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International Journal of Systems ScienceVol. 00, No. 00, 00 Month 20xx, 1–24

RESEARCH ARTICLE

Noise Analysis Compact Differential Evolution

Giovanni Iacca, Ferrante Neri∗, and Ernesto Mininno

Department of Mathematical Information Technology, University of Jyvaskyla, Finland;

(submitted November 2010)

This paper proposes a compact algorithm for optimization in noisy environments. This algorithm has a com-pact structure and employs Differential Evolution search logic. Since it is a compact algorithm, it does notstore a population of solutions but a probabilistic representation of the population. This kind of algorithmicstructure can be implemented in those real-world problems characterized by memory limitations, The degree ofrandomization contained in the compact structure allows a robust behavior in the presence of noise. In additionthe proposed algorithm employs the noise analysis survivor selection scheme. This scheme performs an analysisof the noise and automatically performs a re-sampling of the solutions in order to ensure both reliable pairwisecomparisons and a minimal cost in terms of fitness evaluations. The noise analysis component can be reliablyused in noise environments affected by Gaussian noise which allow an a priori analysis of the noise features.This situation is typical of problems where the fitness is computed by means of measurement devices. Anextensive comparative analysis including four different noise levels has been included. Numerical results showthat the proposed algorithm displays a very good performance since it regularly succeeds at handling diversefitness landscapes characterized by diverse noise amplitudes.

Keywords: differential evolution; noisy fitness landscape; compact differential evolution; noise analysisselection scheme

1. Introduction

The presence of uncertainties in the fitness evaluations is a frequent and pernicious situationwhich affects several and various optimization problems. While the application of exact opti-mization algorithms is completely unsuitable for optimization problems affected by uncertainties,evolutionary algorithms can be employed and adapted in order to handle complex and uncertainsituations.Uncertainties are usually classified into four categories, see (Jin and Branke 2005) and (Neri

and Yang 2010):

(1) Noisy fitness function. Noise in fitness evaluations may come from many different sourcessuch as sensor measurement errors or randomized simulations.

(2) Approximated fitness function. When the fitness function is very expensive to evaluate,or an analytical fitness function is not available, approximated fitness functions are oftenused instead.

(3) Robustness. Often, when a solution is implemented, the design variables or the environ-mental parameters are subject to perturbations or changes. Therefore, a common require-ment is that a solution should still work satisfyingly either when the design variableschange slightly, e.g., due to manufacturing tolerances, or when the environmental param-eters vary slightly. This issue is generally known as the search for robust solutions.

∗Corresponding author. Email: ferrante.neri@jyu.fi

ISSN: 0020-7721 print/ISSN 1464-5319 onlinec© 20xx Taylor & FrancisDOI: 10.1080/00207721.20xx.CATSidhttp://www.informaworld.com

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2 Taylor & Francis and I.T. Consultant

(4) Dynamic fitness function. In a changing environment, it should be possible to continuouslytrack the moving optimum rather than repeatedly re-start the optimization process.

This paper focuses on the first above-mentioned category, i.e. the optimization in the presenceof a noisy fitness function. As summarized in (Di Pietro et al. 2004), the noise in the objectivefunction causes two types of undesirable behavior: 1) a candidate solution may be underestimatedand thus eliminated, 2) a candidate solution may be overestimated, thus saved and allowed tolead towards incorrect search directions. Equivalently, a noise fitness landscape can be seen ascharacterized by false optima which consequently mislead the algorithm search, see (Neri andMakinen 2007).Under these conditions, optimization algorithms can be easily misled by noise and thus detect

unsatisfactory solutions. Although Evolutionary Algorithms (EAs), thanks to their stochasticnature, appear to behave robustly in noisy environments, see (Beyer and Sendhoff 2006), and(Arnold and Beyer 2006), they can still perform poorly when the fitness landscape is affected bynoise. As highlighted in (Branke and Schmidt 2003), the most critical operation is the selectionsince it requires a fitness-based comparison. It is obvious that the presence of noise can jeopardizethe fitness-based selection and thus the entire selection.

1.1. Related work

In order to perform the optimization despite the presence of the noise, various algorithmicsolutions have been proposed in literature. Following the classification reported in (Jin andBranke 2005), two categories of noise handling components for EAs can be distinguished; eachcategory can be divided into two sub-categories:

• Methods which require an increase in the computational cost(1) Explicit Averaging Methods(2) Implicit Averaging Methods

• Methods which perform hypotheses about the noise(1) Averaging by means of approximated models(2) Modification of the Selection Schemes

Explicit averaging methods consider that, in the presence of noise, re-sampling of the fitnessvalues followed by the averaging (for zero-mean noise) of these values is beneficial in order toperform a correct fitness estimation. As a matter of fact, increasing the sample size is equivalentto reducing the variance of the estimated fitness. Thus, ideally, an infinite sample size wouldreduce to zero uncertainties in the fitness estimations, transforming the problem into a non-noisyone.Implicit averaging consists of enlarging the population size in order to give the solutions a

chance to be re-evaluated. In addition, a large population size allows the evaluations of neighborsolutions and thus an estimation of the fitness landscape in a certain portion of decision space.It has been shown in (Fitzpatrick and Grefenstette 1988) that a large population size reducesthe influence of noise on the optimization process. In (Miller and Goldberg 1996), the fact that aGenetic Algorithm (GA) with infinite population size would be noise-insensitive has been proven.The topic, whether a re-sampling or an enlargement of the population size is better when it

comes to noise handling, has been discussed in the literature and various results supporting bothphilosophies have been presented, e.g., in (Fitzpatrick and Grefenstette 1988), (Beyer 1993), and(Hammel and Back 1994).Regardless the selected re-sampling strategy, an increase in the computational overhead is

unavoidable. This fact can turn out being unacceptable in real-world applications characterizedby a computationally expensive fitness function. In order to obtain efficient noise filtering withoutexcessive computational cost, various solutions have been proposed in the literature. In (Aizawa

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and Wah 1993) and (Aizawa and Wah 1994), two variants of adaptive re-sampling systemsbased on the progress of evolution have been proposed. In (Stagge 1998), a variable sample sizeperformed by means of a probabilistic criterion based on the solution quality is presented. In (Neriet al. 2006) and (Neri et al. 2008) both sample and population size are adaptively adjusted onthe basis of a diversity measure. In (Branke and Schmidt 2003), (Branke and Schmidt 2004), and(Cantu-Paz 2004) sequential approaches have been proposed which aim at reducing the samplesize during the tournament selection and performing massive re-sampling only when strictlynecessary. In the context of Evolution Strategies (ES), and an elegant re-sampling mechanism forthe solution sorting has been proposed in (Hansen et al. 2008). An extension and generalizationof the study is then presented in (Hansen et al. 2009).Regarding methods which employ approximated models, the main idea is that the fitness value

estimated by an approximated (and computationally cheap) technique is not less imprecise thanthe original noisy value. For example, Branke et al. (2001) and Sano et al. (2000) propose takingfitness estimates of neighboring individuals in order to predict the fitness value of some candidatesolutions. Paper (Neri et al. 2008), by employing a similar philosophy, proposes constructionof a local linear surrogate model (an approximate model of the true fitness function) whichlocally performs the noise filtering. An interesting approach related to this family is the robustevolutionary design presented in (Ong et al. 2006) and (Lim et al. 2010).Other works make some assumptions regarding the noise in order to propose integration of a

filtering component within the selection schemes so as to perform sorting of the solutions withoutthe use of a large amount of samples. In (Markon et al. 2001), in order to reliably sort the fitnessvalues in an Evolution Strategy (ES), a threshold during deterministic selection is imposed. Atheoretical study about the threshold choice is presented in (Beielstein and Markon 2002). In(Rudolph 2001), under the hypothesis that the noise is bounded, application of a partial orderon the set of noisy fitness values is proposed.In parallel, during the latest years, several methods for handling noise in multi-objective cases

have been proposed. Although multi-objective optimization is not the focus of this article, it isworthwhile mentioning recent works, representative of the field. An in depth analysis of the noiseeffect to the functioning of evolutionary algorithms in multi-objective optimization is given in(Goh and Tan 2007). Some examples of countermeasures for handling are available in (Goh andTan 2006) and (Tan and Goh 2008). The algorithms proposed in (Lim et al. 2005), (Lim et al.2006), and (Lim et al. 2007), handle the uncertainties due to the noise, in the context of singleand multi-objective optimization, without making assumptions on the noise structure by meansof the application of the so-called inverse robust evolutionary design. Paper (Goh et al. 2010)proposes a framework for the construction of robust continuous MO test functions characterizedby different noise-induced features.In recent years, noise filtering components have been designed and integrated into other meta-

heuristics than classical ES and Genetic Algorithms (GAs). For example, in (Ball et al. 2003) amodified Simulated Annealing (SA) has been proposed for noisy problems. In (Gutjahr 2003),an implementation of Ant Colony Optimization (ACO) for noisy environments has been pro-posed. In (Bartz-Beielstein et al. 2007), a sequential sampling procedure for a Particle SwarmOptimization (PSO) has been proposed. In (Pan et al. 2006) and (Klamargias et al. 2008) veryaccurate statistics-based procedures for noise handling are integrated within PSO structures.Regarding Differential Evolution (DE) several studies have been performed in the literature.

