1071E3 ±1.0394E3 ±1.7492E3 ±2.8935E2

F3 4.4019E7 5.4188E7 3.5219E5 2.7270E7 2.1131E7 2.2039E7 1.5659E7

±2.4928E7 ±3.0794E7 ±1.4139E5 ±1.4622E7 ±1.3240E7 ±1.7304E7 ±6.3584E6

F4 1.7259E4 1.5463E4 1.3584E4 1.3253E4 1.3175E4 2.0429E4 1.3743E4

±8.8660E3 ±7.2465E3 ±7.8755E3 ±5.0151E3 ±5.3107E3 ±7.5691E3 ±3.8635E3

F5 3.2493E3 3.2248E3 3.7009E3 3.8371E3 4.0524E3 5.9612E3 5.1906E3

±6.5839E2 ±6.6286E2 ±6.2816E2 ±7.3944E2 ±6.9052E2 ±1.3451E3 ±6.1341E2

F6 5.4271E1 5.5074E1 4.5542E1 4.9854E1 5.0986E1 5.5257E1 5.5256E1

±2.8363E1 ±2.8161E1 ±3.3535E1 ±2.7118E1 ±3.1735E1 ±3.7796E1 ±3.2209E1



F9 1.0753E1 1.1221E1 1.8725E1 9.2031 1.2837E1 3.9583E1 9.1035

±9.8332 ±1.0782E1 ±6.7487 ±6.0851 ±4.7641 ±1.6438E1 ±9.1483

F10 3.1477E2 3.0545E2 9.9900E1 2.6711E2 2.7535E2 2.5700E2 2.3587E2

±4.4268E1 ±4.8868E1 ±2.6497E1 ±7.4154E1 ±7.2031E1 ±8.9553E1 ±7.1292E1

F11 6.5775E1 6.6317E1 4.8459E1 6.2701E1 6.5425E1 6.3768E1 6.0096E1

±3.1030 ±2.8025 ±7.8471 ±5.0429 ±3.6014 ±4.3936 ±4.8993

F12 2.2593E4 2.6521E4 2.6608E4 3.0927E4 3.3237E4 4.0907E4 2.7586E4

±1.5942E4 ±2.2496E4 ±1.9440E4 ±2.1691E4 ±2.5585E4 ±3.1017E4 ±1.5425E4

F13 1.1553E1 1.2690E1 4.6455 6.9814 1.1713E1 6.3711 7.7659

±5.1410 ±5.0542 ±9.1982E−1 ±4.5892 ±4.8206 ±3.0173 ±3.1687

F14 2.2941E1 2.2985E1 2.2756E1 2.2920E1 2.2979E1 2.2983E1 2.2784E1

±2.3215E−1 ±1.7760E−1 ±4.6768E−1 ±2.1602E−1 ±2.1586E−1 ±2.3048E−1 ±2.3233E−1


F16 2.3604E2 2.3961E2 9.5054E1 1.9504E2 2.0239E2 2.4393E2 1.8517E2

±4.5694E1 ±4.1952E1 ±4.0973E1 ±5.6651E1 ±5.3864E1 ±9.9916E1 ±4.9696E1

F17 2.7385E2 2.7376E2 1.4709E2 2.7245E2 2.6210E2 3.0302E2 2.5311E2

±3.2713E1 ±4.1913E1 ±4.8682E1 ±4.3197E1 ±4.6628E1 ±6.8531E1 ±3.6266E1

F18 9.1585E2 9.1588E2 9.2378E2 9.1812E2 9.2063E2 9.3378E2 9.2353E2

±3.0328 ±3.0601 ±2.9886 ±3.1693 ±4.2380 ±1.4088E1 ±1.2936E1

F19 9.1566E2 9.1524E2 9.2358E2 9.1881E2 9.2002E2 9.3426E2 9.2496E2

±3.0053 ±2.9025 ±3.8945 ±3.3849 ±4.8372 ±1.2812E1 ±3.5438

F20 9.1561E2 9.1515E2 9.2352E2 9.1894E2 9.1987E2 9.3436E2 9.2514E2

±3.1676 ±2.8535 ±3.8672 ±3.3860 ±4.5003 ±1.2904E1 ±3.4859

F21 9.1808E2 9.1268E2 5.7645E2 7.1240E2 6.9762E2 6.6868E2 5.0000E2

±1.9018E2 ±1.9433E2 ±1.8290E2 ±2.5086E2 ±2.4840E2 ±2.4517E2 ±7.9599E−13

F22 9.1388E2 9.1477E2 9.3221E2 9.1873E2 9.1948E2 9.4318E2 9.3923E2

±7.3040 ±8.4306 ±2.1768E1 ±1.1708E1 ±1.2064E1 ±2.3724E1 ±1.5972E1

F23 9.7369E2 9.4522E2 6.6278E2 7.6173E2 7.4397E2 7.5931E2 5.3912E2

±1.2880E2 ±1.6435E2 ±2.0967E2 ±2.3705E2 ±2.3447E2 ±2.2819E2 ±1.6727E−3


F25 1.3897E3 1.3900E3 1.3809E3 1.3879E3 1.3878E3 1.3893E3 1.3897E3

±4.5951 ±5.5688 ±7.3188 ±5.9639 ±5.6996 ±6.9835 ±6.8410


Fig. 16. Evaluations performed by the algorithms implementing the regular mutation when solving F1 problem.

TABLE VI

Pairwise Comparison of the Algorithms with GPBX-α Mutation

Operator by Means of Holm Test. Adjusted p-Values

30 VariablesDE- sDE- dDE- cDE- HcDE- randDE-

GPBX-α GPBX-α GPBX-α GPBX-α GPBX-α GPBX-αsDE-GPBX-α 1.0 – – – – –dDE-GPBX-α 1.0 1.0 – – – –cDE-GPBX-α 1.0 1.0 1.0 – – –HcDE-GPBX-α 1.0 1.0 1.0 1.0 – –randDE-GPBX-α 0.0100 1.0774E − 3 1.3454E − 3 7.8502E − 5 0.0390 –swDE-GPBX-α 0.0923 0.4004 0.3679 0.0268 1.5892E − 8

50 VariablesDE- sDE- dDE- cDE- HcDE- randDE-

GPBX-α GPBX-α GPBX-α GPBX-α GPBX-α GPBX-αsDE-GPBX-α 1.0 – – – – –dDE-GPBX-α 1.0 1.0 – – – –cDE-GPBX-α 1.0 1.0 1.0 – – –HcDE-GPBX-α 1.0 1.0 1.0 1.0 – –randDE-GPBX-α 0.0127 9.3800E − 3 6.0323E − 3 3.3409E − 3 0.3019 –swDE-GPBX-α 1.0 1.0 1.0 1.0 0.1164 1.1544E − 5

Fig. 17 shows some examples of the convergence of thebest solution in the population for the different algorithms forproblems F1, F2, F9, and F15. The values plotted for everygeneration are averaged over 100 independent runs. We cansee that the behavior of the two panmictic DEs (DE and sDE)is similar for the four problems. Regarding the decentralizedDEs, randDE is clearly the worst algorithm in the fourproblems, while the cellular algorithms (cDE and HcDE) anddDE are in most cases among the best performing ones.

C. Population Models Comparison with GPBX-α MutationIn this section, we will compare seven new DE algorithms

implementing the novel GPBX-α mutation operator, togetherwith the different population models considered in the previousstudy. The results are summarized in Tables VII and VIIIfor the small and big problem instances, respectively. As inthe previous section, results with light gray background meansignificantly worse solutions than DE-GPBX-α, while darkgray background emphasizes better results (see Tables XVIIIand XIX in the Appendix for the results of the statistical tests).

According to Ivan-Davenport test, there are significant diffe-rences among the algorithms for the two problem sizes (thep-values obtained were 9.4994E − 9 and 1.6569E − 5, forD = 30 and 50 variables). The results of the Holm test areshown in Table VI, where significant results are in bold font.We can state with 95% confidence that randDE is worsethan the other algorithms for the two problem dimensions.The only exception is HcDE for the 50 variables problem,that cannot significantly outperform randDE. Additionally, weobserved that swDE is significantly better than HcDE for the30 variables problems in terms of this comparison.

For the small instances (Table VII), we can see that, as ithappened with the regular mutation, DE-GPBX-α is not ableto outperform all the other algorithms in terms of the qualityof solutions found for any problem (see Table XVIII). In thiscase, the best algorithms are dDE-GPBX-α, swDE-GPBX-α,and cDE-GPBX-α, significantly outperforming all the otherones five (F2, F3, F17, F21, and F25), four (F9, F11, F14,and F15), and three (F13, F18, and F20) times, respectively.The worst algorithm is randDE, outperformed by all the otherones for F5, F9, F18, F19, F20, F21, F22, F23, and F25,followed by HcDE (worse than the others for F3, F11, andF14) and dDE (the worst one for F4 and F8).

With the new GPBX-α operator, now dDE-GPBX-α, cDE-GPBX-α, and swDE-GPBX-α beat DE-GPBX-α, in general.The number of times they outperform DE-GPBX-α is 13(swDE-GPBX-α), nine (dDE-GPBX-α), and six (cDE-GPBX-α), while they are outperformed six (swDE-GPBX-α), eight(dDE-GPBX-α), and three (cDE-GPBX-α) times by DE-GPBX-α.

Regarding the results shown in Table VIII for the big prob-lem instances, we can see that dDE-GPBX-α is significantlybetter than all the other compared algorithms for 8 out of the25 problems (F2, F3, F7, F10, F16, F17, F21, and F23),and swDE is the best algorithm for three ones (F9, F14, F15).There is no other algorithm significantly better than the othersfor some problem in this table.

