DaCruzetal2006 QIEANumericalOptn IEEE CECp2630 7

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Quantum-Inspired Evolutionary Algorithm for NumericalOptimization

Andr e V. Abs da Cruz, Marley M. B. R. Vellasco, Member, IEEE and Marco Aur elio C. Pacheco

Abstract — Since they were proposed as an optimizationmethod, evolutionary algorithms(EA) have been used to solveproblems in several research elds. This success is due, besidesother things, to the fact that these algorithms do not requireprevious considerations regarding the problem to be optimizedand offers a high degree of parallelism. However, some problemsare computationally intensive regarding solution’s evaluation,which makes the optimization by EA’s slow for some situations.This paper proposes a novel EA for numerical optimiza-tion inspired by the multiple universes principle of quantumcomputing. Results show that this algorithm can nd bettersolutions, with less evaluations, when compared with similaralgorithms.

I. INTRODUCTION

Numerical optimization problems are an important eldof research and have applications from the mathematicaloptimization of functions to the optimization of synapticweights of a neural network.

Evolutionary algorithms have been an important tool forsolving this sort of problems [1], [2]. The characteristics of dispensing a rigorous mathematical formulation regarding theproblem one wants to solve, its high parallelism and easy of adaptation to several situations, make this class of algorithmsvery efcient to solve these problems.

Although they are successful to solve several optimization

problems, the evolutionary algorithms present, for someproblems, a slower performance than the one that could beexpected. This happens due to the fact that, time-consumingevaluation functions (for instance, a complex simulator foran industrial plant) can slow down the performance of thealgorithm to unacceptable levels.

In this sense, the quantum-inspired evolutionary algo-rithms [3], [4] represents a novel approach in the eld of evolutionary computation. These algorithms have been usedin combinatorial optimization problems [4] using a binary-based representation and have presented a better performancein solving this kind of problems than the conventionalalgorithms.

For numerical optimization problems, however, a directrepresentation, where real numbers are directly encoded ina chromosome, is usually preferred rather than converting

Andr e V. Abs da Cruz is with the Department of Electrical Engineer-ing, Ponticia Universidade Catolica, Rio de Janeiro, RJ, Brazil (email:[email protected]).

Marley M. B. R. Vellasco is with the Department of Electrical Engi-neering, Ponticia Universidade Catolica, Rio de Janeiro, RJ, Brazil (email:[email protected]).

Marco Aur´elio C. Pacheco is with the Department of Electrical Engi-neering, Ponticia Universidade Catolica, Rio de Janeiro, RJ, Brazil (email:[email protected]).

binary strings to numbers [2]. With a real number representa-tion, the demand for memory is reduced while the numericalprecision is increased.

This paper presents a novel representation for thequantum-inspired genetic algorithms. Besides many otherimportant properties, this model has the ability to nd a goodsolution faster using less individuals. This feature reducesdramatically the number of evaluations needed and is animportant performance factor when the model is being usedin problems where each evaluation takes too much time tobe completed.

This novel representation is based on the previous work about quantum-inspired algorithms for numerical optimiza-tion [5], [6]. In this new model, a simplied representation isused which makes the algorithm easier to implement and tounderstand. In this new representation, only a single pulseis used to represent each quantum-inspired gene and theprobability distribution is created by summing up all thequantum-inspired individuals in the population. Also, in thiswork, the performance of the algorithm is tested with a newset of benchmarks.

The rest of this paper is organized as follows: section 2describes the new simplied representation for the algorithm;section 3 describes the experiments and presents a discussion

regarding the results achieved; section 4 presents someconclusions and points toward future works.

II . T HE Q UANTUM -I NSPIRED E VOLUTIONARY

A LGORITHM USING A R EA L R EPRESENTATION

Figure 1 shows the complete quantum-inspired evolu-tionary algorithm using real representation (QIEA). Thisproposed algorithm is completely explained further in thefollowing sections.