In several papers, e.g. in (Krink et al. 2004), it is empirically shown that the DE is not recom-mended for noisy problems. On the other hand, some algorithmic solutions have been recentlyproposed. As highlighted in (Neri and Tirronen 2010), the DE structure is characterized by alimited number of search moves and a certain degree of randomization usually helps at enhanc-ing the algorithmic performance. When the fitness landscape is noisy, classical DE presents anexcessively deterministic structure thus displaying a poor performance. The fact that a ran-domization is advisable when dealing with noisy optimization problems has been theoretically

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4 Taylor & Francis and I.T. Consultant

shown in (Abramson 1990). In the case of DE, the benefit of the scale factor randomizationin noisy environments has been highlighted in (Das and Konar 2005) and (Das et al. 2005). In(Rahnamayan et al. 2006), an opposition based DE (i.e., a DE which performs extra samplingof symmetrical points) is proposed for noisy environment and shows that generation of the op-position based points beneficially perturbs determinism of a DE structure in the presence of anoisy fitness.In addition, it must be observed that a DE employs, as a survivor selection scheme, the

so called one-to-one spawning of the solutions, i.e., replacement occurs pairwise between theparent and offspring solution. This fact, according to our interpretation, is at the same time theweak and strong point of a DE in a noisy environment. More specifically, if the noise disturbsthe pairwise comparison and leads to an incorrect evaluation of the most promising solutionbetween parent and offspring, the entire process can be jeopardized since incorrect directionsof the search are transmitted over future generations. On the other hand, in the presence ofnoise, a pairwise comparison is easier to perform than a ranking (as in a GA or an ES). Thusa proper component which rather reliably allows this pairwise comparison, despite the noisyfitness function, can significantly improve the DE performance. A component for performing theDE selection in noisy environments has been proposed in (Caponio and Neri 2009) and employedthereafter in a memetic context in (Mininno and Neri 2010).Another emerging topic in optimization, especially regarding DE development is the employ-

ment of compact algorithms. A compact Evolutionary Algorithm (cEA) is an EvolutionaryAlgorithm (EA) belonging to the class of Estimation of Distribution Algorithms (EDAs). Thealgorithms belonging to this class do not store and process an entire population and all the indi-viduals therein, but on the contrary make use of a probabilistic representation of the populationin order to perform the optimization process. In this way, a much smaller amount of parametersmust be stored in the memory. Thus, a run of these algorithms requires much less capaciousmemory devices compared to their correspondent standard population-based algorithms. Thefirst cEA was the compact Genetic Algorithm (cGA) introduced in (Harik et al. 1999). ThecGA simulates the behavior of a standard binary encoded Genetic Algorithm (GA). Paper (Ahnand Ramakrishna 2003) analyzes analogies and differences between cGAs and (1 + 1)-ES andextends a mathematical model of ES (Rudolph 2001) to cGA obtaining useful information onthe performance. Moreover, Ahn and Ramakrishna (2003) introduced the concept of elitism, andpropose two new variants, with strong and weak elitism respectively, that significantly outper-form both the original cGA and (1 + 1)-ES. A real-encoded cGA (rcGA) has been introducedin (Mininno et al. 2008). Some examples of rcGA applications to control engineering are givenin (Cupertino et al. 2006) and (Cupertino et al. 2007).Recently, compact Differential Evolution (cDE) has been introduced, see (Mininno et al. 2011).

The cDE algorithm has important features which makes it unique among compact algorithms.More specifically, Differential Evolution (DE) contains two important features which allow itscompact implementation in a natural way. The first one is the survivor selection scheme, whichemploys the so-called one-to-one spawning logic, i.e. in DE the survivor selection is carried outby performing a pairwise comparison between the performance of a parent solution and its cor-responding offspring. In our view, this logic can be naturally encoded into a compact algorithm,unlike a selection mechanism typical of Genetic Algorithms (GAs), e.g. tournament selection.In other words a DE can be straightforwardly encoded into a compact algorithm without losingthe basic working principles (in terms of survivor selection). The second issue is related to theDE search logic. A DE algorithm contains a limited amount of search moves, see (Neri andTirronen 2010) and (Weber et al. 2010). In order to overcome these algorithmic limitations, apopular modification of the basic DE scheme consists of introducing some randomness into thesearch logic, e.g. in the jDE proposed in (Brest et al. 2006). A compact DE algorithm, due to itsnature, does not hold a full population of individuals but contains its information in distributionfunctions and samples the individuals from it when necessary. Thus, unavoidably some extra

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randomness, with respect to original DE, is introduced. This fact, as shown in (Mininno et al.2011), appears to be beneficial and to produce in some cases even better results with respect tothe population-based version (which is a way more memory-wise expensive). A memetic versionof cDE has been proposed in (Neri and Mininno 2010).

1.2. Summary of the algorithmic proposal

This paper extends the applicability of cDE to noisy environments, on one hand, making useof the implicit randomization due to the solution sampling from a probabilistic distribution,on the other hand, by integrating a noise analysis component which performs an automatic re-sampling of the solutions on the basis of the noise features. This noise analysis component, fora gives Gaussian noise and standard deviation value performs a mathematical analysis of thenoise comparison and establishes the minimum amount of fitness re-evaluations necessary to afair fitness based sorting. Thus, this approach guarantees a fair fitness-based comparison withthe minimum necessary amount of fitness evaluations.The remainder of this paper is organized in the following way. Sections 2 and 3 show the

working principles of DE and cDE respectively including a brief comment on the elitism. Section4 describes the noise analysis survivor selection and its integration within a cDE. Pseudo-codesgiving a detailed explanation of the algorithmic implementation are also included. Section 5experimentally demonstrates the significance of the proposed approach. Finally, Section 6 givesthe conclusion of the present work and presents future developments.

2. Differential Evolution

In order to clarify the notation used throughout this paper we refer to the minimization problemof an objective function f (x), where x is a vector of n design variables in a decision space D.According to its original definition, see (Neri and Tirronen 2010) and (Das and Suganthan

2011), the DE algorithm consists of the following steps. An initial sampling of Np individuals isperformed pseudo-randomly with a uniform distribution function within the decision space D.At each generation, for each individual xk of the Np, three individuals xr, xs and xt are pseudo-randomly extracted from the population. According to the DE logic, a provisional offspring x′offis generated by mutation as:

x′off = xt + F (xr − xs) (1)

where F ∈ [0, 2] is a scale factor which controls the length of the exploration vector (xr − xs)and thus determines how far from point xk the offspring should be generated. The mutationscheme shown in formula (1) is also known as DE/rand/1. Other variants of the mutation rulehave been subsequently proposed in the literature, see (Neri and Tirronen 2010) and (Das andSuganthan 2011):

• DE/best/1: x′off = xbest+ F (xr − xs)

• DE/cur-to-best/1: x′off = xk+ F (xbest − xk)+ F (xr − xs)

• DE/best/2: x′off = xbest+ F (xr − xs)+ F (xu − xv)

• DE/rand/2: x′off = xt+ F (xr − xs)+ F (xu − xv)

• DE/rand-to-best/1: x′off = xt+ F (xbest − xt) +F (xr − xs)

• DE/rand-to-best/2: x′off = xt+ F (xbest − xt) +F (xr − xs)+ F (xu − xv)

where xbest is the solution with the best performance among the individuals of the populationwhere xu and xv are two additional pseudo-randomly selected individuals. It is worthwhile to

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mention the rotation invariant mutation, see (Neri and Tirronen 2010) and (Das and Suganthan2011):

• DE/current-to-rand/1 xoff = xk +K (xt − xk)+ F ′ (xr − xs)

where K is the combination coefficient, which should be chosen with a uniform random distri-bution from [0, 1] and F ′ = K · F . Since this mutation scheme already contains the crossover,the mutated solution does not undergo the crossover operation described below.Recently, in (Price et al. 2005), a new mutation strategy has been defined. This strategy,

namely DE/rand/1/either-or, consists of the following:

x′off =

{

xt + F (xr − xs) if rand (0, 1) < pFxt +K (xr + xs − 2xt) otherwise

(2)

where for a given value of F , the parameter K is set equal to 0.5 (F + 1).When the provisional offspring has been generated by mutation, each gene of the individual

x′off is exchanged with the corresponding gene of xi with a uniform probability and the finaloffspring xoff is generated:

xoff [i] =

{

x′off [i] if rand (0, 1) ≤ Cr

xk, [i] otherwise(3)

where rand (0, 1) is a random number between 0 and 1; i is the index of the gene under exam-ination; Cr is a constant value namely crossover rate. This crossover strategy is well-known asbinomial crossover and indicated as DE/rand/1/bin.For the sake of completeness, we mention that a few other crossover strategies also exist, for

example the exponential strategy, see e.g. (Price et al. 2005). However, in this paper we focuson the binomial strategy since it is the most commonly used and often the most promising.The resulting offspring xoff is evaluated and, according to a one-to-one spawning strategy, it

replaces xk if and only if f(xoff ) ≤ f(xk); otherwise no replacement occurs. It must be remarkedthat although the replacement indexes are saved, one by one, during the generation, the actualreplacements occur all at once at the end of the generation.

3. Compact Differential Evolution and Elitism

Without loss of generality, let us assume that parameters are normalized so that each searchinterval is [−1, 1]. cDE consists of the following. A (2× n) matrix, namely perturbation vectorPV = [µ, σ], is generated. µ values are set equal to 0 while σ values are set equal to a largenumber λ = 10. The value of λ is empirically set in order to simulate a uniform distributionat the beginning of the optimization process. A solution xe is sampled from PV . The solutionxe is called elite. Subsequently, at each step t, some solutions are sampled and an offspring isgenerated by means of DE mutation. For example, if a DE/rand/1 mutation is selected, threeindividuals xr, xs, and xt are sampled from PV .More specifically, the sampling mechanism of a design variable xr [i] associated to a generic

candidate solution xr from PV consists of the following steps. As mentioned above, for eachdesign variable indexed by i, a truncated Gaussian Probability Distribution Function (PDF)characterized by a mean value µ [i] and a standard deviation σ [i] is associated. The formula ofthe PDF is:

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PDF (truncNorm (x)) =e− (x−µ[i])2

2σ[i]2

√

2π

σ [i](

erf(

µ[i]+1√2σ[i]

)

− erf(

µ[i]−1√2σ[i]

)) (4)

where erf is the error function, see (Gautschi 1972).From the PDF, the corresponding Cumulative Distribution Function (CDF) is constructed by

means of Chebyshev polynomials according to the procedure described in (Cody 1969). It mustbe observed that the codomain of CDF is [0, 1]. In order to sample the design variable xr[i] fromPV a random number rand(0, 1) is sampled from a uniform distribution. The inverse function ofCDF, in correspondence of rand(0, 1), is then calculated. This latter value is xr[i]. As mentionedabove, the sampling is performed on normalized values within [−1, 1]. It can be noticed that inorder to obtain the (phenotype) value in the original interval [a, b] here indicated with xphen, thefollowing operation must be performed:

xphen[i] = xr[i](b− a)

2+ a. (5)