If we compare the algorithms with DE-GPBX-α (seeTable XIX), we can see that sDE-GPBX-α is in this case able


TABLE VII

Best and Average Solutions Found by the Different Studied Algorithms with GPBX-α Mutation Operator (30 Variables Problems)

DE-GPBX-α sDE-GPBX-α dDE-GPBX-α cDE-GPBX-α HcDE-GPBX-α randDE-GPBX-α swDE-GPBX-αF1 2.8422E − 15(95%) 2.2737E − 15(96%) 5.6843E − 16(99%) 1.7053E − 15(97%) 0.0(100%) 6.7719(96%) 0.0(100%)

±1.2451E−14 ±1.1195E−14 ±5.6843E−15 ±9.7456E−15 ±0.0 ±5.1387E1 ±0.0

F2 3.3396E1 2.9932E1 1.4630E − 1 4.4741E1 2.8033E1 1.1162E1 7.7053

±5.1656E1 ±3.9837E1 ±1.3522E−1 ±5.3487E1 ±5.0980E1 ±5.6313E1 ±9.4714

F3 8.7039E6 9.7389E6 2.3143E6 9.9011E6 1.3917E7 5.8029E6 4.8435E6

±6.2129E6 ±5.5757E6 ±1.1116E6 ±4.8575E6 ±9.1914E6 ±3.9860E6 ±2.3478E6

F4 7.0128E2 7.6418E2 7.7101E3 1.1174E3 1.0796E3 2.7419E3 2.2900E3

±7.2633E2 ±5.9145E2 ±5.7946E3 ±7.8538E2 ±8.3509E2 ±1.9893E3 ±1.1509E3

F5 2.0312E3 1.9172E3 3.1755E3 2.1334E3 2.2350E3 3.8455E3 3.0210E3

±6.2781E2 ±5.3800E2 ±6.7793E2 ±5.7241E2 ±5.9403E2 ±1.0225E3 ±6.2596E2

F6 5.0369E1 4.3266E1 6.4598E1 4.4504E1 3.4897E1 9.8814E4 2.8899E1

±4.3394E1 ±3.6295E1 ±5.9973E1 ±3.7412E1 ±3.4187E1 ±6.9586E5 ±3.0111E1


F8 2.0940E1 2.0946E1 2.1000E1 2.0952E1 2.0957E1 2.0947E1 2.0946E1

±5.5350E−2 ±4.6982E−2 ±9.5586E−2 ±4.8683E−2 ±4.3312E−2 ±5.0596E−2 ±5.3998E−2

F9 8.1189 6.4871(3%) 6.2682 1.9800(12%) 4.7261(2%) 3.2136E1 1.1741(32%)

±4.3673 ±4.3599 ±2.5670 ±1.5329 ±2.5154 ±1.3764E1 ±1.1694

F10 1.0050E2 1.0195E2 6.8287E1 8.7276E1 1.0608E2 9.1861E1 7.2292E1

±3.9470E1 ±3.7498E1 ±1.8517E1 ±3.7899E1 ±4.1887E1 ±3.5052E1 ±2.1815E1

F11 2.7051E1 2.7334E1 2.6022E1 2.8145E1 3.2208E1 2.9130E1 2.4429E1

±5.6827 ±5.2657 ±5.2148 ±4.0587 ±3.7823 ±4.2189 ±2.9437


F13 2.5625 2.8433 2.3604 1.8325 2.9108 3.1426 2.0745

±1.2766 ±1.3842 ±5.9722E−1 ±4.7275E−1 ±1.2368 ±1.5332 ±4.5557E−1

F14 1.3117E1 1.3142E1 1.3014E1 1.3121E1 1.3209E1 1.3074E1 1.2771E1

±2.2835E−1 ±2.2757E−1 ±4.1624E−1 ±2.0394E−1 ±1.9037E−1 ±2.3929E−1 ±2.6769E−1


F16 1.5752E2 1.5248E2 1.0737E2 1.1893E2 1.1635E2 1.8852E2 1.1740E2

±9.1138E1 ±8.8573E1 ±3.3904E1 ±5.7573E1 ±4.9603E1 ±1.2277E2 ±3.3956E1

F17 2.1738E2 2.1655E2 1.4544E2 2.0179E2 1.9319E2 2.5621E2 1.7673E2

±8.1953E1 ±8.5546E1 ±5.9119E1 ±4.4797E1 ±5.0071E1 ±1.1698E2 ±4.9114E1

F18 9.0589E2 9.0578E2 9.0715E2 9.0526E2 9.0642E2 9.1512E2 9.0471E2

±1.8355 ±1.7706 ±2.0639 ±1.4036 ±1.7421 ±1.0119E1 ±1.5116E1

F19 9.0585E2 9.0603E2 9.0742E2 9.0552E2 9.0627E2 9.1546E2 9.0577E2

±1.8479 ±1.6907 ±2.0393 ±1.4426 ±1.4618 ±1.0840E1 ±1.0820E1

F20 9.0608E2 9.0608E2 9.0737E2 9.0547E2 9.0628E2 9.1527E2 9.0568E2

±1.8081 ±1.6735 ±1.9480 ±1.4137 ±1.5123 ±9.8206 ±1.0807E1

F21 5.0300E2 5.0643E2 5.0000E2 5.0000E2 5.0000E2 5.5657E2 5.0000E2±3.0000E1 ±6.4310E1 ±8.7578E−14 ±7.4707E−14 ±6.9501E−14 ±1.6441E2 ±9.3874E−14

F22 8.9684E2 8.9544E2 9.1038E2 8.9728E2 8.9560E2 9.2967E2 9.0978E2

±1.3833E1 ±1.4033E1 ±2.2041E1 ±1.4164E1 ±1.5677E1 ±3.1237E1 ±1.5200E1

F23 5.3416E2 5.3416E2 5.3416E2 5.3416E2 5.3416E2 5.9752E2 5.3416E2

±3.3099E−3 ±3.8251E−4 ±4.0526E−4 ±4.0822E−4 ±3.0579E−3 ±1.7362E2 ±3.4837E−4

F24 2.0000E2 2.0000E2 2.0000E2 2.0000E2 2.0000E2 2.2398E2 2.0000E2±8.9169E−13 ±8.1923E−13 ±0.0 ±3.2450E−13 ±0.0 ±1.2485E2 ±0.0

F25 1.3854E3 1.3792E3 1.3828E3 1.3781E3 1.3864E3 1.3975E3 1.3880E3

±3.0307E1 ±5.2724E1 ±8.7015 ±5.1911E1 ±2.8189E1 ±1.3516E1 ±6.5463


TABLE VIII

Best and Average Solutions Found by the Different Studied Algorithms with GPBX-α Mutation Operator (50 Variables Problems)

DE-GPBX-α sDE-GPBX-α dDE-GPBX-α cDE-GPBX-α HcDE-GPBX-α randDE-GPBX-α swDE-GPBX-α

F1 5.6843E − 14 5.6275E − 14(1%) 5.5707E − 14(2%) 5.6275E − 14(1%) 5.6843E − 14 7.3625E1(1%) 5.4001E − 14(5%)