A. The Quantum Population

The quantum population Q(t) is the QIEA’s kernel. Thispopulation represents a superposition of states that is ob-served to generate classical individuals that are then, evalu-ated. The quantum population Q(t) is made up of a set of N quantum individuals qi (i = 1 , 2, 3, . . ,N ). Each quantumindividual qi is formed by G genes gij ( j = 1 , 2, 3,...,G )which consist of a pair of values, as shown in equation 1.

qi = [ gij |gij = ( ρij , σ ij )] (1)

The values ρij and σij (where ρij , σ ij ) represents,respectively, the mean and the width of a square pulse,which is used by the algorithm to constrain the set of possible observable values inside the domain of the problem

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2006 IEEE Congress on Evolutionary ComputationSheraton Vancouver Wall Centre Hotel, Vancouver, BC, CanadaJuly 16-21, 2006

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begin1. t ← 12. Create quantum pop. Q(t) with N individuals

with G genes3. while (t < = T )4. Create the PDF’s using the quantum individuals5. E (t) ← generate classical pop. observing

quantum pop. and using CDF’s6. if (t=1)7. then C (t) ← E (t)8. else9. E (t) ← Crossover between E (t)

and C (t)10. evaluate E (t)11. C (t) ← K best individuals

from [E (t) + C (t)]12. end if 13. with the N better individuals from C (t)14. Q(t + 1) ← apply translate operation

to Q(t)

15. Q(t + 1) ← apply resize operationto Q(t + 1)16. t ← t + 117. end with18. end whileend

Fig. 1. Full quantum-inspired evolutionary algorithm using real represen-tation pseudo-code.

being optimized. In other words, each gene in the quantumindividual represents an interval in the search space. Theheight h ij of the pulses is calculated using the quantum

gene and the total number N of quantum individuals in thepopulation, as shown in equation 2.

h ij =1/σ ij

N (2)

This equation guarantees that, lately in the algorithm,the probabilities density functions used to generate classicalindividuals will have a total area equal to 1.

An example of a quantum gene gij = (0 , 2) is shown ingure 2 (in this example, we consider a quantum populationformed by 2 individuals and thus, the height h ij of this geneis 0.25).

The creation of the quantum population, shown in step

2 of gure 1is done by generating N quantum individualswith random mean ρij inside the problem domain and withwidth σij equal to the total domain width. For example,considering that the function f (x) with x [− 100, 100]is going to be minimized, the quantum individuals q1 =(0, 200), q2 = ( − 99.8, 200), q3 = (14 .172637823, 200) andq4 = (100 , 200) are examples of valid individuals in theinitial population. It is important to notice that, usually, thesquare pulses in the initial population will be partially out of the domain bounds. In the latter example, the individual q1

is the only one that is completely inside the initial domain

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Fig. 2. An example of a quantum gene.

bounds ( [− 100, 100]). The individual q2 , for instance, denesa square pulse in the interval [− 199.8, 0.2] being partiallyoutside the domain bounds. However, this issue is not aproblem and will be treated in the next steps of the algorithm.

B. Quantum Individuals Observation

After initializing the quantum population in step 2, thealgorithm enters the main evolutionary process loop. Thisloop will be executed for a number T of generations and isformed by several tasks.

Step 4 is one of the most important in the algorithm.In this step, the interference process between the quantumindividuals is executed to generate a Probability DensityFunction (PDF). This process basically consists in summingup the individuals that form the quantum population. In other

words, in this step, the phenotype of the rst gene of allindividuals are summed. Next, the square pulses from thesecond gene for all individuals are also summed, and so on,until all phenotypes from all individuals have been summedup. Mathematically, the probability density function of gene j , on generation t of the algorithm is given by the equation3.

P DF j =N

i

gij (3)

Where gij represents the phenotype (i.e. the square pulsewith width σij and center ρij ) of the j − th gene of the i − th

quantum individual.As an example, let’s consider another quantum population,

at some generation t , to be formed by two individuals, eachone made of 2 genes. The conguration of these individualsis given in table I.

The graphical representation of these individuals is shownin gure 3. The resultant PDF’s of the individuals shown intable I are shown in gure 4.

Thus, the PDF’s are related to a specic gene of thequantum individuals. The observations made using thesePDF’s will be able to generate the classical individuals.

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Individual Genesq1 g11 = ( −5, 20) , g12 = (0 , 20)q2 g21 = (5 , 20) , g22 = (5 , 20)

TABLE I

EXAMPLE OF QUANTUM INDIVIDUALS FORMING POPULATION Q( T ).