A provisional offspring x′off is then generated by mutation, according to a DE logic, as shown

in eq. (1) or the other mutation schemes shown above, see (Neri and Tirronen 2010). When theprovisional offspring has been generated by mutation, the crossover (e.g. binomial or exponential) between x′off and the elite solution xe is performed; the final offspring xoff is thus generated.Both offspring and elite solutions are considered with their corresponding fitness values and theone-to-one spawning selection is applied. On the basis of their fitness value a winner solution(solution displaying the best fitness) and a loser solution (solution displaying the worst fitness)are detected.The winner solution biases the virtual population by affecting the PV values. The update rule

for µ values is given by:

µt+1 = µt +1

Np(winner − loser) , (6)

where Np is virtual population size. The update rule for σ values is given by:

(

σt+1)2

=(

σt)2

+(

µt)2 −

(

µt+1)2

+1

Np

(

winner2 − loser2)

(7)

where Np is a parameter, namely virtual population size. Details for constructing formulas(6) and (7) are given in (Mininno et al. 2008) while details about the dynamic of the virtualpopulation in cDE can be found in (Mininno et al. 2011). This set of operations is repeated overtime for a given budget.It is worthwhile mentioning that, according to the definition given in (Ahn and Ramakr-

ishna 2003), cEAs can employ persistent and non-persistent elitism. The first allows the elitereplacement only when a better solution is detected while the second imposes a periodic eliterefreshment (every η comparisons) even if the elite has not been outperformed. The selection ofthe elite strategy plays a fundamental role in noisy optimization. Since a compact algorithm storein memory only one solution, the elite, if the elite selection has been performed over-estimatingthe quality of the solution, the entire search can be jeopardized. For this reason, in noisy en-vironments the non-persistent scheme appears to be more efficient than the persistent one, asshown in (Mininno et al. 2011).

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4. Noise Analysis Survivor Selection

A noisy fitness function (affected by Gaussian noise), see (Jin and Branke 2005) can mathemat-ically be expressed as:

f (x, z) =

∞∫

−∞

[f(x) + z] dz = f(x), z ∼ N(0, σ2) (8)

where x is the design vector, f (x) is a time-invariant function and z is an additive noise normallydistributed with 0 mean and variance σ2.In principle, search of the optimum is referred to as f (x); however since only fitness values

related to the f (x) are available, the noisy optimization problem consists of optimizing f (x)in a decision space D where x is defined. Without loss of generality, this paper will refer tominimization problems.As mentioned above, a common approach for reducing the noise bandwidth which affects the

fitness function f (x) is to run multiple evaluations of f (x) and compute their average f (x). Itcan be proven that:

f (x) = limn→∞

1

n

n∑

i=1

f (x, zi) =f (x) (9)

and that a finite increase in the sample size leads to a reduction in the noise variance. Morespecifically, if a candidate solution has been sampled n times, the standard deviation associatedwith the noise is reduced by

√n times.

This operation, although beneficial, is clearly computationally expensive, and can significantlyslow down an optimization algorithm. A trivial n time re-sampling would slow down the optimiza-tion of n times. This can make the optimization time unacceptable for several kinds of real-worldapplications. For this reason, in this paper we propose the integration, within cDE frameworks,of an adaptive re-sampling strategy. As shown in Section 3, cDE sequentially performs pairwisecomparisons between the fitness of the elite and the fitness of the newly generated offspring inorder to select the new elite and continue the optimization. When each pairwise comparison isperformed, the following procedure has been integrated.When the offspring xoff is generated, the value δ =

∣

∣f (xe)− f (xoff )∣

∣ is computed. If δ > 2σthe candidate solution displaying the best performance value is simply chosen for the subsequentgeneration. This choice can be justified considering that for a given Gaussian distribution, 95.4%of the samples fall into an interval whose amplitude is 4σ and has at its center the mean valueof the distribution, see (NIST/SEMATECH 2003). In this case, if the difference between twofitness values is greater than 2σ, it is likely that the point which seems to have a better fitnesstruly is the best performing of the two candidate solutions.On the other hand, if δ < 2σ, noise bands related to the two candidate solutions do overlap,

and determining a ranking based on only one fitness evaluation is impossible. In this case, usingindications of α = min

{

f (xe) , f (xoff )}

and β = max{

f(xe), f (xoff )}

, the following index iscalculated:

υ =α+ 2σ − (β − 2σ)

β + 2σ − (α− 2σ). (10)

The index υ represents the intersection of two intervals, characterized by a center in the fitnessvalue and semi-amplitude 2σ, with respect to their union. In other words, υ is a normalizedmeasure of the noise band overlap. This index can vary between 0 and 1. The limit condition

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Figure 1. Graphical representation of the parameter υ

υ ≈ 0 means that the overlap is limited and thus pairwise ranking given by the single sampleestimations is most likely correct. The complementary limit condition, υ ≈ 1 means that theinterval overlap is almost total and the two fitness values are too close to be distinguished in thenoisy environment. In other words, υ can be seen as a reliability measure of a pairwise solutionranking in the presence of noisy fitness.For the sake of clarity, a graphical representation of υ is given in Fig. 1.On the basis of the calculated value of υ, a set of additional samples ns is performed for both

parent and offspring solutions (the fitness function of parent and offspring is computed ns times)and their respective values of nsi and fi updated. These samples are determined by calculating:

ns =

⌈

(

1.96

2 · (1− υ)

)2⌉

(11)

where 1.96 is the upper critical value of a normal distribution associated to a confidence levelequal to 0.975, see (NIST/SEMATECH 2003). Thus, ns represents the minimum amount of sam-ples which ensure a reliable characterization of the noise distribution, i.e., the amount of sampleswhich allows consideration of the average fitness values as the mean value of a distribution.However, as shown in eq. (11), since for υ → 1 it would result in ns → ∞, a saturation value

for ns has been set in order to avoid infinite loops. It must be remarked that this saturationvalue is the only extra parameter to be set, with respect to a standard DE. In addition, settingof this parameter can be intuitively carried out on the basis of the global computational budgetavailable and the precision requirement in the specific application.When additional samples are performed, the average fitness values f are updated and the solu-

tion characterized by the most promising average fitness is selected for subsequent generation. Itmust be highlighted that the noise analysis survivor selection assumes that the noise is Gaussianand that the standard deviation of the noice can be estimated. This situation obviously does notcover all the optimization problems in noisy environments. On the other hand this is the typicalsituation in industial applications when the noise is due to a set of measurement devices whichmeasures quantities contributing to the construction of the fitness, see (Caponio et al. 2007).The resulting algorithm, consisting of a cDE employing non-persistent elitism and noise analy-

sis survivor selection scheme is named Noise Analysis compact Differential Evolution (NAcDE).The pseudo-code of NAcDE is given in Figs. 2 and 3.

5. Numerical Results

In order to test the algorithmic performance of NAcDE, the base test problems contained in(Suganthan et al. 2005), have been considered in this study. More specifically, the following testfunctions have been included:

• f1 Shifted sphere function

• f2 Shifted Schwefel’s Problem 1.2

• f3 Shifted Rotated High Conditioned Elliptic Function• f4 Shifted Schwefel’s Problem 1.2 with Noise

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10 Taylor & Francis and I.T. Consultant

counter t = 0for i = 1 : n do

{** PV initialization **}initialize µ [i] = 0initialize σ [i] = λ = 10

end for

generate elite xe by means of PVθ = 0while budget condition do

{** Mutation **}generate 3 individuals xr, xs, and xt by means of PVcompute x′

off = xt + F(xr − xs)

{** Crossover **}xoff = x′

off

for i = 1 : n do

generate rand(0, 1)if rand(0, 1) > Cr then

xoff [i] = elite [i]end if

end for

{** Elite Selection **}[winner, loser] = compete (xoff , xe)θ = θ + 1if xoff == winnerOR θ ≥ η then

elite = xoff

θ = 0end if

{** PV Update **}for i = 1 : n do

µt+1[i] = µt[i] + 1Np

(winner[i] − loser[i])

σt+1 =√

(σt[i])2 + (µt[i])2 − (µt+1[i])2 + 1Np

(winner2[i] − loser2[i])

end for

t = t + 1end while

Figure 2. NAcDE pseudo-code

{**Compete function with Noise Analysis**}[winner, loser] = compete (xoff , xe){********}winner = xe and loser = xoff

if∣

∣f (xe) − f (xoff )∣

∣ > 2σ then

if f (xoff ) ≤ f (xe) then

winner = xoff and loser = xe

end if

else

α = min{

f (xe) , f (xoff )}

β = max{

f(xe), f (xoff )}

compute υ = α+2σ−(β−2σ)β+2σ−(α−2σ)

compute ns =

⌈

(

1.962·(1−υ)

)2⌉

perform re-samplingupdate f (xe) and f (xoff )update fitness counterif f (xoff ) ≤ f (xe) then

winner = xoff and loser = xe

end if

end if

Figure 3. Pseudo-code of the noise analysis survivor selection in NAcDE

• f5 Schwefel’s Problem 2.6 with Global Optimum on Bounds

• f6 Shifted Rosenbrock’s Function

• f7 Shifted rotated Griewank’s Function with Bounds

• f8 Shifted rotated Ackley’s Function with Global Optimum on the Bounds

• f9 Shifted Rastrigin’s Function

• f10 Shifted rotated Rastrigin’s Function

• f11 Shifted Rotated Weierstrass Function

• f12 Schwefel’s Problem 2.13• f13 Expanded Extended Griewank’s plus Rosenbrock’s Function (F8F2)

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International Journal of Systems Science 11

• f14 Shifted Rotated Expanded Scaffer’s F6

In order to make a preliminary analysis of the algorithmic performance also in terms of scal-ability, each test problem has been considered with dimensionality n = 10 and n = 30.The noise simulation has been performed in the following way. For each test problem, codomain

amplitude C has been estimated as the difference between the fitness value in the global optimum(when analytically known, otherwise the best fitness value detected in the literature) and theaverage fitness value computed over 100 points pseudo-randomly generated within the decisionspace. Then, for each test problem, in addition to the stationary case three noisy test cases (levelsof noise) have been generated by adding to the time-invariant function a zero-mean Gaussiannoise characterized by a standard deviation equal to 5%, 10% and 20% of C, respectively. Thesepercentage values are referred to here as noise levels. It must be remarked that the noise levelof 5% is of great interest in real world applications, since it is the typical noise bandwidthfor industrial measurement instrumentation. In the following tables the various noise levels areindicated with 0, 0.05, 0.1, and 0.2, respectively.The proposed NAcDE has been tested for the 14 × 4 × 2 = 112 problems and compared

with modern algorithms designed for handling uncertainties. More specifically, the followingalgorithms have been considered in this study (see original papers for in depth explanations ofthe parameters):

• Differential Evolution with Randomized Scale Factor and Threshold based Selection (DE-RSF-TS) proposed in (Das et al. 2005). The DE-RSF-TS algorithm has been run with Fpseudo-randomly selected between 0.5 and 1, and CR = 0.3. The threshold value τ hasbeen set equal to 2σ.