±0.0 ±5.6843E−15 ±7.9982E−15 ±5.6843E−15 ±0.0 ±5.0988E2 ±1.2451E−14

F2 4.1715E3 4.7085E3 3.3051E1 5.4375E3 6.1383E3 1.6587E3 1.4901E3

±3.2261E3 ±3.2214E3 ±1.5885E1 ±3.6148E3 ±5.8476E3 ±1.9499E3 ±9.8544E2

F3 2.3293E7 2.5549E7 3.7751E6 1.8027E7 3.1862E7 6.7535E6 1.0414E7

±1.6538E7 ±1.7624E7 ±1.4350E6 ±1.0635E7 ±2.6563E7 ±4.0839E6 ±3.4380E6

F4 1.7819E4 1.8183E4 5.5729E4 2.7180E4 2.8051E4 2.7925E4 2.7447E4

±6.8399E3 ±7.2290E3 ±1.8043E4 ±8.5263E3 ±9.1078E3 ±9.1437E3 ±7.6363E3

F5 4.2975E3 4.3847E3 8.1065E3 5.3982E3 5.3964E3 8.6784E3 7.3132E3

±9.1665E2 ±1.0323E3 ±1.5947E3 ±1.2174E3 ±1.1295E3 ±2.2303E3 ±1.2290E3

F6 7.3810E1 7.6386E1 1.0456E2 7.7839E1 6.1197E1 5.4491E4 6.4355E1

±3.7877E1 ±3.7599E1 ±6.3353E1 ±3.6803E1 ±3.7637E1 ±3.7835E5 ±4.6361E1



F9 2.1830E1 1.6825E1 2.0785E1 1.0696E1 1.9412E1 9.4795E1 5.1838

±9.2006 ±7.5934 ±5.8148 ±4.4189 ±5.7697 ±4.0724E1 ±3.0896

F10 2.6071E2 2.6341E2 1.2321E2 2.0803E2 2.3422E2 2.1472E2 1.8001E2

±6.8916E1 ±7.5481E1 ±3.1260E1 ±8.7655E1 ±7.8511E1 ±8.5304E1 ±5.2390E1

F11 5.9078E1 5.8421E1 5.0933E1 5.8538E1 6.3572E1 5.9951E1 5.0524E1

±6.5492 ±7.9733 ±6.9605 ±5.2518 ±4.3999 ±5.3181 ±4.9020

F12 2.6298E4 2.8564E4 4.6239E4 3.1211E4 3.2640E4 5.1128E4 2.9392E4

±2.1791E4 ±2.1712E4 ±3.6293E4 ±1.9982E4 ±2.7635E4 ±4.4906E4 ±1.8680E4

F13 5.1234 5.2699 5.1339 3.8837 4.3499 8.0601 4.7253

±2.9114 ±3.2542 ±1.0643 ±7.1721E−1 ±1.3606 ±3.7048 ±9.4196E−1

F14 2.2932E1 2.2881E1 2.2761E1 2.2901E1 2.2983E1 2.2911E1 2.2551E1

±1.9589E−1 ±2.4836E−1 ±4.4310E−1 ±2.4159E−1 ±2.0361E−1 ±2.0972E−1 ±2.7590E−1

F15 2.8345E2 2.8243E2 3.1319E2 2.7441E2 2.6081E2 3.5353E2 2.1933E2

±9.2922E1 ±9.3979E1 ±9.9589E1 ±9.5048E1 ±8.5601E1 ±7.9662E1 ±8.3598E1

F16 2.2694E2 2.1094E2 1.1490E2 1.6806E2 1.7372E2 2.4018E2 1.5420E2

±7.4356E1 ±7.6932E1 ±4.7595E1 ±5.7388E1 ±5.8299E1 ±1.1542E2 ±5.3600E1

F17 2.7866E2 2.7614E2 1.9780E2 2.6936E2 2.5965E2 3.0860E2 2.5020E2

±7.0621E1 ±6.4782E1 ±6.8184E1 ±4.1959E1 ±5.2466E1 ±7.8553E1 ±5.5301E1

F18 9.2209E2 9.2196E2 9.2873E2 9.2275E2 9.2658E2 9.4982E2 9.2788E2

±3.6478 ±3.8656 ±6.1689 ±3.5456 ±4.4416 ±2.0498E1 ±4.6304

F19 9.2197E2 9.2167E2 9.2957E2 9.2262E2 9.2723E2 9.5101E2 9.2797E2

±3.4918 ±2.6209 ±5.8980 ±3.2044 ±4.3930 ±2.0525E1 ±4.6441

F20 9.2272E2 9.2169E2 9.2890E2 9.2263E2 9.2720E2 9.5073E2 9.2798E2

±5.7631 ±2.6976 ±5.4041 ±3.2010 ±4.3800 ±2.0525E1 ±4.6317


F22 9.2512E2 9.2665E2 9.3877E2 9.2221E2 9.2368E2 9.6700E2 9.3817E2

±2.0261E1 ±1.8196E1 ±2.2482E1 ±1.6176E1 ±1.7481E1 ±2.8550E1 ±1.5127E1

F23 7.1310E2 6.9077E2 5.7710E2 8.8967E2 8.1899E2 7.2534E2 5.4862E2

±2.2850E2 ±2.2218E2 ±1.2942E2 ±2.0884E2 ±2.3417E2 ±2.4589E2 ±6.6835E1


F25 1.3853E3 1.3848E3 1.3805E3 1.3821E3 1.3830E3 1.3910E3 1.3847E3

±6.2215 ±7.3940 ±9.2925 ±6.8560 ±6.6720 ±1.3618E1 ±5.6160


Fig. 17. Evolution of the best solution for the algorithms implementing the regular mutation when solving some example problems.

to outperform it for one function, while it is never worse. Re-garding the algorithms with decentralized populations, dDE-GPBX-α, cDE-GPBX-α, and swDE-GPBX-α are better thanDE-GPBX-α for eleven, five, and ten problems, respectively,and they are worse for ten, six, and eight functions. The be-havior of HcDE-GPBX-α and randDE-GPBX-α is much worsethan DE-GPBX-α, since they outperform it in three and fourproblems, respectively, and they are worse in 12 and 10 cases.

We show in Fig. 18 the average (on 100 runs) of theevolution of the best fitness value in the population for theseven studied DEs with GPBX-α operator in some exampleproblems. We can see that, as it happened with the algorithmsimplementing the regular mutation operator, the two panmicticalgorithms offer a similar behavior. Additionally, it can be seenhow randDE-GPBX-α gets stuck in a local optimal solutionvery quickly (in around 50 000 evaluations) for some problems(F1 and F9). Regarding swDE-GPBX-α, it shows a goodperformance for functions F1 and F9, but on the contraryit is among the worst algorithms for the other two problemsshowed. Once again, the algorithms with cellular populations(cDE-GPBX-α and HcDE-GPBX-α), as well as dDE-GPBX-α, are in most cases among the best ones.

D. Evaluation of the New GPBX-α MutationWe compare in this section the behavior of the differ-

ent studied DE algorithms with the new GPBX-α operator(Tables VII and VIII) and the standard one (Tables IV and V).The Holm test gives adjusted p-values > 0.05 for both the 30and 50 variables problem, as it is shown in Table IX, meaningthat there are not significant differences between the equivalentalgorithms with the standard mutation operator and GPBX-αfor all the problems.

If we compare the effects of using the standard mutationoperator or GPBX-α with the different population topologiesfor every problem (results for the Wilcoxon test in Tables XXand XXI), we can see that, in general, the algorithms im-plementing GPBX-α behave better than the equivalent oneswith the standard mutation operator for some unimodal (F3),multimodal basic (F7, F10, and F11), multimodal expanded(F13 and F14), and composition functions (F16 and F25,the latter one only for the 50 variables case). The GPBX-αpromotes a deeper local exploitation than the compared one,but still the algorithms implementing it behave better than theequivalent ones with the regular mutation operator for someof the complex multimodal functions studied.


Fig. 18. Evolution of the best solution for the algorithms implementing GPBX-α when solving some example problems.

In general, we noticed that dDE, cDE, HcDE, and randDEperform significantly better with the regular mutation operatorthan with GPBX-α. On the contrary, swDE-GPBX-α generallyoutperforms (with statistical significance) swDE. For the otheralgorithms we cannot conclude that one operator is better thanthe other.

We compare in Fig. 19 the convergence speed of several al-gorithms using both the regular mutation operator and GPBX-α. We can see that for some problems, like the unimodalones in the figure (functions F1 and F2) the algorithmsimplementing GPBX-α converge slightly slower with respectto the equivalent DEs with the regular mutation. For someother problems, like the multimodal F9 and F15 shown, theyconverge faster to better solutions.

Summarizing our results, we observed a similar behavioron the two panmictic DE algorithms, DE and sDE, withthe two studied mutation operators for all the consideredproblems. The distributed DE (dDE) is highly competitive withthe regular mutation operator, but its behavior worsens whenimplementing the new GPBX-α operator. On the contrary,swDE has resulted to be more competitive with GPBX-α than

TABLE IX

Pairwise Comparison of the Algorithms with the Standard

Mutation and GPBX-α by Means of Holm Test. Adjusted

p-Values

GPBX-α vs. 30 Variables 50 VariablesDE 1.0 1.0sDE 1.0 1.0dDE 0.3432 0.8653cDE 1.0 1.0HcDE 1.0 1.0randDE 1.0 1.0swDE 1.0 1.0

with the other operator. Finally, the cDE algorithm was highlycompetitive for the two recombination operators.

We present in Table X a ranking with the best algorithmsfor every problem size (i.e., from the results in Sections VI-Band VI-C). It was built by adding the position in the clas-sification from better (position 1) to worse (position 14) ofall the algorithms for every problem. The algorithms are


Fig. 19. Comparison of the evolution of the best solution in the population during the run for DE, dDE, cDE, and swDE with regular mutation and GPBX-αwhen solving some example problems.

then arranged in the ranking in terms of the values obtainedin this sum, being those with lower values located in thefirst positions. Additionally, the overall ranking of the sevencompared population topologies for all the tests performedwith the two mutation operators was built by adding theposition of every algorithm in the two rankings. This finalclassification is provided in the third column in Table X.

We can see in Table X that the best algorithm for the twoproblem sizes is dDE. Additionally, it is very interesting to seethe good behavior of swDE-GPBX-α and swDE, which areranked as the second and third best algorithms, respectively.The small-world topology was the only case in which theGPBX-α operator outperformed the regular one. The fouralgorithms implementing the cellular populations (hierarchicalor not) are in the middle of the ranking. The algorithms withpanmictic populations were the worst ones for every mutationscheme, with the exception of the DEs with quasi-randomtopology (randDE and randDE-GPBX-α). These algorithmswere the worst ones in every classification. We believe that thereason for this poor performance lies in the frequent changesperformed on the population topology (after every generation

with no improvement of any solution). This supposes anextremely aggressive migration process of the solutions in thepopulation in which all the neighborhoods are modified withno criterion. Therefore, the advantage of using a decentral-ized population, allowing different areas of the population toconverge toward distinct regions of the search space, is lost.

E. Heterogeneous Distributed DE

We discuss in this section the results obtained by theheterogeneous distributed DE (HdDE) algorithm presented inSection IV-H. As in the previous studies, they were obtainedafter 100 independent runs, and they are shown in Table XI forthe two problem sizes. We compared HdDE with dDE, the bestalgorithm among the studied ones in Sections VI-B and VI-C.The values that were favorable to HdDE in this comparisonare emphasized with bold font. The results of the Wilcoxontest between HdDE and dDE for every problem are given inTable XXII in the Appendix. Those cases in Table XI thatare statistically better than dDE are marked with a dark graybackground, while the results favorable to dDE are emphasizedwith light gray background.


Fig. 20. Comparison of the evolution of the best solution for HdDE and the canonical, distributed, and small-world algorithms with regular mutation andGPBX-α when solving some example problems.