−20 0 200

0.02

0.04

g11

−20 0 200

0.02

0.04

g12

−20 0 200

0.02

0.04

g21

−20 0 200

0.02

0.04

g22

Fig. 3. Genes of 2 quantum individuals.

These classical individuals are formed by a vector of realnumbers (with as many elements as the number of genesin the quantum individuals) and the values that form thesevectors are chosen randomly using the PDF’s as a probabilityfunction. To carry out this random choice, one needs theCumulative Distributive Function (CDF), which is given byequation 4.

CDF j (x) =

L s

j

L ij

P DF j (x)dx (4)

Where L ij e Ls

j are the lower and upper limits of functionP DF j . As mentioned before, thanks to the formulation givento the height h ij in equation 2 for each quantum gene, thetotal area of each of the PDF curves P DF j is 1. Besides

−20 −10 0 10 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

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0.1

PDF 1(x)

−20 −10 0 10 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

PDF 2(x,t)

Fig. 4. An example of some resulting PDF’s.

that, as the PDF’s are made of a sum of square pulses, thePDF area can be easily computed by dividing the functioncurve in rectangles and by summing up the area of each of these rectangles. Using the example from gure 4, the PDFcan be divided in rectangles, as shown in gure 5.

−20 −10 0 10 200

0.01

0.02

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0.06

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0.08

0.09

0.1

−20 −10 0 10 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

R1 R3R2 R4 R5 R6

Fig. 5. PDF’s division in rectangles.

The CDF’s can then be computed from these PDF’s basedon the rectangles R1, R2, R3, R4, R5 and R6. The resultantCDF of the rst gene varies from 0 to 0.25 (total area of rectangle R1) in the interval [− 15, 5], from 0.25 to 0.75 (totalarea of rectangle R1 + rectangle R2) and from 0.75 to 1.0(total curve area). The CDF’s can be represented graphicallyas shown in gure 6.

−20 −10 0 10 200

0.1

0.2

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0.7

0.8

0.9

1

CDF 1(x)

−20 −10 0 10 200

0.1

0.2

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0.7

0.8

0.9

1

CDF 2(x)

Fig. 6. CDF’s examples.

After generating the CDF’s, it is possible to generate aset of classical individuals by using these curves (step 5from the algorithm). This population created is made byuniformly choosing a random number between 0 and 1 andby identifying this point in the CDF. Mathematically, thevalue of a gene in the classical population is generated asshown in equation 5.

x = CDF − 1 (r ) (5)

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Where r is a random number in the [0, 1] interval. Throughthis process, a temporary classical population E (t), with K individuals is, then, generated automatically. As explainedbefore, the PDF’s (and so the CDF’s) can be outside thedomain bounds of the problem. If the chosen individuals areoutside the domain bounds, they are simply corrected to beequal to the nearest domain bound. Though this can create asmall pressure toward individuals lying on the edges of thedomain, this can be disregarded due to the fact that, usually,much less than 50% of the pulse will be outside the domainbounds.

In the rst generation (step 6), the classical populationC (t) is an exact copy of the temporary classical populationE (t) created in step 5. If the algorithm is not in the rstgeneration, it should apply the crossover operator amongthe individuals of the temporary classical population E (t)and the classical population C (t). This proposed crossoveroperator works as shown in the algorithm in gure 7.

begin

for i = 1 to Kselect i-th individual ei from E (t)select i-th individual ci from C (t)for j = 1 to G

r ← choose random number in [0, 1)if r < ξ

eij ← eij

elseeij ← cij

end if end for

end forend

Fig. 7. Crossover algorithm.

In this algorithm, K is the number of individuals inclassical population C (t), G is the number of genes ineach individual and the value ξ is the crossover rate usedduring the process. A rate equal to 1 will copy all thegenes in the created individual to the offspring. A rate equalto 0 does not modify the individuals and no evolution isobserved. An example of how this algorithm works canbe seen in gure 8, where the chosen numbers r are(0.9, 0.8, 0.1, 0.2, 0.7, 1.0, 0.15, 0.1) (the rst number here isthe random number for the rst element in the vectors, the

second number is the random number for the second elementand so on) and the crossover rate is ξ = 0 .6.It is worth pointing out that for the crossover to work

correctly, the number of temporary classical individualscreated from the CDF’s must be equal to the number K of individuals in population C (t).