• Opposition-Based Differential Evolution (ODE) for noisy problems proposed in (Rahna-mayan et al. 2006). The ODE has been run with DE/rand/1/bin, F = 0.75, Cr = 0.9,jumping rate constant Jr = 0.3, differential amplification factor F ′ = 0.1.

• Noise Analysis Differential Evolution (NADE) proposed in (Caponio and Neri 2009). TheNADE algorithm has been run with DE/rand/1/bin scheme, with F pseudo-randomlyselected between 0.5 and 1, and CR = 0.9. The maximum number of evaluations perindividual has been set equal to 30.

• Noise Analysis compact Genetic Algorithm (NAcGA) proposed in (Neri et al. 2010) hasbeen run with Np = 100, and η = 66. The maximum number of evaluations per individualhas been set equal to 30.

All these algorithms have been run with a population size of 100 individuals. Regarding theNAcDE, the following setting has been made: F = 0.75, Cr = 0.9, Np = 100, and η = 66 inaccordance with the theorem in (Ahn and Ramakrishna 2003) . Also for NAcDE, the maximumnumber of evaluations per individual has been fixed equal to 30.For each algorithm and each problem, 30 independent runs have been performed. Each run has

been continued for 5000×n fitness evaluations. It must be remarked that, for a fair comparison,each fitness re-evaluation is counted as a fitness evaluation on the final budget (e.g. if for assessingthe superiority of a solution the noise analysis component imposes 20 extra function calls, theglobal fitness evaluation counter is updated accordingly). Table 1 displays the average fitnessvalues ± corresponding standard deviationIn order to strengthen the statistical significance of the results, the Wilcoxon Rank-Sum test

has also been applied according to the description given in (Wilcoxon 1945), where the confidencelevel has been fixed to 0.95. Table 2 summarizes the results of the Wilcoxon test for n = 10.A “+” indicates the case in which NAcDE statistically outperforms, for the corresponding testproblem, the algorithm indicated in column; a “=” indicates that a pairwise comparison leadsto success of the Wilcoxon Rank-Sum test, i.e., the two algorithms have the same performance;a “-” indicates that NAcDE is outperformed.

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12 Taylor & Francis and I.T. Consultant

Table 1. Final average fitness ± standard deviation for n = 10

Test Problem DE-RSF-TS ODE NADE NAcGA NAcDE

f1 0 1.17e-09±5.13e-10 1.05e+02±1.63e+02 1.23e-09±5.32e-10 5.00e+00±4.36e+00 1.86e-08±1.45e-08

0.05 4.00e+03±9.96e+02 5.74e+03±1.86e+03 2.04e+03±1.26e+03 2.12e+03±6.03e+02 1.86e+03±6.85e+02

0.1 7.05e+03±2.45e+03 9.62e+03±3.72e+03 3.42e+03±1.36e+03 4.18e+03±1.02e+03 2.84e+03±1.03e+03

0.2 1.30e+04±4.30e+03 1.41e+04±4.82e+03 5.99e+03±1.75e+03 6.39e+03±2.10e+03 4.63e+03±1.19e+03

f2 0 1.44e+00±5.41e-01 2.53e+03±1.80e+03 1.44e+00±6.80e-01 5.12e+01±3.40e+01 3.43e-02±9.26e-02

0.05 1.76e+04±5.96e+03 1.92e+04±6.86e+03 1.43e+04±6.62e+03 1.08e+04±3.37e+03 9.80e+03±3.18e+03

0.1 1.82e+04±6.86e+03 1.93e+04±6.88e+03 1.68e+04±6.71e+03 1.64e+04±5.17e+03 1.39e+04±4.20e+03

0.2 1.92e+04±6.79e+03 1.81e+04±6.79e+03 1.73e+04±5.94e+03 1.57e+04±5.13e+03 1.44e+04±3.95e+03

f3 0 1.32e+06±4.21e+05 2.56e+07±7.64e+06 1.28e+06±4.55e+05 1.68e+06±7.35e+05 1.42e+06±8.39e+05

0.05 1.62e+08±8.15e+07 2.33e+08±1.45e+08 1.24e+08±9.08e+07 8.17e+07±4.10e+07 6.04e+07±3.25e+07

0.1 2.26e+08±1.37e+08 1.99e+08±1.14e+08 2.34e+08±1.43e+08 1.22e+08±7.22e+07 9.68e+07±5.75e+07

0.2 2.48e+08±1.37e+08 2.68e+08±1.74e+08 2.57e+08±1.70e+08 1.70e+08±7.64e+07 1.36e+08±7.56e+07

f4 0 1.07e+04±5.43e+03 3.03e+03±1.60e+03 1.36e+03±7.20e+02 5.67e+03±3.51e+03 1.53e+04±5.60e+03

0.05 2.32e+04±8.64e+03 2.66e+04±8.76e+03 2.12e+04±7.73e+03 5.48e+04±3.17e+04 3.82e+04±2.00e+04

0.1 2.44e+04±5.81e+03 2.48e+04±6.76e+03 2.42e+04±6.79e+03 5.00e+04±3.26e+04 4.00e+04±2.56e+04

0.2 2.54e+04±7.91e+03 2.49e+04±6.87e+03 2.43e+04±6.30e+03 6.30e+04±4.37e+04 4.27e+04±1.89e+04

f5 0 7.06e-07±4.00e-07 4.69e+03±2.33e+03 7.11e-07±4.31e-07 5.87e+01±2.90e+01 2.76e+01±6.26e+01

0.05 1.75e+03±6.97e+02 6.25e+03±1.91e+03 1.19e+03±3.25e+02 3.55e+03±1.22e+03 1.04e+03±4.04e+02

0.1 4.25e+03±1.48e+03 7.18e+03±1.75e+03 2.74e+03±8.59e+02 6.79e+03±1.10e+03 1.92e+03±6.59e+02

0.2 1.01e+04±2.48e+03 8.28e+03±8.48e+02 4.44e+03±1.54e+03 1.03e+04±1.91e+03 4.29e+03±1.41e+03

f6 0 5.19e+00±5.41e-01 5.30e+06±1.79e+07 5.30e+00±4.81e-01 2.35e+03±2.08e+03 1.01e+03±1.01e+03

0.05 1.65e+09±8.44e+08 2.27e+09±1.36e+09 9.90e+08±6.64e+08 5.87e+08±4.41e+08 2.89e+08±1.36e+08

0.1 2.33e+09±1.20e+09 3.24e+09±1.77e+09 1.34e+09±6.98e+08 9.96e+08±6.67e+08 3.43e+08±1.97e+08

0.2 3.79e+09±2.33e+09 3.82e+09±1.57e+09 2.19e+09±1.26e+09 1.70e+09±1.04e+09 7.66e+08±6.32e+08

f7 0 1.27e+03±5.08e-02 1.27e+03±2.27e-01 1.27e+03±4.65e-13 1.27e+03±4.35e-01 1.27e+03±8.73e-07

0.05 1.36e+03±5.48e+01 1.27e+03±4.65e-13 1.32e+03±2.65e+01 1.63e+03±6.71e+01 1.32e+03±2.44e+01

0.1 1.49e+03±1.04e+02 1.27e+03±1.45e-01 1.37e+03±7.80e+01 1.88e+03±1.24e+02 1.38e+03±6.26e+01

0.2 1.99e+03±3.89e+02 1.28e+03±3.39e+01 1.47e+03±1.08e+02 2.22e+03±1.55e+02 1.52e+03±7.10e+01

f8 0 2.04e+01±7.81e-02 2.06e+01±1.20e-01 2.04e+01±7.84e-02 2.04e+01±6.63e-02 2.04e+01±8.64e-02

0.05 2.04e+01±1.12e-01 2.06e+01±1.14e-01 2.04e+01±1.04e-01 2.05e+01±7.56e-02 2.04e+01±8.37e-02

0.1 2.05e+01±1.30e-01 2.06e+01±1.56e-01 2.05e+01±1.05e-01 2.05e+01±9.11e-02 2.04e+01±9.12e-02

0.2 2.07e+01±1.72e-01 2.07e+01±1.08e-01 2.06e+01±1.75e-01 2.06e+01±1.04e-01 2.05e+01±9.70e-02

f9 0 7.41e+00±1.58e+00 1.87e+01±6.45e+00 7.51e+00±1.35e+00 4.59e+00±1.09e+00 1.02e+01±5.08e+00

0.05 5.01e+01±1.07e+01 3.90e+01±9.72e+00 3.22e+01±8.37e+00 2.52e+01±5.55e+00 2.11e+01±5.80e+00

0.1 7.35e+01±1.74e+01 7.57e+01±2.05e+01 6.20e+01±1.37e+01 4.29e+01±1.01e+01 4.26e+01±1.07e+01

0.2 1.04e+02±1.99e+01 1.08e+02±1.58e+01 9.05e+01±1.35e+01 6.69e+01±1.04e+01 6.43e+01±1.33e+01

f10 0 7.99e+00±1.78e+00 1.63e+01±6.28e+00 7.84e+00±1.27e+00 4.19e+00±1.09e+00 1.02e+01±4.42e+00

0.05 4.52e+01±5.82e+00 3.79e+01±8.61e+00 3.43e+01±7.73e+00 2.29e+01±4.32e+00 2.15e+01±8.12e+00

0.1 7.90e+01±1.59e+01 7.20e+01±1.53e+01 5.23e+01±1.30e+01 4.20e+01±9.69e+00 4.34e+01±8.94e+00