TABLE X

Ranking of All the Studied Algorithms From Better (Number

1) to Worse (14) for the Two Studied Problem Sizes

Small Problems Big Problems Overall Ranking1. dDE 1. dDE 1. Small-world2. swDE-GPBX-α 2. swDE-GPBX-α3. swDE 3. swDE 2. Distributed4. HcDE 4. cDE5. cDE 5. HcDE 3. Cellular6. cDE-GPBX-α 6. cDE-GPBX-α7. sDE 7. dDE-GPBX-α 4. Hierarchical8. DE 8. DE9. dDE-GPBX-α 9. DE-GPBX-α 5. Panmictic asynchronous9. sDE-GPBX-α 10. sDE-GPBX-α11. DE-GPBX-α 11. sDE 6. Panmictic synchronous12. HcDE-GPBX-α 12. HcDE-GPBX-α13. randDE 13. randDE 7. Random14. randDE-GPBX-α 14. randDE-GPBX-α

An overall ranking is provided in the last column.

We first applied the Holm test to look for significantdifferences between the two algorithms (HdDE and dDE) forall the problems. The p-values obtained were 0.5485 and0.6892 for the 30 and 50 variables problems, respectively.Therefore, we cannot conclude that one algorithm is betterthan the other for all the problems. However, we performnext a more detailed comparison of the algorithms for everyproblem using the unpaired Wilcoxon test.

It can be seen that HdDE beats dDE for the consideredproblems. HdDE was better than dDE in 25 problems (12small and 13 big ones), while dDE outperformed HdDE for 20problems (nine small and 11 big ones). The HdDE algorithmis better than dDE for the two problem sizes in 11 out ofthe 20 multimodal problems: two basic ones (F7 and F8),the two studied expanded functions (F13 and F14), and sixcomposition functions (F17, F18, F19, F20, F22, and F25).

We would like to emphasize that the benefits of the combi-nation of the two mutation operators in HdDE go further thansimply the application of the two algorithms separately. For thesmall problems, HdDE is better (with statistical significance,as it is shown in Table XXII in the Appendix) than dDE anddDE-GPBX-α in 12 and 17 occasions, respectively, while itis beat by them nine (dDE) and three (dDE-GPBX-α) times.In the case of the big problems, the obtained numbers aresimilar, since HdDE outperforms dDE for 13 problems, anddDE-GPBX-α for 16. On the contrary, it is worse than dDEin 11 problems, and only four in the case of dDE-GPBX-α.

We also performed the ranking of HdDE, dDE, and dDE-GPBX-α for the two problem sizes. The result is that HdDEwas the best algorithm for the two problem sizes, followed bydDE and dDE-GPBX-α, in that order.

We plot in Fig. 20 the evolution of the best solution in thepopulation during the run for HdDE compared to the DEswith canonical, distributed, and small-world populations withregular mutation and GPBX-α. Therefore, we have chosen forthis comparison the canonical DE plus the two best topologiesfound in previous sections. We can see that the convergencecurve performed by HdDE is, in general, among those of the

TABLE XI

Best and Average Solutions Found by HdDE

Problem 30 Variables 50 Variables

F1 2.2737E − 15(96%) 4.8317E − 14(15%)

±1.1195E−14 ±2.0399E−14

F2 8.1263 3.3020E2

±5.6012 ±1.7433E2

F3 2.3077E6 4.4442E6

±1.1569E6 ±1.5854E6

F4 1.3259E2 2.9429E3

±8.9040E1 ±1.3783E3

F5 7.6639E2 3.2201E3

±3.5578E2 ±4.6823E2

F6 3.1942E1 5.7372E1

±2.7471E1 ±3.3465E1

F7 4.6963E3 6.1953E3±4.2874E−13 ±9.2769E−13

F8 2.0924E1 2.1093E1±8.5023E−2 ±1.4042E−1

F9 3.9102(3%) 1.8705E1

±2.0162 ±6.1415

F10 6.0091E1 1.1281E2

±1.7517E1 ±2.6180E1

F11 2.5724E1 5.1934E1

±5.5315 ±8.1031

F12 7.8639E3 3.8415E4

±7.1472E3 ±2.5274E4

F13 2.0401 4.3640

±4.5017E−1 ±8.2732E−1

F14 1.2687E1 2.2303E1±3.8957E−1 ±4.7943E−1

F15 3.1676E2 2.6174E2

±9.4320E1 ±9.3950E1

F16 8.5840E1 1.0470E2

±1.9057E1 ±4.8172E1

F17 1.0075E2 1.1467E2±3.2027E1 ±3.4067E1

F18 9.0336E2 9.1677E2

±1.0464E1 ±3.8088

F19 9.0424E2 9.1619E2

±5.2655E−1 ±3.2055

F20 9.0423E2 9.1620E2

±5.2193E−1 ±3.4034

F21 5.0000E2 7.3700E2

±8.5121E−14 ±2.5295E2

F22 8.7579E2 9.0107E2±2.0036E1 ±1.1351E1

F23 5.3416E2 7.8510E2

±3.3741E−4 ±2.3721E2

F24 2.0000E2 2.0000E2

±7.6551E−13 ±1.5264E−12

F25 1.2762E3 1.3732E3±1.2556E2 ±7.3975


Fig. 21. Comparison of the evolution of the best solution for HdDE and the compared state-of-the-art DE algorithms when solving some example problems.


TABLE XII

Pairwise Comparison of HdDE and Some State-of-the-Art DEs

by Means of Holm Test. Adjusted p-Values

30 VariablesSaDE DEGL JADE HdDE

DEGL 1.0 – – –JADE 0.08227 0.1496 – –HdDE 1.0 1.0 0.08227 –

50 VariablesSaDE DEGL JADE HdDE

DEGL 1.0 – – –JADE 1.0 1.0 – –HdDE 0.3314 0.6836 0.6256 –

best algorithms for the studied problems. Additionally, HdDEperforms a better convergence to more accurate results forseveral problems (F4, F14, and F17 are some examples).Problems F4, F14, and F17 are good examples of the fruitfulcollaboration between the two subpopulations in HdDE: theelbow in the convergence curve of this algorithm points outhow it was able to escape from a local optimal solution thanksto the collaboration of the two populations with the differentmutation schemes (please note that the plotted values areaveraged after 100 independent runs). We would also like toemphasize that this elbow can be located at late stages of thesearch, meaning that the algorithm keeps enough diversity tobe able to create new solutions that will allow it to escapefrom local optima.

F. Comparison with Other Algorithms from the Literature

In order to show the competitiveness of the proposedapproaches, we compare in this section the best algorithm pro-posed in this paper (HdDE) versus some other DEs belongingto the state-of-the-art algorithms in continuous optimization.They are SaDE [79], DEGL [29], and JADE [13]. Thesealgorithms were already demonstrated to be highly competitivewith the best existing algorithms in the literature in continuousoptimization [13], [29], [79].

All the state-of-the-art compared algorithms were imple-mented in the JCell framework for this paper (freely avail-able at http://jcell.gforge.uni.lu/), and their parameters weretaken from their original papers. For every problem, all thealgorithms were run with the same 100 seeds of the randomnumber generator for the 100 independent runs. Thereforethe starting populations are composed by exactly the samesolutions for all the algorithms. In the case of HdDE andDEGL (with larger population sizes), 50 solutions are the sameas for the other algorithms, and the other ones are randomlygenerated (but still the same ones for all the problems). Asrecommended in [29], the population size of DEGL was setto ten times the problem dimension, and the neighborhood sizeto 10% the population size (i.e., to the problem dimension).We use the JADE algorithm with archive, because it wasconcluded to be better in the original paper.

As in the rest of the paper, we compare the algorithms forthe small and big problems, and the results are provided inTables XIII and XIV, respectively. The best solution found bythe algorithms for every problem is emphasized in bold font.

TABLE XIII

Comparison with Other Approaches in the Literature for the

30 Variables Problems

SaDE DEGL JADE HdDE

F1 0.0(100%) 0.0(100%) 1.2506E − 14(78%) 2.2737E − 15(96%)

±0.0 ±0.0 ±2.3666E−14 ±1.1195E−14

F2 7.0790E2 2.4430E − 4 1.4893E − 13 8.1263

±1.5289E3 ±1.0191E−4 ±4.6257E−14 ±5.6012

F3 3.1588E6 7.2602E4 1.5055E4 2.3077E6

±5.8459E6 ±3.2078E4 ±3.5296E4 ±1.1569E6

F4 2.8137E2 7.0579E − 3 3.2663E3 1.3259E2

±4.2464E2 ±3.5729E−3 ±3.7898E3 ±8.9040E1

F5 4.2171E3 7.4464E − 2 3.7219E2 7.6639E2

±8.1045E2 ±3.3019E−2 ±3.4806E2 ±3.5578E2

F6 7.1687E1 8.8349E − 6 6.4396(8%) 3.1942E1

±5.3781E1 ±9.6506E−6 ±2.3845E1 ±2.7471E1

F7 4.6963E3 4.6963E3 4.6963E3 4.6963E3±6.9614E−13 ±6.3985E−13 ±2.2390E−13 ±4.2874E−13

F8 2.0796E1 2.0945E1 2.1162E1 2.0924E1

±2.7363E−1 ±5.5328E−2 ±5.6892E−2 ±8.5023E−2

F9 0.0(100%) 1.5246E2 1.3074E − 14(77%) 3.9102(3%)