After using the crossover operator, the temporary popu-lation E (t) must be evaluated. Then, the individuals in thispopulation and in population C (t) are ordered in a single setand the K best individuals from this new set are selected toform the new population C (t).

6

5

4

3

79

8

1

4

6

5

9

87

3

1

e ij c ij e' ij

4

6

4

3

87

8

1

Fig. 8. Crossover Example.

C. Updating the Quantum Population

After generating the classical individuals, it is necessaryto update the quantum individuals in the population Q(t).This procedure is done in two steps. The rst step (step 14in the algorithm) modies the center ρ of the quantum genes.This procedure is simple and is done by making the meanvalues of each gene equal to the values of the genes fromthe classical individuals. Mathematically, this is representedby equation 6.

ρij = cij (6)

Where ρij is the value that denes the center of the j − thgene of the i − th quantum individuals in Q(t) and cij is thevalue of the j − th gene of the i − th classical individual inC (t).

The second step in the process of updating the quantumpopulation (step 15 of the algorithm) consists in shrinkingor enlarging the width of the quantum genes. This change inthe width is done homogeneously for all the quantum genesand for all the quantum individuals. The heuristic used todetermine if the new width should be enlarged or shrinked isthe 1/ 5th rule [2]: if at most 20% of the classical populationcreated in the current generation has improved, the genewidth is reduced; if this rate is higher than 20% the widthis enlarged; if the rate is exactly 20%, no changes are made.Mathematically, this can be represented as shown in equation7.

σij =σij · δ < 1/ 5σij /δ > 1/ 5σij = 1 / 5

(7)

Where σij is the width of the j − th gene of the i − thquantum individual in Q(t), δ is an arbitrary value, usually inthe interval [0, 1] and is the rate of how many individualsof the new population have their overall evaluation improved.

This rule is based on the following heuristics: if more than20% of the classical individuals have improved, the width

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should be enlarged in order to try to improve the globalsearch; if less than 20% of the individuals have improved,the width should be shrinked in order to try to improve thelocal search; if the rate is exactly 20% nothing should bedone, as this rate is considered to be an ideal rate for theoptimization process. Also, this rule can be applied at everygeneration of the algorithm or it can be applied at largersteps (for instance, every ve generations of the algorithm).

All these steps of the algorithm should be repeated for anumber T of generations, according to what was shown ingure 1.

Besides the algorithm, it’s possible to dene a diagramwhich shows the relationship between the parts that formthe complete model. This diagram can be seen in gure 9.

QuantumIndividual

q1(t)

ClassicalPopulation

C(t)

EvaluationFunction

QuantumGene

g1

QuantumGene

g2

QuantumGene

gn

QuantumOperators

PopulationQuantum

Q(t)

QuantumIndividual

q2(t)

QuantumIndividual

qn(t)

Fig. 9. Complete quantum-inspired evolutionary algorithm block diagram.

This diagram shows that the quantum population Q(t)is composed of quantum individuals that, in their turn, aremade of quantum genes. This quantum population is usedto generate a classical population which is then evaluatedand used to modify the quantum individuals in an interactiveprocess.

III. E XPERIMENTS

The benchmark functions used here is a set of 4 differentfunctions selected from [7]. These functions are widely usedas benchmark in numerical optimization and the cited paperallows the comparison between results using the followingmethods:

• Stochastic Genetic Algorithms (StGA) [7];• Fast Evolutionary Programming (FEP) [8];• Fast Evolutionary Strategy (FES) [9];• Particle Swarm Optimization (PSO) [10].The functions are listed in table II.All these functions should be minimized. Functions f 1 and

f 2 are unimodal and relatively easy to minimize but, forhigher dimensions, they become harder to optimize. Functionf 3 and f 4 are multi-modal functions and have lots of local

Equation Domain

f 1 (x ) =30

j =1

x 2j x j [−30, 30]

f 2 ( x ) =30

j =1|x j | +

30

j =1|x j | x j [−10, 10]

f 3 (x ) =1

4000

30

j =1

x j2

−30

j =1

cosx j

√i+ 1 x j [−600, 600]

f 4 ( x ) = −20 + exp −0.2 130

30

j =1

x 2j −

exp1

30

30

j =1

cos2 πx j x j [−10, 10]

TABLE II

FUNCTIONS TO BE OPTIMIZED .

minima, representing a harder class of functions to optimize.The minimum of all these functions is 0. Figures 10, 11,12 and 13 shows a graphical view of these functions with 2

variables.