0.2 1.00e+02±1.75e+01 1.05e+02±1.47e+01 8.38e+01±1.70e+01 6.52e+01±1.15e+01 6.58e+01±6.47e+00

f11 0 9.30e+00±5.75e-01 1.02e+01±8.17e-01 9.17e+00±7.44e-01 6.48e+00±6.19e-01 5.63e+00±1.61e+00

0.05 9.93e+00±1.01e+00 1.02e+01±8.41e-01 9.47e+00±5.45e-01 7.45e+00±7.29e-01 7.81e+00±1.56e+00

0.1 1.05e+01±9.29e-01 1.11e+01±1.07e+00 1.03e+01±1.11e+00 7.89e+00±7.66e-01 9.66e+00±5.48e-01

0.2 1.18e+01±1.47e+00 1.17e+01±1.24e+00 1.16e+01±1.53e+00 9.45e+00±6.45e-01 1.03e+01±8.40e-01

f12 0 7.94e+03±2.24e+03 1.24e+04±6.21e+03 7.86e+03±3.00e+03 2.10e+03±7.40e+02 9.30e+03±6.23e+03

0.05 5.89e+04±1.77e+04 4.98e+04±1.57e+04 4.85e+04±1.22e+04 2.26e+04±5.77e+03 3.45e+04±1.26e+04

0.1 9.41e+04±2.75e+04 9.36e+04±3.22e+04 6.59e+04±3.17e+04 3.68e+04±1.07e+04 4.41e+04±1.50e+04

0.2 1.17e+05±4.64e+04 1.30e+05±4.66e+04 1.16e+05±4.28e+04 5.71e+04±2.52e+04 5.90e+04±1.72e+04

f13 0 2.18e+00±3.35e-01 2.56e+00±8.73e-01 2.13e+00±3.42e-01 7.97e-01±3.83e-01 9.00e-01±3.19e-01

0.05 4.90e+01±3.74e+01 4.05e+01±2.69e+01 3.62e+01±1.96e+01 2.71e+01±1.42e+01 1.90e+01±4.86e+00

0.1 6.05e+01±4.67e+01 4.83e+01±3.76e+01 4.91e+01±3.71e+01 2.85e+01±2.02e+01 2.59e+01±1.08e+01

0.2 6.10e+01±4.82e+01 6.07e+01±4.83e+01 5.61e+01±4.20e+01 3.79e+01±1.82e+01 2.73e+01±1.35e+01

f14 0 3.86e+00±2.02e-01 4.17e+00±2.27e-01 3.89e+00±1.51e-01 3.46e+00±2.43e-01 3.32e+00±4.04e-01

0.05 3.94e+00±1.39e-01 4.27e+00±8.76e-02 3.99e+00±9.06e-02 3.70e+00±1.49e-01 3.33e+00±3.90e-01

0.1 4.01e+00±1.95e-01 4.28e+00±8.99e-02 4.06e+00±1.48e-01 3.75e+00±2.36e-01 3.87e+00±1.93e-01

0.2 4.19e+00±2.77e-01 4.33e+00±1.34e-01 4.16e+00±2.27e-01 4.00e+00±1.60e-01 4.15e+00±1.13e-01

Results displayed in Table 1 show that, for n = 10, DE-RSF-TS and ODE obtain the bestresults in only 3 cases, NADE in 6 cases, NAcGA in 14 cases, and the proposed NAcDE in theremaining 30 cases, out of the 56 test problems considered in this study. Thus, counting only thebold entries in Table 1 we can conclude that, for a low dimensional case (n = 10), the proposedNAcDE displays a very good performance since it obtains the best results in more than 53%of the test problems in a comparison with other four algorithms. The second best algorithm,the NAcGA, appears to behave worse than NAcDE, as it obtained the best performance inonly 25% of the cases. The statistical analysis confirms the superiority of NAcDE, for n = 10and for the considered problems, with respect to the other algorithms considered in this study.More specifically, NAcDE is significantly outperformed by DE-RSF-TS and ODE in only 7 caseswhile it outperformed these two algorithms for at least 70% of the problems. The comparisonwith NADE and NAcGA shows that the proposed NAcDE algorithm outperforms the otheralgorithms employing noise analysis components in about half of the cases and is outperformedfor less than 20% of comparisons.In order to draw some statistically significant conclusions regarding the performance of the

NAcDE algorithm, the Holm procedure, see (Holm 1979) and (Garcia et al. 2008), for the five

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International Journal of Systems Science 13

Table 2. Wilcoxon test for n = 10 (“+” means that NAcDE outperforms DE, “-” means that NAcDE is outperformed, and “=”

means that the algorithms have the same performance)

Test Problem DE-RSF-TF OBDE NADE NArCGAf1 0 - + - +

0.05 + + = =0.1 + + = +0.2 + + + +

f2 0 + + + +0.05 + + + =0.1 + + = =0.2 + + = =

f3 0 = + = =0.05 + + + =0.1 + + + =0.2 + + + =

f4 0 - - - -0.05 - = - =0.1 - - - =0.2 - - - =

f5 0 - + - +0.05 + + = +0.1 + + + +0.2 + + = +

f6 0 - + - +0.05 + + + +0.1 + + + +0.2 + + + +

f7 0 + + - +0.05 + - = +0.1 + - = +0.2 + - - +

f8 0 = + = =0.05 = + = +0.1 = + = =0.2 + + = =

f9 0 = + = -0.05 + + + +0.1 + + + =0.2 + + + =

f10 0 = + = -0.05 + + + =0.1 + + + =0.2 + + + =

f11 0 + + + +0.05 + + + =0.1 + + + -0.2 + + + -

f12 0 = = = -0.05 + + + -0.1 + + + =0.2 + + + =

f13 0 + + + =0.05 + + + =0.1 + + + =0.2 + + + +

f14 0 + + + =0.05 + + + +0.1 + + + -0.2 = + = -

algorithms and 14 × 4 = 56 problems under consideration has been performed. The Holm pro-cedure consists of the following. Considering the results in the tables above, the five algorithmsunder analysis have been ranked on the basis of their average performance calculated over thefifty-six test problems. More specifically, a score Ri for i = 1, . . . , NA (where NA is the number ofalgorithms under analysis, NA = 5 in our case) has been assigned. The score has been assignedin the following way: for each problem, a score of 5 is assigned to the algorithm displaying thebest performance, 4 is assigned to the second best, 3 to the third and so on. The algorithmdisplaying the worst performance scores 1. For each algorithm, the scores outcoming by eachproblem are summed up averaged over the amount of test problems (5 in our case). On the basisof these scores the algorithms are sorted (ranked). With the calculated Ri values, NAcDE hasbeen taken as a reference algorithm. Indicating with R0 the rank of NAcDE, and with Rj forj = 1, . . . , NA − 1 the rank of one of the remaining nine algorithms, the values zj have been

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14 Taylor & Francis and I.T. Consultant

Table 3. Holm procedure for 10 dimensionsNA − j Algorithm zNA

− j pNA− j δ/(NA − j) Hypothesis

4 OBDE -7.59e+00 1.60e-14 1.25e-02 Rejected3 DERSF -5.44e+00 2.69e-08 1.67e-02 Rejected2 NADE -2.39e+00 8.41e-03 2.50e-02 Rejected1 NArcGA -1.85e+00 3.20e-02 5.00e-02 Accepted

calculated as

zj =Rj −R0

√

NA(NA+1)6NTP

where NTP is the number of test problems in consideration (NTP = 56 in our case). By meansof the zj values, the corresponding cumulative normal distribution values pj have been calcu-lated. These pj values have then been compared with the corresponding δ/(NA − j) where δis the level of confidence, set to 0.05 in our case. Table 3 displays zj values, pj values, andcorresponding δ/(NA − j). The values of zj and pj are expressed in terms of zNA−j and pNA−j

for j = 1, . . . , NA − 1. Moreover, it is indicated whether the null-hypothesis (that the two algo-rithms have indistinguishable performances) is “Rejected” i.e., NAcDE statistically outperformsthe algorithm under consideration, or “Accepted” if the distribution of values can be consideredthe same (there is no outperformance).It can be observed that on average the proposed NAcDE, in the 10-dimensional cases under

examination, has a comparable performance with respect to NArcGA and significantly outper-forms all the other algorithms considered in this study .Numerical results for n = 30, in terms of average final fitness and corresponding Wilcoxon

test are displayed in Tables 4 and 5, respectively.Numerical results for n = 30 confirm the capability of NAcDE to perform well in noisy

environments. When the dimensionality grows, as expected, the performance of population-based algorithms increases with respect to compact algorithms, see (Mininno et al. 2008) and(Mininno et al. 2011). In this case NADE obtains a respectable performance with respect toNAcDE. On the other hand, NAcDE still outperforms all the other algorithms considered in thisstudy also for n = 30. Table 4 shows that NAcDE performs better than the other algorithms for27 test problems while the second best algorithm in the table is NADE which reaches the bestperformance in only 11 cases. In addition, the superiority of NAcDE with respect to DE-RSF-TSand ODE is more clear for n = 30 rather than n = 10. As shown in Table 5 DE-RSF-TS andODE succeed at outperforming NAcDE in only a few isolated cases, 7 and 5 times respectively,while for the absolute majority of numerical experiments are outperformed by NAcDE. Thecomparison with NADE shows that NAcDE outperforms NADE in 27 cases and is outperformed9 times. A similar situation occurs in the comparison between NAcDE and NAcGA, whereNAcDE outperforms NAcGA 26 times and is outperformed 11 times.As a general trend, it can be appreciated from numerical results that algorithms employing

the noise analysis selection display a high performance in various noise condition as they tend toavoid a waste of computational effort by re-sampling the solution only when necessary and guar-antee that over-estimations of the solutions do not mislead the search of promising directions.With respect to NADE, NAcDE employs a virtual population and consequently a randomizedgeneration of the solutions composing the offspring. This mechanism appears to be more promis-ing than the randomized scale factor integrated into the NADE. According to our interpretation,while the randomized scale factor is a value which has the same meaning throughout the entireevolution, the employment of virtual population allows the generation of randomized individu-als in a large portion of the decision space, at the beginning of the optimization process, andprogressively generates individuals only in the neighbourhood of the elite. In this sense NAcDE

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Table 4. Final average fitness ± standard deviation for n = 30