±0.0 ±9.9517 ±2.4042E−14 ±2.0162

F10 8.6953E1 1.6712E2 1.2302E2 6.0091E1

±1.6434E1 ±9.3799 ±6.6935E1 ±1.7517E1

F11 1.8952E1 3.9297E1 3.5629E1 2.5724E1

±4.8759 ±1.0505 ±3.1777 ±5.5315

F12 7.6120E3 1.9836E3 4.1569E3 7.8639E3

±6.4375E3 ±2.3922E3 ±5.4296E3 ±7.1472E3

F13 1.1360 1.4339E1 3.5055 2.0401

±1.8689E−1 ±9.7278E−1 ±1.8670 ±4.5017E−1

F14 1.2285E1 1.3110E1 1.3811E1 1.2687E1

±5.0020E−1 ±1.9593E−1 ±2.1482E−1 ±3.8957E−1

F15 3.4552E2(1%) 3.9100E2 3.4233E2(2%) 3.1676E2

±1.3974E2 ±9.6499E1 ±1.0769E2 ±9.4320E1

F16 1.1663E2 2.0174E2 2.2434E2 8.5840E1±4.5449E1 ±4.2645E1 ±1.4513E2 ±1.9057E1

F17 9.9163E1 2.3417E2 3.4203E2 1.0075E2

±3.9783E1 ±4.7890E1 ±1.2771E2 ±3.2027E1

F18 8.6898E2 9.0341E2 9.0682E2 9.0336E2

±5.9810E1 ±2.1004E−1 ±1.9490 ±1.0464E1

F19 8.6778E2 9.0336E2 9.0683E2 9.0424E2

±6.0942E1 ±3.2852E−1 ±1.6842 ±5.2655E−1

F20 8.7650E2 9.0336E2 9.0688E2 9.0423E2

±5.9749E1 ±3.3055E−1 ±1.9489 ±5.2193E−1

F21 5.1929E2 5.0000E2 5.4002E2 5.0000E2

±1.0177E2 ±7.9982E−14 ±1.2299E2 ±8.5121E−14

F22 9.1984E2 8.7180E2 9.2344E2 8.7579E2

±3.3138E1 ±1.0896E1 ±1.7553E1 ±2.0036E1

F23 5.4047E2 5.3819E2 5.7404E2 5.3416E2

±6.3076E1 ±4.0283E1 ±1.2025E2 ±3.3741E−4

F24 2.0000E2 2.0000E2 2.4240E2 2.0000E2±0.0 ±0.0 ±1.8740E2 ±7.6551E−13

F25 1.3823E3 1.1263E3 1.4006E3 1.2762E3

±8.2150 ±7.1553 ±2.5917E1 ±1.2556E2


TABLE XIV

Comparison with Other Approaches in the Literature for the

50 Variables Problems

SaDE DEGL JADE HdDE

F1 1.0232E − 14(82%) 2.9240E − 12 5.6843E − 14 4.8317E − 14(15%)

±2.1949E−14 ±1.1938E−12 ±0.0 ±2.0399E−14

F2 3.3254E3 2.9487E2 5.9231E − 13 3.3020E2

±9.5433E3 ±6.2552E1 ±2.8106E−13 ±1.7433E2

F3 9.0116E6 7.4639E6 9.2985E4 4.4442E6

±1.8093E7 ±1.3587E6 ±9.0424E4 ±1.5854E6

F4 6.4853E3 1.1999E3 3.6563E4 2.9429E3

±2.8287E3 ±2.7915E2 ±1.4322E4 ±1.3783E3

F5 1.1157E4 3.3191E2 3.6946E3 3.2201E3

±1.4652E3 ±1.1557E2 ±7.1625E2 ±4.6823E2

F6 2.0409E2 3.0722E1 2.4886 5.7372E1

±1.8589E2 ±1.3427E1 ±1.0970E1 ±3.3465E1

F7 6.1953E3 6.1953E3 6.1953E3 6.1953E3±4.4368E−12 ±2.6232E−9 ±7.1974E−13 ±9.2769E−13

F8 2.0797E1 2.1139E1 2.1299E1 2.1093E1

±3.8228E−1 ±3.5500E−2 ±4.1087E−2 ±1.4042E−1

F9 9.6634E − 15(83%) 3.2573E2 5.6275E − 14(1%) 1.8705E1

±2.1460E−14 ±1.4297E1 ±5.6843E−15 ±6.1415

F10 2.4334E2 3.4349E2 2.6441E2 1.1281E2

±3.5398E1 ±1.6255E1 ±1.1469E2 ±2.6180E1

F11 4.1910E1 7.2690E1 6.8649E1 5.1934E1

±4.7434 ±1.5193 ±4.3846 ±8.1031

F12 3.5814E4 9.5027E3 2.0048E4 3.8415E4

±2.4712E4 ±7.9459E3 ±1.9215E4 ±2.5274E4

F13 2.1154 2.9904E1 7.7854 4.3640

±2.9108E−1 ±1.1750 ±9.3908E−1 ±8.2732E−1

F14 2.1815E1 2.2942E1 2.3671E1 2.2303E1

±6.1830E−1 ±1.6824E−1 ±2.1198E−1 ±4.7943E−1

F15 3.0571E2 3.3302E2 3.1848E2(2%) 2.6174E2

±1.5229E2 ±9.4317E1 ±9.2035E1 ±9.3950E1

F16 1.5704E2 2.5378E2 2.2870E2 1.0470E2±5.7039E1 ±2.4010E1 ±1.1575E2 ±4.8172E1

F17 1.2077E2 2.8284E2 3.7569E2 1.1467E2

±2.1617E1 ±2.1617E1 ±6.9206E1 ±3.4067E1

F18 9.8646E2 9.1224E2 9.3598E2 9.1677E2

±3.1410E1 ±1.0894 ±1.7285E1 ±3.8088

F19 9.8590E2 9.1230E2 9.3532E2 9.1619E2

±4.0976E1 ±9.5186E−1 ±1.6130E1 ±3.2055

F20 9.8785E2 9.1226E2 9.3590E2 9.1620E2

±3.1503E1 ±9.1633E−1 ±1.6098E1 ±3.4034

F21 7.6478E2 7.1593E2 6.8545E2 7.3700E2±3.3996E2 ±2.4986E2 ±2.4168E2 ±2.5295E2

F22 9.6737E2 9.1650E2 9.6502E2 9.0107E2

±1.7584E1 ±3.9815 ±2.4867E1 ±1.1351E1

F23 7.0505E2 7.6527E2 6.8462E2 7.8510E2

±2.8793E2 ±2.3657E2 ±2.2106E2 ±2.3721E2

F24 1.0216E3 2.0000E2 9.9060E2 2.0000E2

±3.8723E2 ±1.5836E−12 ±1.8249E2 ±1.5264E−12

F25 1.4056E3 1.3952E3 1.4019E3 1.3732E3

±1.7135E1 ±4.3774 ±7.3243 ±7.3975

TABLE XV

Ranking of HdDE and the State-of-the-Art Algorithms From

Better (Number 1) to Worse (4) for the Two Studied Problem

Sizes

Small Problems Big Problems Overall Ranking1. HdDE 1. HdDE 1. HdDE1. SaDE 2. DEGL 2. SaDE3. JADE 3. JADE 3. DEGL4. DEGL 4. SaDE 3. JADE

An overall ranking is provided in the last column.

TABLE XVI

Results of Wilcoxon Test for Comparing DE Versus the Other

Approaches with Regular Mutation for the 30 Variables

Problems

DE vs. sDE dDE cDE HcDE randDE swDE

F2 0.5103 4.008E − 18 1.294E − 13 4.864E − 11 1.228E − 13 1.33E − 17

F3 0.7065 4.008E − 18 5.306E − 8 1.309E − 9 1.56E − 5 1.944E − 14

F4 0.1674 4.342E − 16 0.0001169 0.001807 1.766E − 5 0.06262

F5 0.7014 0.5717 5.187E − 14 2.799E − 13 1.845E − 16 4.131E − 18

F6 0.9383 2.877E − 7 0.3146 0.003939 0.02576 0.05569

F7 0.6947 2.042E − 6 0.2918 0.00565 2.923E − 5 0.5004

F8 0.02399 2.255E − 11 0.5977 0.4321 0.02213 0.1674

F9 0.7451 4.247E − 8 0.08573 0.01327 4.668E − 12 0.008194

F10 0.9575 5.755E − 18 6.584E − 7 0.01694 2.115E − 8 1.754E − 12

F11 0.5147 1.146E − 17 6.562E − 10 0.3113 3.789E − 6 1.625E − 15

F12 0.2888 0.2126 0.5858 0.03177 0.005496 0.8865

F13 0.4005 7.77E − 18 2.514E − 5 0.8352 1.249E − 11 9.192E − 5

F14 0.6512 3.917E − 6 0.004114 0.6964 0.03961 1.556E − 10

F15 0.7656 0.05149 0.6136 0.1208 0.7758 2.374E − 9

F16 0.6193 5.419E − 18 1.355E − 5 0.0003311 0.1027 7.639E − 10

F17 0.4734 1.291E − 17 0.01572 0.005793 0.002006 0.00309

F18 0.1919 1.905E − 13 0.0315 1.631E − 6 6.707E − 12 9.56E − 16

F19 0.9849 3.264E − 8 0.01679 2.439E − 5 5.849E − 11 2.538E − 14

F20 0.4798 3.578E − 7 0.1551 0.0002625 4.137E − 11 1.888E − 12

F21 0.9031 9.503E − 9 6.31E − 5 0.0004402 0.1459 2.492E − 11

F22 0.9192 0.05703 0.00157 0.1325 1.053E − 5 4.668E − 12

F23 0.8302 0.004087 0.5241 0.8122 3.241E − 11 0.5975

F24 1.0 1.0 1.0 1.0 0.05906 1.0

F25 0.3331 1.552E − 9 0.4971 0.7846 1.25E − 7 2.244E − 12

After applying the Iman-Davenport test, we obtainedp-values of 0.03515 and 0.2441 for the small and big prob-lems, respectively. Therefore, this test is pointing out somesignificant differences among the algorithms for the smallproblems, while it did not find significant differences for thebig ones. However, after performing a pairwise comparison ofthe algorithms for all the problems with the Holm test, wedid not find significant differences at 95% confidence, thereare only significant differences between JADE and SaDE, andbetween JADE and HdDE with 90% confidence. The resultsare given in Table XII.