Fig. 10. Function f 1 .

The parameter conguration for the QIEA for all theproblems above is given in table III.

f 1 f 2 f 3 f 4Quantum Population Q (t ) Size 10 5 5 4

Classical Observed Population Size 10 5 10 4Number of generations k

before checking improvement 10 8 5 20

Crossover Rate 0.1 0.1 0.1 1Number of Genera tions 3000 3520 5250 2500

TABLE III

PARAMETERS FOR THE QIEA.

For each function 50 experiments were made. The numberof generations was chosen in order to make the total numberof evaluations equal to the number of evaluations for theStochastic Genetic Algorithm in [7]. The mean number of evaluations for each experiment is shown in table IV.

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QIEA StGA FEP FES PSOf 1 30000 30000 150000 n.a. 250000f 2 17600 17600 200000 n.a. n.a.f 3 52500 52500 200000 200030 250000f 4 10000 10000 150000 150030 n.a.

TABLE IV

M EAN NUMBER OF FUNCTION EVALUATIONS FOR EACH EXPERIMENT .

The results of this comparison are shown in tables V VIwith respect to the mean best value found for each function.The results for all the algorithms are taken from [7].

QIEA StGA FEPf 1 7.8 ×10− 17 2.45 ×10− 15 5.7 ×10− 4

f 2 3.0 ×10− 10 2.03 ×10− 7 8.1 ×10− 3

f 3 0 2.44 ×10− 17 1.6 ×10− 2

f 4 1.36

×10− 8 3.52

×10− 8 1.8

×10− 2

FES PSOf 1 n.a. 11 .175f 2 n.a. n.a.f 3 3.7 ×10− 2 0.4498f 4 1.2 ×10− 2 n.a.

TABLE V

COMPARISON RESULTS BETWEEN QIEA, S TGA, FEP, FES AND PSO.

QIEA StGA FEPf 1 1.0 ×10− 16 5.25 ×10− 16 1.3 ×10− 4

f 2 2.3 ×10− 10 2.95 ×10− 8 7.7 ×10− 4

f 3 1.4 ×10− 17 4.54 ×10− 17 2.2 ×10− 2

f 4 4.6 ×10− 8

3.51 ×10− 9

2.1 ×10− 3

FES PSOf 1 n.a. 1.3208f 2 n.a. n.a.f 3 5.0 ×10− 2 0.0566f 4 1.8 ×10− 3 n.a.

TABLE VI

COMPARISON OF STANDARD DEVIATIONS BETWEEN QIEA, S TGA, FE P,

FES AN D PSO.

The results show that QIEA was able to reach better resultswith much less evaluations than FEP, FES and PSO. Also,the algorithm was able to nd better results than StGA with

the same number of function evaluations. A. Discussion

There are two main aspects regarding the QIEA that shouldbe discussed: why the evolution is faster than with conven-tional genetic algorithms and the impact of the congurationparameters on the performance of the algorithm.

The rst important aspect of the QIEA is that the quantumpopulation is able to directly describe schemata, differentlyfrom the conventional genetic algorithm where the individ-uals represent, exclusively, a set of points in the searchspace. The schemata represent a whole set of individualsand thus, the evaluation of some individuals inside the region

represented by the schemata provides, not only an evaluationfor each of them, but an approximate evaluation of the regionof the search space that is covered by the schemata (as longas this region does not have abrupt changes). The abilityof evaluating whole regions of the search space instead of single points allows the algorithm to identify the promisingregions of the search space faster than conventional geneticalgorithms.

Another way of understanding how the QIEA is capable of converging is through an analogy with evolutionary strategies[2]. In the original algorithm ( (1 + 1) − ES , the population

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consists in a single individual represented by a vector of realnumbers v and a vector of variances σ . The only operatorin evolutionary strategies is a mutation operator that changesthe genes in v by a small quantity δi (where i is the indexof each one of the genes that make v ), chosen randomlyfrom a Gaussian distribution with mean zero and varianceσi . The quantum population can be seen as the individualfrom the evolutionary strategy but, in this case, the set of individuals that forms the quantum population is capable of forming more complex curves than a Gaussian one. Besides,as the population is formed by several independent pulses,the curves can split between several regions of the searchspace. As the evolution goes on, the more promising regionsare identied (as the tendency is that these regions will createmore individuals with better evaluations) and the pulses willbe ”attracted” to these regions, converging to an optimumsolution. These characteristics helps the algorithm to avoidbeing trapped for a long time in local minima and also speedup the algorithm.