Test Problem DE-RSF-TS OBDE NADE NArCGA NAcDE

f1 0 2.05e-05±8.89e-06 7.02e+03±3.24e+03 1.80e-05±7.38e-06 1.88e-05±1.92e-05 3.34e+02±4.08e+02

0.05 1.82e+04±4.86e+03 3.28e+04±9.04e+03 9.01e+03±2.73e+03 1.54e+04±2.42e+03 1.61e+04±2.62e+03

0.1 3.99e+04±8.11e+03 6.56e+04±2.07e+04 1.63e+04±4.30e+03 2.03e+04±2.99e+03 2.16e+04±3.16e+03

0.2 6.83e+04±1.71e+04 9.38e+04±1.36e+04 3.11e+04±7.54e+03 4.41e+04±7.27e+03 2.85e+04±5.32e+03

f2 0 1.36e+04±2.51e+03 6.98e+04±9.27e+03 1.43e+04±2.90e+03 1.15e+03±1.08e+03 1.47e+04±5.98e+03

0.05 1.56e+05±3.92e+04 1.64e+05±4.24e+04 1.51e+05±3.34e+04 9.90e+04±1.27e+04 1.00e+05±1.67e+04

0.1 1.62e+05±3.96e+04 1.63e+05±4.04e+04 1.52e+05±3.69e+04 1.04e+05±1.19e+04 1.13e+05±2.61e+04

0.2 1.58e+05±3.48e+04 1.64e+05±4.24e+04 1.55e+05±4.41e+04 9.77e+04±2.10e+04 1.24e+05±2.28e+04

f3 0 1.45e+08±3.50e+07 4.25e+08±1.36e+08 1.50e+08±2.24e+07 9.58e+06±5.18e+06 5.06e+07±2.04e+07

0.05 1.42e+09±4.50e+08 1.74e+09±4.83e+08 7.13e+08±2.61e+08 5.59e+08±7.71e+07 3.96e+08±1.48e+08

0.1 1.79e+09±5.69e+08 2.18e+09±5.42e+08 1.24e+09±4.49e+08 9.74e+08±1.97e+08 6.14e+08±1.60e+08

0.2 2.32e+09±6.03e+08 2.28e+09±5.08e+08 1.29e+09±5.90e+08 1.15e+09±2.90e+08 1.01e+09±3.22e+08

f4 0 1.01e+05±3.59e+04 7.59e+04±1.17e+04 3.96e+04±1.31e+04 9.23e+04±3.34e+04 1.27e+05±3.10e+04

0.05 1.99e+05±4.72e+04 2.08e+05±4.52e+04 1.87e+05±5.05e+04 2.94e+05±1.69e+05 2.58e+05±1.56e+05

0.1 1.96e+05±3.85e+04 2.07e+05±4.38e+04 2.04e+05±3.78e+04 2.97e+05±1.48e+05 3.09e+05±1.84e+05

0.2 2.01e+05±3.94e+04 2.09e+05±4.79e+04 2.06e+05±4.76e+04 3.29e+05±1.55e+05 2.75e+05±1.72e+05

f5 0 2.07e+03±7.48e+02 1.85e+04±2.40e+03 1.95e+03±4.25e+02 3.71e+03±1.51e+03 6.31e+03±1.37e+03

0.05 1.81e+04±1.89e+03 2.10e+04±1.70e+03 1.15e+04±1.16e+03 1.19e+04±1.26e+03 1.16e+04±1.41e+03

0.1 2.74e+04±4.01e+03 2.72e+04±5.80e+03 1.60e+04±2.61e+03 1.68e+04±2.58e+03 1.45e+04±1.79e+03

0.2 3.64e+04±4.91e+03 3.76e+04±6.82e+03 2.23e+04±2.17e+03 2.81e+04±3.09e+03 1.84e+04±3.34e+03

f6 0 3.82e+01±3.64e+01 5.70e+08±3.96e+08 3.64e+01±3.86e+01 4.82e+03±4.61e+03 4.32e+06±1.05e+07

0.05 1.61e+10±5.76e+09 2.96e+10±9.28e+09 5.80e+09±2.70e+09 4.62e+09±1.05e+09 4.16e+09±1.40e+09

0.1 2.75e+10±9.39e+09 4.90e+10±1.76e+10 1.20e+10±5.82e+09 9.90e+09±3.80e+09 6.31e+09±2.43e+09

0.2 5.56e+10±1.88e+10 7.19e+10±2.03e+10 1.94e+10±7.23e+09 2.70e+10±6.77e+09 1.07e+10±4.33e+09

f7 0 4.70e+03±7.81e-12 4.78e+03±2.36e+01 4.70e+03±1.92e-12 4.70e+03±5.54e-01 4.70e+03±1.99e-06

0.05 5.50e+03±1.59e+02 4.79e+03±0.00e+00 5.04e+03±9.11e+01 5.27e+03±8.31e+01 5.00e+03±1.07e+02

0.1 6.39e+03±4.41e+02 4.82e+03±9.07e+01 5.33e+03±1.26e+02 6.11e+03±3.35e+02 5.28e+03±1.20e+02

0.2 8.64e+03±9.03e+02 5.38e+03±3.57e+02 5.83e+03±3.37e+02 8.48e+03±5.01e+02 5.83e+03±3.55e+02

f8 0 2.10e+01±4.81e-02 2.10e+01±5.89e-02 2.10e+01±6.75e-02 2.10e+01±6.33e-02 2.09e+01±1.16e-01

0.05 2.10e+01±8.67e-02 2.10e+01±5.47e-02 2.10e+01±5.37e-02 2.10e+01±6.48e-02 2.10e+01±7.24e-02

0.1 2.11e+01±9.23e-02 2.10e+01±8.92e-02 2.10e+01±1.12e-01 2.10e+01±4.65e-02 2.10e+01±6.13e-02

0.2 2.11e+01±1.18e-01 2.11e+01±1.33e-01 2.11e+01±9.13e-02 2.10e+01±5.22e-02 2.10e+01±6.94e-02

f9 0 4.62e+01±6.93e+00 1.61e+02±2.59e+01 4.69e+01±5.63e+00 7.21e+01±1.34e+01 6.26e+01±1.38e+01

0.05 2.47e+02±2.50e+01 2.42e+02±3.13e+01 1.67e+02±1.97e+01 1.93e+02±1.61e+01 1.36e+02±1.73e+01

0.1 3.49e+02±3.05e+01 3.78e+02±6.10e+01 2.41e+02±3.37e+01 2.52e+02±2.03e+01 2.08e+02±3.66e+01

0.2 4.84e+02±5.38e+01 5.09e+02±4.44e+01 3.22e+02±3.49e+01 3.36e+02±3.36e+01 2.82e+02±2.72e+01

f10 0 4.73e+01±4.65e+00 1.57e+02±2.43e+01 4.79e+01±5.18e+00 6.99e+01±1.24e+01 6.39e+01±1.20e+01

0.05 2.16e+02±2.03e+01 2.08e+02±2.92e+01 1.44e+02±2.42e+01 2.03e+02±1.24e+01 1.47e+02±2.41e+01

0.1 3.27e+02±3.38e+01 3.61e+02±3.48e+01 2.43e+02±3.19e+01 2.57e+02±2.07e+01 2.41e+02±2.84e+01

0.2 4.52e+02±4.89e+01 4.82e+02±5.86e+01 3.01e+02±4.31e+01 3.57e+02±3.00e+01 2.92e+02±2.68e+01

f11 0 4.00e+01±1.18e+00 4.10e+01±1.08e+00 4.03e+01±1.49e+00 3.37e+01±2.25e+00 2.94e+01±3.05e+00

0.05 4.11e+01±1.11e+00 4.15e+01±1.76e+00 4.14e+01±1.60e+00 3.90e+01±1.22e+00 3.31e+01±2.77e+00

0.1 4.27e+01±2.05e+00 4.29e+01±1.98e+00 4.21e+01±2.01e+00 3.99e+01±1.19e+00 3.53e+01±3.33e+00

0.2 4.55e+01±2.12e+00 4.52e+01±2.28e+00 4.39e+01±2.22e+00 4.13e+01±1.70e+00 4.01e+01±2.07e+00

f12 0 4.02e+05±5.99e+04 3.51e+05±8.64e+04 4.10e+05±6.06e+04 2.46e+05±7.47e+04 3.21e+05±7.99e+04

0.05 9.72e+05±1.22e+05 8.38e+05±1.05e+05 7.49e+05±1.53e+05 6.78e+05±1.30e+05 5.70e+05±1.52e+05

0.1 1.34e+06±2.67e+05 1.22e+06±1.99e+05 9.91e+05±1.87e+05 9.91e+05±1.31e+05 9.24e+05±1.71e+05

0.2 1.78e+06±2.96e+05 1.62e+06±2.27e+05 1.28e+06±2.40e+05 1.32e+06±1.67e+05 1.37e+06±2.06e+05

f13 0 1.45e+01±1.08e+00 2.87e+01±9.19e+00 1.45e+01±1.07e+00 9.93e+00±2.34e+00 1.80e+01±1.17e+01

0.05 7.64e+02±2.95e+02 9.97e+02±4.87e+02 3.04e+02±1.11e+02 2.17e+02±7.89e+01 2.95e+02±8.90e+01

0.1 1.15e+03±4.80e+02 1.28e+03±6.05e+02 7.30e+02±4.38e+02 3.04e+02±7.99e+01 3.90e+02±1.46e+02

0.2 1.42e+03±5.40e+02 1.31e+03±5.59e+02 1.06e+03±4.98e+02 5.06e+02±2.20e+02 5.59e+02±2.19e+02

f14 0 1.36e+01±1.63e-01 1.39e+01±1.60e-01 1.37e+01±1.26e-01 1.32e+01±2.50e-01 1.25e+01±4.88e-01

0.05 1.37e+01±1.51e-01 1.39e+01±1.95e-01 1.37e+01±1.41e-01 1.34e+01±1.84e-01 1.26e+01±4.96e-01

0.1 1.38e+01±2.14e-01 1.40e+01±1.86e-01 1.38e+01±2.00e-01 1.35e+01±2.34e-01 1.29e+01±4.40e-01