In a more detailed study, the state-of-the-art algorithms arecompared versus HdDE using the Wilcoxon test (the resultsof the test are in Tables XXIII and XXIV) for every problem.Those figures with dark gray background in Tables XIIIand XIV mean that the algorithm in the corresponding columnis significantly better (according to the unpaired Wilcoxontest) than HdDE for the indicated problem, while the lightgray color stands for significantly better performance ofHdDE.


Fig. 22. Evaluations performed by HdDE and the compared state-of-the-art algorithms when solving F1 problem.

For the small instances (the results are in Table XIII), wecan see that HdDE significantly outperforms SaDE, DEGL,and JADE for 11, 10, and 18 problems, respectively, while it isworse for 11, 12, and 6. In the case of the large scale problems,whose results are presented in Table XIV, the comparison ismore favorable to HdDE, since it is significantly better thanSaDE, DEGL, and JADE for 15, 14, and 17 problems, while itis worse only for seven, ten, and seven functions, respectively.Therefore, we can say that attending to pairwise comparisonsof HdDE with the other state-of-the-art algorithms for everyproblem (considering the two problem dimensions), HdDE isoutperforming all of them.

We show in Fig. 21 the convergence plots of the comparedstate-of-the-art DEs, together with HdDE. As we can see,the behavior of the algorithms is very different dependingon the considered problem. JADE performs very well forproblems F1 and F2 but it is the worst algorithm for theother four displayed problems. SaDE is not outstanding in anyof these plots, and it is only clearly outperformed by DEGLand HdDE for F24. The final solution provided by DEGL isnever far from the best algorithm, with the exception of F2,for which all the algorithms perform similarly except JADE,outperforming all the others. The proposed HdDE algorithmis highly competitive with the compared ones. It is always thebest (for F10 and F24) or second best algorithm (in the otherfour cases) for all the problems.

We compare now the efficiency of the different algorithmsin terms of the number of evaluations performed to reach theoptimal solution for F1 (the only function for which they findit) in the boxplots in Fig. 22. As we can see, SaDE is the fastestalgorithm, followed by HdDE and JADE (with no significantdifferences between them). The slowest algorithm, is DEGLwith significant differences with respect to all the other ones.For the big problems, SaDE converges again faster than HdDE(the other two algorithms could not find the optimal solution).

To end this section, we classify in Table XV the consideredstate-of-the-art algorithms and HdDE according to their rankfrom better (position 1) to worse (with value 4) for everyproblem in the two dimensions. As we can see, HdDE is thebest ranked algorithm for the two problem dimensions, so it isin the first position of the overall ranking in the right column

(made by adding the position of every algorithm in the othertwo columns). SaDE is the second best algorithm, althoughit did not scale well from 30 to 50 variables: it went fromsharing the first position with HdDE for the small problemsto the last one for the big functions. Finally, DEGL and JADEshare the third position in the overall ranking.

VII. Conclusion and Future Works

We studied in this paper the influence of several populationschemes in differential evolution algorithms. Our motivationfor this paper was that, despite the high influence the popu-lation scheme traditionally has in EAs, it was never analyzedfor DEs before. We have considered two panmictic populationsand five decentralized ones. On the one hand, the panmicticpopulations are the canonical DE implementation, and thesame one but with synchronous update of all the solutions.On the other hand, the five decentralized populations aredistributed, cellular, hierarchical cellular, random, and small-world. Five of them (panmictic synchronous, cellular, hierar-chical cellular, random, and small-world) were implementedfor the first time in a DE algorithm in this paper. Thealgorithms were evaluated on the benchmark of 25 functionsdesigned for the CEC’05 competition on real-parameter opti-mization. The best overall algorithm was dDE, both in termsof efficiency and accuracy.

Additionally, a new mutation strategy, called GPBX-α,was proposed for DE, and it was tested on all the studiedpopulation models. As a general result, we can say that theDEs with decentralized population are more accurate (i.e., theyfind better results) than the panmictic (or non-decentralized)ones when using GPBX-α. The exception is randDE, whichis clearly the worst performing algorithm. The algorithmsimplementing the new mutation scheme are highly competitivewith the equivalent ones with the regular mutation operator,outperforming them for many problems, independently of thepopulation scheme used.

Finally, as an attempt to provide an algorithm with com-petitive performance for all the problems, we designed aheterogeneous distributed algorithm with two islands (calledHdDE), one executing the classical DE/rand/1/bin algorithm,


TABLE XVII

Results of Wilcoxon Test for Comparing DE Versus the Other Approaches with Regular Mutation for the 50 Variables Problems

DE vs. sDE dDE cDE HcDE randDE swDEF1 0.03619 1.074E − 6 2.429E − 11 5.228E − 11 0.000332 7.812E − 12F2 0.3418 4.008E − 18 2.058E − 13 1.409E − 14 6.714E − 11 4.008E − 18F3 0.009386 4.008E − 18 4.135E − 7 1.06E − 10 1.588E − 12 5.8E − 17F4 0.2114 0.006106 0.0006437 0.0007208 0.08342 0.007986F5 0.8325 1.609E − 5 3.451E − 7 1.488E − 10 1.868E − 11 4.523E − 18F6 0.6413 0.05227 0.2064 0.02273 0.3929 0.7505F7 0.4876 2.87E − 15 0.2408 0.5173 0.003048 0.001237F8 0.8541 4.625E − 10 0.6611 0.7349 0.1126 0.8487F9 0.7774 2.137E − 9 0.5683 0.001031 2.144E − 15 0.3165F10 0.2445 4.131E − 18 5.515E − 8 3.605E − 6 3.529E − 8 4.898E − 13F11 0.1463 5.419E − 18 9.204E − 7 0.7297 1.04E − 6 5.467E − 14F12 0.1165 0.04197 0.005048 0.006706 0.000468 0.01202F13 0.09506 7.855E − 16 5.346E − 9 0.8085 1.369E − 12 6.432E − 8F14 0.2983 0.0007672 0.6339 0.3195 0.7065 9.204E − 7F15 0.7661 6.655E − 5 0.4068 0.9254 0.2685 0.1396F16 0.1931 3.425E − 17 2.036E − 6 1.739E − 5 0.2631 7.748E − 12F17 0.5623 7.102E − 18 0.7426 0.04784 0.1728 0.0002921F18 0.8649 8.006E − 18 8.887E − 7 1.475E − 12 1.137E − 8 1.037E − 16F19 0.2229 1.79E − 17 7.578E − 9 8.946E − 12 1.258E − 8 4.661E − 18F20 0.2515 1.33E − 17 1.961E − 9 1.249E − 11 1.336E − 8 4.008E − 18F21 0.2705 1.339E − 10 5.28E − 5 0.0001039 4.015E − 8 3.046E − 16F22 0.3331 7.881E − 11 4.13E − 5 8.199E − 5 1.373E − 7 1.589E − 17F23 0.1308 1.66E − 11 8.284E − 7 1.022E − 6 3.389E − 7 6.49E − 18F24 0.8172 2.759E − 5 3.119E − 9 5.982E − 10 0.003042 3.713E − 9F25 0.5214 9.472E − 14 0.01376 0.01167 0.05838 0.7583

TABLE XVIII

Results of Wilcoxon Test for Comparing DE Versus the Other Approaches with GPBX-α for the 30 Variables Problems

DE vs. sDE dDE cDE HcDE randDE swDEF1 0.7998 0.173 0.5754 0.05906 0.5541 0.05906F2 0.7978 4.008E − 18 0.07747 0.04557 2.42E − 11 4.728E − 10F3 0.05227 1.543E − 17 0.03177 1.238E − 6 5.904E − 6 1.507E − 7F4 0.1264 1.08E − 17 1.016E − 7 6.417E − 5 2.749E − 14 8.789E − 16F5 0.1591 5.17E − 15 0.1157 0.007135 1.051E − 13 3.224E − 14F6 0.574 0.05748 0.3595 0.002552 0.2268 8.696E − 6F7 0.8832 0.08974 0.02432 0.04936 0.481 0.261F8 0.6438 4.365E − 7 0.129 0.03317 0.9986 0.5623F9 0.01239 0.001572 1.376E − 17 8.871E − 9 1.203E − 16 1.013E − 17F10 0.8031 1.056E − 9 0.009576 0.2242 0.01429 1.565E − 7F11 0.5717 0.1531 0.1622 3.712E − 10 0.04941 0.0002556F12 0.1664 0.004438 0.846 0.05525 0.001109 0.1013F13 0.07747 0.8085 2.293E − 6 0.0326 0.001786 0.01499F14 0.3263 0.1181 0.6586 0.004535 0.09438 3.057E − 14F15 0.6278 0.833 0.462 0.4687 0.2038 2.02E − 10F16 0.6313 7.858E − 7 0.001081 7.309E − 5 0.2295 5.795E − 5F17 0.4608 4.234E − 11 0.6812 0.105 0.1273 3.203E − 5F18 0.6862 0.0001748 0.01429 0.03043 1.754E − 12 9.729E − 5F19 0.4863 4.049E − 6 0.2601 0.03798 1.936E − 12 0.0004275F20 0.7767 4.318E − 5 0.005979 0.3297 1.512E − 12 0.005979F21 0.7535 6.315E − 8 0.204 0.9092 0.0003495 0.0229F22 0.3488 1.609E − 5 0.7714 0.8058 2.436E − 10 8.74E − 9F23 0.2147 0.06359 0.763 0.1404 3.012E − 12 0.2177F24 0.8339 0.02249 0.0687 0.02249 1.0 0.02249F25 0.7819 3.666E − 6 0.5977 0.5623 5.154E − 5 0.4734