The other important aspect to be discussed regards the

parameters of the QIEA and how these parameters mod-ify the evolutionary process. There is no ”magic” formulafor choosing these values and experimentation is the mosttrustable way to identify them. However, some importantobservations can be made regarding the parameters.

Regarding the number of individuals in the quantumpopulation, it is important to take in consideration that avery small number of individuals can lead to a prematureconvergence of the algorithm. This happens due to the factthat the pulses that form the population will quickly moveto a local minima and will be locked in these regionsindenitely. On the other hand, a large number of quantumindividuals will make the width of the sum of PDF’s larger.

Thus, the algorithm will have problems to converge to theoptimal solution, even if it is able to nd the region where theoptimal point is. The differences between the smoothness of the PDF’s’ sum are shown in gure 14 and 15 where the rstone is the result of a population of 5 quantum individualsand the latter of 20 individuals.

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0.5

1

1.5

2

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0.5

1

1.5

2

−2 −1 0 1 20

0.5

1

1.5

2

−2 −1 0 1 20

0.5

1

1.5

2

Fig. 14. PDF’s of a population with 5 quantum individuals.

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1

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2

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2

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0.5

1

1.5

2

Fig. 15. PDF’s of a population with 20 quantum individuals.

The size of the classical population (the observed popula-tion) is not critical for the algorithm performance. Fewerindividuals in a generation are, usually, enough for thealgorithm to converge. Too many individuals will not makea big difference in the performance and will enlarge thenumber of evaluations, reducing the overall performance.Thus, it is recommended to use a small number of individuals(but, it is important to remember that this number should be,at least, equal to the number of quantum individuals).

The number of generations that should pass before theimprovement rate (used with the 1/ 5th rule) of the algorithmmust be veried is also very important and should alsobe found by a process of experimentation. Each functionthat to be optimized has a different behavior regarding thisparameter. Small values makes the pulses width changes very

quickly which, in some cases, make the time the algorithmhas to explore a given region too small to allow it to ndgood individuals. On the other hand, large values will makethe algorithm lose to much time exploring a given region.Figure 16 shows how the pulses’ width changes during theevolutionary process when the values of k are 2, 5 and 10.It is important to notice how the curve oscillates much morewith small values for k.

Figure 17 shows the variation of the pulses’ width (withk = 10 ) and the improvement rate during the evolutionaryprocess.

Finally, the crossover rate also has an important role inthe evolutionary process. Usually, unimodal problems, with

soft evaluation landscapes and few variables, can use a largecrossover rate (values between 0.9 and 1.0 should be usedfor the rst trials). Problems with lots of local minima or toomuch variables should use small crossover rates (between 0.1and 0.2).

IV. C ONCLUSIONS AND F UTURE W ORKS

This paper presented a new quantum-inspired evolutionaryalgorithm with real representation that is better suited fornumerical optimization problems than using binary represen-tation.

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0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

7

8

Generations

P u l s e

W i d t h

k = 2k = 5k = 10

Fig. 16. How the pulses’ width changes during the evolutionary processwith different values of k .

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Generations

Success RatePulse Width

Fig. 17. Variation of pulses’ width and the improvement rate during theevolutionary process.

The new algorithm has been evaluated in several bench-mark problems and showed very promising results, withbetter performance than other well-established algorithms.Further tests are needed in order to evaluate its robustnesswith other kinds of problems.

Future works include the use of the algorithm to optimizenew benchmark functions and the use of the algorithm inneuroevolution and other practical numerical optimizationproblems. Also, it would be interesting to have a study aboutthe relationship and the differences between this algorithmand other probabilistic models like Ant Colony Optimization.

V. A CKNOWLEDGMENTS

We would like to thank the Conselho Nacional de Desen-volvimento Cient ıco e Tecnol ogico (CNPq) for funding thisresearch.

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