0.2 1.41e+01±2.13e-01 1.42e+01±1.82e-01 1.40e+01±2.04e-01 1.36e+01±1.72e-01 1.35e+01±2.98e-01

progressively focuses the search while it is converging to promising genotypes. However, this pro-gressive focus of the search is performed by a randomization strategy. In other words, NAcDEtends to generate randomized solutions in progressively smaller areas of the decision space. Thus,in a noisy fitness landscape, the progressive focus allows a search in the most interesting areasand, on the other hand, the randomization tends to filter the noise. Finally the fine-tuning of thesearch directions is performed by the noise analysis component. Regarding the comparison withNAcGA, the advantage of NAcDE is due to the search logic and the nature of the algorithmrather than the capability of handling noise. Briefly, as shown in (Mininno et al. 2011), the DElogic in a compact algorithm appears to be beneficial to prevent premature convergence and,most importantly, the one-to-one spawning typical of DE can be naturally encoded in a compactalgorithm. In DE the survivor selection is performed by performing a pairwise comparison be-tween the performance of a parent solution and its corresponding offspring. In our opinion thislogic can be naturally encoded into a compact algorithm unlike the case of a selection mechanismtypical of Genetic Algorithms (GAs), e.g. tournament selection. In other words, we believe thatwhile a cGA is a GA which lost part of its functionality to become a compact and fit withina limited memory, a DE can be straightforwardly encoded into a compact algorithm without

June 2, 2011 13:50 International Journal of Systems Science NAcDE2

16 Taylor & Francis and I.T. Consultant

Table 5. Wilcoxon test for n = 30 (“+” means that NAcDE outperforms DE, “-” means that NAcDE is outperformed, and “=”

means that the algorithms have the same performance)

Test Problem DE-RSF-TS OBDE NADE NArCGAf1 0 - + - -

0.05 + + - =0.1 + + - =0.2 + + = +

f2 0 = + = -0.05 + + + =0.1 + + + =0.2 + + + -

f3 0 + + + -0.05 + + + +0.1 + + + +0.2 + + = =

f4 0 - - - -0.05 = = = =0.1 - - - =0.2 = = = =

f5 0 - + - -0.05 + + = =0.1 + + = +0.2 + + + +

f6 0 - + - -0.05 + + + =0.1 + + + +0.2 + + + +

f7 0 - + - +0.05 + - = +0.1 + - = +0.2 + - = +

f8 0 = + = =0.05 = = = =0.1 + + + +0.2 + + + =

f9 0 - + - +0.05 + + + +0.1 + + + +0.2 + + + +

f10 0 - + - +0.05 + + = +0.1 + + = =0.2 + + = +

f11 0 + + + +0.05 + + + +0.1 + + + +0.2 + + + =

f12 0 + = + -0.05 + + + +0.1 + + = =0.2 + + = =

f13 0 = + = -0.05 + + = -0.1 + + + -0.2 + + + =

f14 0 + + + +0.05 + + + +0.1 + + + +0.2 + + + =

Table 6. Holm procedure for 30 dimensionsNA − j Algorithm zNA

− j pNA− j δ/(NA − j) Hypothesis

4 OBDE -7.17e+00 3.71e-13 1.25e-02 Rejected3 DERSF -5.14e+00 1.38e-07 1.67e-02 Rejected2 NADE -1.37e+00 8.46e-02 2.50e-02 Accepted1 NArcGA -1.31e+00 9.43e-02 5.00e-02 Accepted

losing the basic working principles. In our opinion, the combination of these elements makeNAcDE a valid and robust algorithm for handling noisy environments. The Holm procedure forthe 30-dimensional case is reported in Table 6.On the basis of the reported numerical results, we can conclude that NAcDE performs well in

low dimensional problems. However, in these cases simpler search principles, like the offspring

June 2, 2011 13:50 International Journal of Systems Science NAcDE2

International Journal of Systems Science 17

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

95

96

97

98

99

100

101

102

103

104

105

Fitness function evaluations

Fitn

ess

func

tion

valu

es

DERSFOBDENADENAcGANAcDE

Figure 4. f11 for n = 10 and no noise

generation of NAcGA, seem to be competitive. When the dimensionality is higher (even only30 dimensions), as a general trend, the cDE logic appears to be more promising than that ofcGA. On the other hand, as shown in other works, see (Mininno et al. 2008) and (Mininno et al.2011), in high dimensions population-based algorithms appear to display a better performancethan their compact versions. For the dimensionality levels considered in this article, the proposedNAcDE appears to have a good performance on a various set of problems regardless the modalityand separability of the problems. The only landscape where the proposed NAcDE appears tohave quite a poor performance is the Schwefel problem 1.2 (fitness f4). For this landscape thesimple neighbourhood search of NAcGA appears to be promising as it would behave like a localsearch component descending the only basin of attraction present in the landscape. On theother hand also NADE performs well on this problem. Probably for this fitness landscape theemployment of a population appears to be efficient (in low dimensions) for detecting the mostpromising areas of the decision space.Some examples of performance trends are shown in Fig.s 4, 5, 6, and 7.

5.1. Validation of the noise analysis component over the compact differential evolution

framework

In order to clearly show the advantages of the noise analysis component in this context wecompared the performance of NAcDE and standard cDE. In other words we compared theperformance of the same algorithmic structure where, in one case, the noise analysis componentis integrated and, in the other case, without the noise analysis according to the description inSection 3. Numerical results and Wilcoxon test are given Tables 7 and 8.As expected, for both the sets of problems, at 10 and 30 dimensions, the noise analysis compo-

nent slightly worsens the performance in the absence of noise. The noise analysis requires somefitness evaluations for the minimum re-sampling thus slowing down the algorithmic convergence.On the other hand, the effectiveness of the noise analysis is evident in all the noisy cases. Thisclearly shows that for cDE frameworks the noise analysis is a valid component for handling thenoise.

June 2, 2011 13:50 International Journal of Systems Science NAcDE2

18 Taylor & Francis and I.T. Consultant

0 1 2 3 4 5 6

x 104

−296.8

−296.6

−296.4

−296.2

−296

−295.8

−295.6

−295.4

−295.2

−295

−294.8

Fitness function evaluations

Fitn

ess

func

tion

valu

es

Expanded Rotated Extended Scaffe’s F6 (F14:CEC2005) Noise:0.05

DERSFOBDENADENAcGANAcDE

Figure 5. f14 for n = 10 and 5% of noise

0 5 10 15

x 104

−150

−100

−50

0

50

100

150

200

250

Fitness function evaluations

Fitn

ess

func

tion

valu

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DERSFOBDENADENAcGANAcDE

Figure 6. f9 for n = 30 and 10% of noise

6. Conclusion

This paper proposes a novel DE based compact algorithm for optimization in noisy environments.The proposed algorithm employs a virtual population, i.e. a probabilistic model, from whichsolutions composing the offspring are sampled. The virtual population converges towards themost promising genotype suggested by the elite. In this way, the algorithm progressively focusesthe search in the most promising areas of the decision space while the randomization performsa first level of noise filtering. To avoid that overestimated solutions mislead the search the noiseanalysis survivor selection and the non-persistent elitism perform a fine-tuning in the noisefiltering and guarantee a periodic refreshment of the genotypes leading the search. The proposedalgorithm has been compared with four modern algorithms designed for handling noisy fitnesslandscapes. Three of these algorithms are population-based while one is a compact algorithmemploying a different search logic. An extensive problem benchmark including two levels of

June 2, 2011 13:50 International Journal of Systems Science NAcDE2

REFERENCES 19

0 5 10 15

x 104

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6x 10

4

Fitness function evaluations

Fitn

ess

func

tion

valu

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DERSFOBDENADENAcGANAcDE

Figure 7. f5 for n = 30 and 20% of noise

dimensionality and four noise levels has been set.Numerical results show that for the considered problems, the proposed algorithms displays a

very good performance and appears to be robust under various conditions in terms of featuresof fitness landscape (multi-modality, separability etc.), for both the dimensionality values, andmost importantly for the various noise levels considered in this study. This feature is due to anefficient noise analysis mechanism which guarantees that only the strictly necessary number offitness re-evaluations are performed. This component, combined with the search logic and thespecific degree of randomization make the proposed NAcDE an efficient alternative for solvingoptimization problems characterized by noise.The proposed algorithm, due to its compact nature, can be run on devices characterized by

a limited hardware and a fitness coming from measurements. For this reason, NAcDE can beof a great use for on-line training of moving robots since they would greatly benefit from anoptimization performed within their control card rather than an external computer. A futuredevelopment of this work will consider the engineering implementation on a micro-control cardfor moving robots.

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Table 7. Validation of the noise analysis component in 10 dimensions

Test Problem cDE NAcDE Wilcoxonf1 0 2.176e-06 ± 4.03e-06 1.860e-08 ± 1.45e-08 +

0.05 7.908e+03 ± 4.44e+03 1.860e+03 ± 6.85e+02 +0.1 1.351e+04 ± 7.27e+03 2.840e+03 ± 1.03e+03 +0.2 2.756e+04 ± 1.05e+04 4.630e+03 ± 1.19e+03 +

f2 0 1.969e-02 ± 6.80e-02 3.430e-02 ± 9.26e-02 -0.05 4.914e+04 ± 2.90e+04 9.800e+03 ± 3.18e+03 +0.1 4.961e+04 ± 2.26e+04 1.390e+04 ± 4.20e+03 +0.2 8.142e+04 ± 6.64e+04 1.440e+04 ± 3.95e+03 +

f3 0 2.784e+05 ± 2.40e+05 1.420e+06 ± 8.39e+05 -0.05 7.577e+08 ± 8.41e+08 6.040e+07 ± 3.25e+07 +0.1 1.526e+09 ± 1.29e+09 9.680e+07 ± 5.75e+07 +0.2 2.033e+09 ± 1.29e+09 1.360e+08 ± 7.56e+07 +

f4 0 2.476e+02 ± 6.30e+02 1.530e+04 ± 5.60e+03 -0.05 7.865e+04 ± 2.88e+04 3.820e+04 ± 2.00e+04 +0.1 1.032e+05 ± 7.24e+04 4.000e+04 ± 2.56e+04 +0.2 2.015e+05 ± 1.76e+05 4.270e+04 ± 1.89e+04 +