TABLE XIX

Results of Wilcoxon Test for Comparing DE Versus the Other Approaches With GPBX-α for the 50 Variables Problems

DE vs. sDE dDE cDE HcDE randDE swDEF1 1.0 0.3711 1.0 4.008E − 18 0.518 0.05906F2 0.1473 4.008E − 18 0.004159 0.009672 8.635E − 11 8.534E − 14F3 0.1783 4.008E − 18 0.008661 0.04557 1.037E − 16 6.976E − 15F4 0.8622 4.008E − 18 5.269E − 12 9.658E − 13 6.702E − 7 1.711E − 12F5 0.6388 4.131E − 18 8.33E − 10 1.27E − 10 3.522E − 13 4.388E − 18F6 0.567 0.002077 0.704 0.05703 0.4341 0.08159F7 0.4718 2.066E − 11 0.2687 0.1401 0.09774 4.558E − 6F8 0.3113 3.801E − 13 0.7117 0.4587 0.06763 0.2101F9 6.725E − 5 0.587 1.871E − 15 0.08621 3.74E − 17 4.523E − 18F10 0.5881 3.326E − 17 5.88E − 5 0.01632 6.944E − 7 4.556E − 12F11 0.8325 4.538E − 11 0.3797 6.702E − 7 0.5693 9.228E − 14F12 0.5929 6.508E − 6 0.04632 0.08973 0.0001307 0.1541F13 0.6913 0.05793 0.05227 0.6913 1.253E − 5 0.2164F14 0.1541 0.0022 0.5236 0.04095 0.005102 4.509E − 15F15 0.8565 0.01769 0.6958 0.2483 0.001643 1.603E − 5F16 0.2911 2.602E − 16 3.394E − 8 3.156E − 6 0.5037 5.804E − 12F17 0.5191 1.06E − 10 0.7091 0.4525 0.2935 0.01213F18 0.3212 9.296E − 16 0.1206 2.436E − 10 1.161E − 8 6.923E − 14F19 0.4044 3.621E − 15 0.1157 9.903E − 13 3.077E − 8 2.05E − 14F20 0.1473 1.079E − 13 0.4525 3.684E − 11 5.515E − 8 5.846E − 13F21 0.9805 0.0004628 3.303E − 6 0.0001245 0.6984 0.02608F22 0.4083 5.809E − 6 0.4161 0.6512 1.421E − 6 1.992E − 7F23 0.7925 0.001724 2.951E − 6 0.0005967 0.5191 8.558E − 6F24 0.2079 0.5455 0.692 0.1764 0.3027 0.3214F25 0.612 0.0002455 0.0001898 0.007283 0.3834 0.2558

and the second one using the new mutation strategy proposedin this paper (GPBX-α). As a result, HdDE is the best of allthe compared algorithms for most problems.

In the last part of the paper, the best studied algorithm(HdDE) was compared versus three DE algorithms fromthe state of the art: SaDE, DEGL, and JADE. These threealgorithms were demonstrated to outperform other well-known DE algorithms, as well as other kinds of evolution-ary algorithms. We conclude from this paper that HdDEis the best algorithm for the two different problem sizestested.

Along further research lines, we are working on the designof new, more sophisticated, accurate and efficient, hetero-geneous models for DE algorithms. Additionally, we havean special interest on the study and design of other newdifferent ways for decentralizing the population, since wehave demonstrated in this paper that it is a major issue onthe behavior of DE algorithms, as it is the case in otherfamilies of EAs. We are also considering the implementationin DE of some techniques that are performing well on otherEAs. Some examples are importing from PSO the conceptsof multiswarms [80], the incorporation of perturbations intosome of the algorithm parameters [81], or using some learningstrategy during the search [82], among others [83], [84].Finally, the parallelization of the new decentralized mod-els and their application to the resolution of complex real-world problems seems a natural step forward from thispaper.

Acknowledgment

The authors would like to acknowledge that Prof. Suganthanshared his SaDE code with us and facilitated our implemen-tation of it.

Appendix

STATISTICAL TESTS

In order to make this paper more clear and comprehensible,we show in this appendix the results we obtained after ap-plying our unpaired Wilcoxon statistical tests [77], [85], [86].These tests were performed in all the comparisons made in thepaper in order to assess reliability to our results and conclu-sions. This test is a non-parametric alternative to the studentt-test. This method is used to check whether two data sam-plings belong to different populations or not. Therefore, wecan use it to compute if there are statistically significant dif-ferences between the data reported by two different algorithmsafter the independent runs.

The null hypothesis for this test is that the median differencebetween pairs of observations in the underlying populationsrepresented by the samples of results provided by the algo-rithms is zero.

Tables XVI–XXIV present the results of the Wilcoxontest in the comparisons made in this paper. All these tablescompare one single base algorithm (specified in the firstcolumn) versus each one of the other algorithms in the othercolumns.

Tables XVI and XVII present the comparisons made be-tween DE (the base algorithm) and all the other studied al-gorithms implementing the regular mutation operator, namelysDE, dDE, cDE, and HcDE. Table XVI presents the resultsfor the 30 variables problems, while Table XVII is dedicatedto the big (50 variables) ones.

The same study is performed in Tables XVIII and XIXfor the small and big instances, respectively, but in this casefor the algorithms implementing the new GPBX-α mutationoperator.

In Tables XX and XXI, we are comparing every algorithmwith the new GPBX-α mutation operator versus the equivalent


TABLE XX

Results of Wilcoxon Test for Comparing the DEs with GPBX-α and Regular Mutation for the 30 Variables Problems

GPBX-α vs. DE sDE dDE cDE HcDE randDE swDEF1 0.05906 0.1003 1.0 0.1814 4.008E − 18 0.1814 4.008E − 18F2 0.01089 0.03123 4.008e − 18 1.36e − 10 0.02253 0.1365 8.906e − 15F3 1.538E − 8 7.631E − 5 4.008e − 18 0.3595 2.158E − 8 2.055E − 7 3.451E − 7F4 0.5905 0.02442 4.008e − 18 3.263e − 13 6.392e − 12 8.172E − 7 1.845e − 16F5 5.258e − 18 1.253e − 17 4.008e − 18 9.296e − 16 5.17e − 15 1.209e − 15 9.472e − 14F6 3.505E − 5 5.628E − 5 4.234e − 11 0.0005817 0.1967 8.363E − 5 0.7531F7 4.75E − 9 1.92e − 10 0.001705 2.468E − 7 2.429e − 11 0.4041 1.123E − 8F8 0.1361 0.4161 0.1057 0.9247 0.2645 0.4549 0.8568F9 3.938e − 12 2.088e − 10 0.002391 0.003341 2.662e − 12 9.11e − 17 0.2447F10 6.148e − 13 2.977e − 14 0.0008526 4.546E − 8 1.507E − 7 0.0001521 3.712e − 10F11 5.012e − 17 1.384e − 16 4.383E − 5 2.865E − 9 0.0001121 4.909E − 7 1.196e − 15F12 0.2645 0.2431 1.443E − 5 0.2015 0.963 0.1174 0.5834F13 1.198e − 16 7.425e − 16 0.2631 6.789e − 15 8.211e − 15 0.8968 3.966e − 17F14 1.115E − 6 2.179E − 6 0.7609 0.02057 0.2268 1.253E − 7 2.537e − 12F15 0.1608 0.05353 0.003547 0.2389 0.3069 0.8686 0.1926F16 0.0001074 5.795E − 5 1.02E − 5 1.769E − 8 1.915E − 8 0.9607 0.001533F17 0.1133 0.1379 0.0003608 0.1943 0.04941 0.4078 3.721E − 5F18 3.844e − 12 1.961E − 9 3.682E − 9 4.211E − 7 2.203e − 11 1.317e − 13 3.778E − 7F19 3.712e − 10 1.136e − 13 4.035e − 12 3.209E − 7 1.422e − 10 3.366e − 14 5.308E − 5F20 7.038e − 12 4.313e − 14 3.241e − 12 1.425E − 7 1.329e − 10 1.658e − 14 5.88E − 5F21 0.2357 0.2897 0.6303 0.06925 0.02833 0.05328 1.176E − 7F22 0.002277 0.004734 3.301E − 5 0.8433 0.2795 8.942e − 11 0.07407F23 0.05332 0.5612 0.007588 0.04842 0.0008323 0.0727 0.5387F24 0.0687 0.03603 4.008E − 18 1.0 4.008E − 18 0.1003 4.008E − 18F25 6.983E − 9 2.604e − 10 0.3488 2.029E − 7 1.769E − 8 0.001326 4.525E − 7

TABLE XXI

Results of Wilcoxon Test for Comparing the DEs with GPBX-α and Regular Mutation for the 50 Variables Problems

GPBX-α vs. DE sDE dDE cDE HcDE randDE swDEF1 1.757E − 7 5.323E − 10 3.71E − 14 1.832E − 17 2.679E − 17 9.693E − 11 5.73E − 17F2 0.3418 0.8272 4.008E − 18 1.746E − 14 2.136E − 12 0.01453 2.266E − 15F3 4.728E − 10 1.547E − 11 4.008E − 18 2.184E − 7 0.0092 8.607E − 15 6.702E − 9F4 0.3613 0.008317 4.008E − 18 2.552E − 17 2.457E − 16 7.61E − 7 2.335E − 17F5 1.179E − 12 8.314E − 14 4.008E − 18 3.523E − 15 5.925E − 15 1.738E − 12 1.181E − 17F6 1.488E − 5 1.717E − 6 4.035E − 12 5.346E − 9 0.04784 0.003695 0.3246F7 1.566E − 12 3.782E − 12 0.08413 3.661E − 12 1.267E − 10 0.05241 4.814E − 12F8 0.6837 0.911 0.3263 0.6512 0.9083 0.2422 0.2983F9 3.829E − 10 1.196E − 6 0.06737 0.003017 3.433E − 12 2.908E − 15 0.0001907F10 1.337E − 9 6.199E − 6 2.959E − 8 1.686E − 5 0.0001461 0.001619 5.956E − 8F11 7.518E − 13 6.148E − 13 0.01743 2.901E − 8 0.0004802 7.457E − 7 8.789E − 16F12 0.2349 0.6169 5.005E − 5 0.9383 0.7349 0.2147 0.574F13 1.409E − 14 6.517E − 17 0.001626 2.69E − 9 1.79E − 17 0.0003235 3.522E − 13F14 0.7426 0.001377 0.8947 0.7531 0.6339 0.04252 2.528E − 8F15 0.8656 0.773 0.08889 0.2726 0.1179 0.0009177 0.004133F16 0.05482 0.000195 8.422E − 6 0.002552 0.001044 0.5678 1.42E − 5F17 0.8272 0.4005 4.626E − 9 0.4381 0.6862 0.5356 0.6512F18 1.23E − 15 7.77E − 18 4.9E − 12 1.754E − 12 2.749E − 14 1.167E − 9 4.95E − 7F19 4.591E − 17 3.136E − 17 1.797E − 12 2.436E − 10 1.011E − 15 1.016E − 10 1.935E − 6F20 1.687E − 17 3.045E − 17 1.083E − 11 6.42E − 10 9.507E − 17 1.666E − 10 1.747E − 6F21 7.396E − 8 2.492E − 6 0.1096 1.569E − 5 0.002515 0.6423 0.1166F22 6.999E − 5 1.659E − 6 0.04902 0.2164 0.1674 1.695E − 7 0.5693F23 1.88E − 9 1.782E − 7 0.278 1.488E − 5 0.02076 0.5304 1.841E − 8F24 7.03E − 9 7.954E − 9 1.393E − 14 6.216E − 16 1.51E − 17 4.542E − 12 1.333E − 17F25 2.577E − 7 5.04E − 7 0.9329 2.735E − 8 3.488E − 6 0.7054 1.096E − 6


TABLE XXII

Results of Wilcoxon Test for Comparing HdDE Versus dDE and

dDE-GPBX-α

HdDE vs. 30 Variables 50 VariablesdDE dDE-GPBX-α dDE dDE-GPBX-α

F1 0.1003 0.2807 8.357e − 11 0.003445F2 4.008e − 18 4.008e − 18 4.008e − 18 4.008e − 18F3 4.008e − 18 0.9712 4.008e − 18 0.005048F4 1.511e − 13 4.008e − 18 6.298e − 18 4.008e − 18F5 0.03928 4.008e − 18 6.433e − 09 4.008e − 18F6 1.115e − 06 1.373e − 06 0.004734 4.94e − 10F7 0.002083 0.902 2.556e − 06 5.41e − 05F8 3.706e − 13 4.045e − 08 2.179e − 10 8.436e − 15F9 0.0006389 2.35e − 08 0.5375 0.005206F10 0.5103 0.004659 0.002125 0.01156F11 0.0002264 0.8514 0.0005817 0.482F12 0.0006198 0.008402 0.001123 0.1006F13 0.01068 0.001096 0.01965 8.581e − 07F14 1.968e − 05 5.711e − 07 2.115e − 08 1.456e − 09F15 0.004272 0.8522 2.712e − 07 0.0002139F16 0.1664 1.227e − 07 0.0383 0.009017F17 0.004297 5.161e − 10 1.565e − 07 1.625e − 15F18 1.281e − 09 9.236e − 17 6.635e − 16 7.102e − 18F19 6.267e − 11 9.871e − 18 5.012e − 17 4.008e − 18F20 1.013e − 10 8.249e − 18 2.755e − 16 4.258e − 18F21 1.616e − 06 6.452e − 07 0.01955 1.044e − 05F22 1.034e − 09 1.384e − 16 2.916e − 16 6.517e − 17F23 0.1173 1.098e − 05 0.007745 2.439e − 07F24 0.03603 0.03603 3.063e − 09 1.622e − 05F25 2.245e − 08 2.115e − 08 8.888e − 10 9.48e − 09

TABLE XXIII

Results of Wilcoxon Test for Comparing HdDE Versus the

Selected State-of-the-Art Algorithms From the Literature

for the 30 Variables Problems

HdDE vs. SaDE DEGL JADEF1 0.1003 0.1003 0.000826F2 0.006235 4.008E − 18 4.008E − 18F3 0.0003334 4.008E − 18 4.008E − 18F4 0.6169 4.008E − 18 4.008E − 18F5 4.008E − 18 4.008E − 18 2.203E − 11F6 1.69E − 9 4.131E − 18 7.15E − 13F7 2.623E − 16 4.008E − 18 0.09527F8 0.001588 0.05569 4.008E − 18F9 1.253E − 17 4.008E − 18 8.837E − 18F10 2.799E − 13 4.008E − 18 5.558E − 13F11 3.241E − 12 4.008E − 18 8.006E − 18F12 0.9083 2.658E − 11 8.682E − 5F13 4.388E − 18 4.008E − 18 1.956E − 17F14 7.354E − 8 9.658E − 13 4.008E − 18F15 0.0157 5.68E − 8 0.04768F16 6.944E − 8 4.008E − 18 1.7E − 14F17 0.7168 4.008E − 18 4.008E − 18F18 0.002411 2.321E − 16 5.766E − 15F19 0.0007672 1.099E − 16 6.149E − 17F20 0.0441 5.8E − 17 1.164E − 16F21 1.382E − 15 6.96E − 11 6.054E − 9F22 1.645E − 16 0.05144 4.661E − 18F23 0.0001078 0.1181 2.877E − 7F24 0.03603 0.03603 0.0003451F25 3.529E − 8 1.261E − 13 5.602E − 16

TABLE XXIV

Results of Wilcoxon Test for Comparing HdDE Versus the

Selected State-of-the-Art Algorithms from the Literature

for the 50 Variables Problems

HdDE vs. SaDE DEGL JADEF1 4.97E − 12 4.008E − 18 0.0007266F2 3.644E − 7 0.2935 4.008E − 18F3 0.0001121 8.224E − 17 4.008E − 18F4 3.156E − 15 1.253E − 17 4.008E − 18F5 4.008E − 18 4.008E − 18 2.308E − 7F6 7.905E − 13 6.42E − 10 9.871E − 18F7 5.86E − 18 4.008E − 18 0.455F8 2.201E − 8 0.0005321 4.131E − 18F9 4.008E − 18 4.008E − 18 4.008E − 18F10 4.008E − 18 4.008E − 18 1.017E − 17F11 5.024E − 13 4.008E − 18 4.008E − 18F12 0.7349 9.56E − 16 3.2E − 8F13 4.008E − 18 4.008E − 18 4.258E − 18F14 3.917E − 7 6.33E − 17 4.008E − 18F15 0.0001014 2.005E − 11 3.558E − 6F16 1.655E − 14 1.216E − 17 2.395E − 15F17 0.09643 4.008E − 18 4.008E − 18F18 7.112E − 17 1.425E − 16 6.112E − 18F19 1.372E − 14 2.457E − 16 6.49E − 18F20 7.76E − 17 1.693E − 16 5.931E − 18F21 0.05569 0.2633 0.3068F22 4.131E − 18 2.755E − 16 4.131E − 18F23 0.3686 0.00377 0.04594F24 1.257E − 16 2.58E − 6 9.398E − 18F25 1.017E − 17 4.131E − 18 4.008E − 18

algorithm with the regular mutation scheme. Therefore, we arenot comparing here one base algorithm versus some differentones. The base algorithm is different for every column: it isthe equivalent version of the algorithm in every column, butimplementing the GPBX-α mutation.

The behavior of the new HdDE algorithm is statisticallycompared versus the other distributed algorithms studied inthis paper, namely dDE and dDE-GPBX-α, in Table XXII.The results for the two problems are provided in differentcolumns of the same table.

Finally, we show in Tables XXIII and XXIV the results weobtained after applying the Wilcoxon test to the comparisonof the best proposed algorithm in this paper, HdDE, versus theother DEs we took from the state of the art for the small andbig problem sizes, respectively.

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Bernabe Dorronsoro received the Degree in Engi-neering and the Ph.D. degree in computer sciencefrom the University of Malaga, Malaga, Spain, in2002 and 2007, respectively.

He is currently a Scientific Collaborator with theFaculty of Sciences, Technology and Communica-tion, University of Luxembourg, Luxembourg City,Luxembourg. Among his main successful publica-tions, he has several articles in impact journals andone book. His current research interests include gridcomputing, ad hoc networks, the design of new

efficient meta-heuristics, and their application for solving complex real-worldproblems in the domains of logistics, telecommunications, bioinformatics,combinatorial, multiobjective, and global optimization.

Dr. Dorronsoro has been a member of the organizing committees of severalconferences and workshops, and he usually serves as a reviewer for leadingimpact journals and conferences.

Pascal Bouvry received the Ph.D. degree in com-puter science from the University of Grenoble,Grenoble, France, in 1994.

He is currently a Professor with the Faculty ofSciences, Technology and Communication, Univer-sity of Luxembourg, Luxembourg City, Luxem-bourg, and is heading the Computer Science andCommunication Research Unit (http://csc.uni.lu). Hespecializes in parallel and evolutionary computing.His current research interests include the applicationof nature-inspired computing for solving reliability,

security, and energy-efficiency problems in clouds, grids, and ad hoc networks.

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