f5 0 1.438e+02 ± 3.82e+02 2.760e+01 ± 6.26e+01 +0.05 6.050e+03 ± 2.61e+03 1.040e+03 ± 4.04e+02 +0.1 1.288e+04 ± 6.00e+03 1.920e+03 ± 6.59e+02 +0.2 1.945e+04 ± 5.49e+03 4.290e+03 ± 1.41e+03 +

f6 0 9.353e+01 ± 2.31e+02 1.010e+03 ± 1.01e+03 -0.05 7.816e+09 ± 6.42e+09 2.890e+08 ± 1.36e+08 +0.1 1.428e+10 ± 1.08e+10 3.430e+08 ± 1.97e+08 +0.2 2.007e+10 ± 1.38e+10 7.660e+08 ± 6.32e+08 +

f7 0 1.267e+03 ± 8.18e-06 1.270e+03 ± 8.73e-07 =0.05 1.605e+03 ± 2.57e+02 1.320e+03 ± 2.44e+01 +0.1 2.321e+03 ± 3.97e+02 1.380e+03 ± 6.26e+01 +0.2 2.968e+03 ± 7.89e+02 1.520e+03 ± 7.10e+01 +

f8 0 2.013e+01 ± 1.04e-01 2.040e+01 ± 8.64e-02 =0.05 2.042e+01 ± 1.91e-01 2.040e+01 ± 8.37e-02 +0.1 2.056e+01 ± 2.03e-01 2.040e+01 ± 9.12e-02 +0.2 2.077e+01 ± 1.70e-01 2.050e+01 ± 9.70e-02 +

f9 0 2.612e+00 ± 1.08e+00 1.020e+01 ± 5.08e+00 -0.05 5.007e+01 ± 1.80e+01 2.110e+01 ± 5.80e+00 +0.1 8.345e+01 ± 2.12e+01 4.260e+01 ± 1.07e+01 +0.2 1.354e+02 ± 2.91e+01 6.430e+01 ± 1.33e+01 +

f10 0 1.803e+00 ± 1.32e+00 1.020e+01 ± 4.42e+00 -0.05 4.972e+01 ± 1.69e+01 2.150e+01 ± 8.12e+00 +0.1 9.905e+01 ± 2.67e+01 4.340e+01 ± 8.94e+00 +0.2 1.413e+02 ± 5.10e+01 6.580e+01 ± 6.47e+00 +

f11 0 6.852e+00 ± 1.67e+00 5.630e+00 ± 1.61e+00 =0.05 8.485e+00 ± 1.51e+00 7.810e+00 ± 1.56e+00 +0.1 9.737e+00 ± 1.89e+00 9.660e+00 ± 5.48e-01 +0.2 1.354e+01 ± 2.36e+00 1.030e+01 ± 8.40e-01 +

f12 0 3.036e+02 ± 4.15e+02 9.300e+03 ± 6.23e+03 -0.05 6.260e+04 ± 2.46e+04 3.450e+04 ± 1.26e+04 +0.1 1.228e+05 ± 5.45e+04 4.410e+04 ± 1.50e+04 +0.2 2.327e+05 ± 1.14e+05 5.900e+04 ± 1.72e+04 +

f13 0 5.319e-01 ± 1.58e-01 9.000e-01 ± 3.19e-01 =0.05 3.322e+02 ± 2.68e+02 1.900e+01 ± 4.86e+00 +0.1 8.781e+02 ± 6.15e+02 2.590e+01 ± 1.08e+01 +0.2 1.172e+03 ± 1.16e+03 2.730e+01 ± 1.35e+01 +

f14 0 3.672e+00 ± 2.94e-01 3.320e+00 ± 4.04e-01 =0.05 3.795e+00 ± 2.55e-01 3.330e+00 ± 3.90e-01 =0.1 3.988e+00 ± 2.94e-01 3.870e+00 ± 1.93e-01 =0.2 4.331e+00 ± 3.48e-01 4.150e+00 ± 1.13e-01 =

quential Sampling in Noisy Environments,” (Vol. 39, ed. K. F. Doerner et al., Springer,chap. 14, pp. 261–273.

Beielstein, T., and Markon, S. (2002), “Threshold selection, hypothesis test, and DOE methods,”in Proceedings of the IEEE Congress on Evolutionary Computation, pp. 777–782.

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on Evolutionary Computation, 10(5), 507–526.Branke, J., and Schmidt, C. (2003), “Selection in the presence of noise,” in Proceedings of

Genetic and Evolutionary Computation Conference ed. Cantu-Paz E. et. al., Lecture Notesin Computer Science, Springer, pp. 766–777.

June 2, 2011 13:50 International Journal of Systems Science NAcDE2

REFERENCES 21

Table 8. Validation of the noise analysis component in 30 dimensions

Test Problem cDE NAcDE Wilcoxonf1 0 5.584e-07 ± 6.70e-07 3.340e+02 ± 4.08e+02 -

0.05 4.621e+04 ± 1.42e+04 1.610e+04 ± 2.62e+03 +0.1 8.127e+04 ± 2.23e+04 2.160e+04 ± 3.16e+03 +0.2 1.156e+05 ± 2.54e+04 2.850e+04 ± 5.32e+03 +

f2 0 5.539e+03 ± 3.02e+03 1.470e+04 ± 5.98e+03 -0.05 3.996e+05 ± 2.78e+05 1.000e+05 ± 1.67e+04 =0.1 6.091e+05 ± 3.63e+05 1.130e+05 ± 2.61e+04 +0.2 8.344e+05 ± 7.43e+05 1.240e+05 ± 2.28e+04 +

f3 0 1.812e+07 ± 9.15e+06 5.060e+07 ± 2.04e+07 -0.05 2.661e+09 ± 1.17e+09 3.960e+08 ± 1.48e+08 +0.1 3.898e+09 ± 1.59e+09 6.140e+08 ± 1.60e+08 +0.2 5.778e+09 ± 2.76e+09 1.010e+09 ± 3.22e+08 +

f4 0 5.453e+04 ± 1.63e+04 1.270e+05 ± 3.10e+04 -0.05 5.807e+05 ± 3.09e+05 2.580e+05 ± 1.56e+05 +0.1 1.051e+06 ± 1.06e+06 3.090e+05 ± 1.84e+05 +0.2 1.314e+06 ± 1.04e+06 2.750e+05 ± 1.72e+05 +

f5 0 8.243e+03 ± 1.07e+03 6.310e+03 ± 1.37e+03 =0.05 2.783e+04 ± 7.61e+03 1.160e+04 ± 1.41e+03 +0.1 3.888e+04 ± 8.22e+03 1.450e+04 ± 1.79e+03 +0.2 4.860e+04 ± 9.78e+03 1.840e+04 ± 3.34e+03 +

f6 0 1.142e+02 ± 9.96e+01 4.320e+06 ± 1.05e+07 -0.05 4.473e+10 ± 1.62e+10 4.160e+09 ± 1.40e+09 +0.1 7.394e+10 ± 2.84e+10 6.310e+09 ± 2.43e+09 +0.2 1.393e+11 ± 3.37e+10 1.070e+10 ± 4.33e+09 +

f7 0 4.696e+03 ± 7.61e-07 4.700e+03 ± 1.99e-06 -0.05 6.162e+03 ± 4.00e+02 5.000e+03 ± 1.07e+02 +0.1 9.040e+03 ± 1.26e+03 5.280e+03 ± 1.20e+02 +0.2 1.168e+04 ± 1.54e+03 5.830e+03 ± 3.55e+02 +

f8 0 2.050e+01 ± 1.45e-01 2.090e+01 ± 1.16e-01 =0.05 2.086e+01 ± 1.68e-01 2.100e+01 ± 7.24e-02 =0.1 2.100e+01 ± 1.39e-01 2.100e+01 ± 6.13e-02 =0.2 2.123e+01 ± 1.01e-01 2.100e+01 ± 6.94e-02 +

f9 0 5.308e+01 ± 1.39e+01 6.260e+01 ± 1.38e+01 =0.05 2.651e+02 ± 4.85e+01 1.360e+02 ± 1.73e+01 +0.1 4.238e+02 ± 5.50e+01 2.080e+02 ± 3.66e+01 +0.2 5.778e+02 ± 7.30e+01 2.820e+02 ± 2.72e+01 +

f10 0 5.782e+01 ± 1.64e+01 6.390e+01 ± 1.20e+01 =0.05 2.870e+02 ± 4.25e+01 1.470e+02 ± 2.41e+01 +0.1 4.151e+02 ± 8.00e+01 2.410e+02 ± 2.84e+01 +0.2 5.349e+02 ± 5.75e+01 2.920e+02 ± 2.68e+01 +

f11 0 2.822e+01 ± 3.54e+00 2.940e+01 ± 3.05e+00 =0.05 3.563e+01 ± 2.63e+00 3.310e+01 ± 2.77e+00 +0.1 4.186e+01 ± 4.02e+00 3.530e+01 ± 3.33e+00 +0.2 4.858e+01 ± 3.46e+00 4.010e+01 ± 2.07e+00 +

f12 0 4.080e+04 ± 2.75e+04 3.210e+05 ± 7.99e+04 -0.05 1.070e+06 ± 3.37e+05 5.700e+05 ± 1.52e+05 +0.1 1.690e+06 ± 4.20e+05 9.240e+05 ± 1.71e+05 +0.2 2.467e+06 ± 4.68e+05 1.370e+06 ± 2.06e+05 +

f13 0 6.669e+00 ± 2.12e+00 1.800e+01 ± 1.17e+01 -0.05 2.054e+03 ± 8.62e+02 2.950e+02 ± 8.90e+01 +0.1 2.907e+03 ± 1.64e+03 3.900e+02 ± 1.46e+02 +0.2 5.879e+03 ± 3.11e+03 5.590e+02 ± 2.19e+02 +

f14 0 1.300e+01 ± 4.66e-01 1.250e+01 ± 4.88e-01 =0.05 1.342e+01 ± 3.58e-01 1.260e+01 ± 4.96e-01 =0.1 1.375e+01 ± 4.53e-01 1.290e+01 ± 4.40e-01 =0.2 1.434e+01 ± 3.96e-01 1.350e+01 ± 2.98e-01 =

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