Do Evolutionary Algorithms Indeed Require Randomness?

2013 IEEE Congress on Evolutionary Computation June 20-23, Cancún, México

Do Evolutionary Algorithms Indeed Require Randomness? Ivan Zelinka

Roman Senkerik, Michal Pluhacek

Department of Computer Science, FEI VSB Technical University of Ostrava Tr. 17. Listopadu 15, Ostrava Czech Republic Email: [email protected] http://www.ivanzelinka.eu

Department of Informatics and Artificial Intelligence FAI, Tomas Bata Univerzity in Zlin Nad Stranemi 4511 Zlin 76001 Czech Republic Email: [email protected]

Abstract—Inherent part of evolutionary algorithms that are based on Darwin theory of evolution and Mendel theory of genetic heritage, are random processes. In this participation we discuss whether random processes really are needed in evolutionary algorithms. We use n periodic deterministic processes instead of random number generators and compare performance of evolutionary algorithms powered by those processes and by pseudo-random number generators. Deterministic processes used in this participation are based on deterministic chaos and are used to generate periodical series with different length. Results presented here are numerical demonstration rather than mathematical proofs. We propose that certain class of deterministic processes can be used instead of random number generators without lowering the performance of evolutionary algorithms.

I.

I NTRODUCTION

The term “chaos” covers a rather broad class of phenomena whose behavior may seem erratic, chaotic at first glance. Often, this term is used to denote phenomena which are of a purely stochastic nature, such as the motion of molecules in a vessel with gas and so. The discovery of the phenomenon of deterministic chaos brought about the need to identify manifestations of this phenomenon also in experimental data. Deterministically chaotic systems are necessarily nonlinear, and conventional statistical procedures, which are mostly linear, are insufficient for their analysis. If the output of a deterministically chaotic system is subjected to linear methods, such signal will appear as the result of a random process. Examples include Fourier spectral analysis, which will disclose nonzero amplitudes at all frequencies in a chaotic system, and so chaos can be easily mistaken for random noise. Till now was chaos observed in many of various systems (including evolutionary one) and in the last few years is also used to replace pseudo-number generators (PRGNs) in evolutionary algorithms (EAs). Lets mention for example research papers like [1] (a comprehensive overview of mutual intersection between EAs and chaos is discussed here), one of the first use of chaos inside EAs [2], [3]-[7] discussing use of deterministic chaos inside particle swarm algorithm instead of PRGNs, [15] - [18] investigating relations between chaos and randomness or the latest one [19], [20], [14] and [13] using chaos with EAs in applications, amongst the others. Another research joining deterministic chaos and pseudo-

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random number generator has been done for example in [15]. Possibility of generation of random or pseudorandom numbers by use of the ultra weak multidimensional coupling of p 1dimensional dynamical systems is discussed there. Another paper [8] deeply investigate logistic map as a possible pseudorandom number generator and is compared with contemporary pseudo-random number generators. A comparison of logistic map results is made with conventional methods of generating pseudorandom numbers. The approach used to determine the number, delay, and period of the orbits of the logistic map at varying degrees of precision (3 to 23 bits). Logistic map, we are using here, was also used in [9] like chaos-based true random number generator embedded in reconfigurable switchedcapacitor hardware. Another paper [16] proposed an algorithm of generating pseudorandom number generator, which is called (couple map lattice based on discrete chaotic iteration) and combine the couple map lattice and chaotic iteration. Authors also tested this algorithm in NIST 800-22 statistical test suits and is used in image encryption. In [17] authors exploit interesting properties of chaotic systems to design a random bit generator, called CCCBG, in which two chaotic systems are cross-coupled with each other. For evaluation of the bit streams generated by the CCCBG, the four basic tests are performed: monobit test, serial test, auto-correlation, Poker test. Also the most stringent tests of randomness: the NIST suite tests have been used. A new binary stream-cipher algorithm based on dual one-dimensional chaotic maps is proposed in [18] with statistic proprieties showing that the sequence is of high randomness. Similar studies are also done in [10], [2], [11] and [12]. This publication is focused on use of deterministic chaos to generate n periodic deterministic series, that are used inside evolutionary algorithms instead of pseudo-random number generator. For long time various PRNGs were used inside evolutionary algorithms. During last few years deterministic chaos systems (DCHS) are used instead of PRNGs. As was demonstrated in [3]-[7], very often is performance of EAs using DCHS better or fully comparable with EAs using PRNGs. See for example [3]. Used EAs (we do not discuss here special cases, modified for special experiments) of different kind like genetic algorithms [25], differential evolution [22], particle swarm [26], SOMA [21], scatter search [23], evo-

lutionary strategies [24], etc... do not analyze whether used pseudo-random numbers are really random one and do not use information about its randomness. Random numbers are only simply used. On the other side, as demonstrated in mentioned references, EAs with DCHS gives the same or often better performance. Difference between series from PRNGs and DCHS is that in the case of DCHS one can easily reconstruct/calculate whole series generated by DCHS from arbitrary time series point (of course with very high precision only, otherwise trajectories will diverge soon). In the PRNGs it is impossible (it can be done only for the case that kind of used PRNGs and its seed is known). Because DCHS generate periodical series (thanks to final numerical precision - numbers behind decimal point) it is obvious that EAs performance shall be from certain numerical precision of DCHS comparable with performance of classical EAs. II.

Fig. 1. Time series of period 36 (precision = 4) based on Eq. 1 for A = 4, see Tab. II

mapping function is ”converted” to the stepwise mapping function. That is the source of the appearance of many periodic orbits, later on used in our experiments.

E XPERIMENT D ESIGN

Experiments done in this research can be divided into 2 parts. The first one reports how existence of periodicity generated by deterministic chaos systems depends on numerical precision and the second one is using n periodical time series generated by chaotic systems with given numerical precision inside EAs instead of PRGNs. Both parts are discussed here. A. Chaos or Periodicity In this part the central object of this study is simple logistic equation (Eq. 1). Several experiments were performed and some figures, exhibiting results, were generated, see Fig. 1 - 10. It is good to remember that also another chaotic generators of different mathematical description can be used like Lozi, Henon, Ikeda or another like for example artificially synthesized and reported in [1] . xn+1 = Axn (1 − xn )

(1)

Logistic equation has been used in few levels of investigation, i.e. •

Identification of n periodic orbit with dependence on numerical precision, see Fig. 1 - 7.

•

Global view on logistic equation behavior with fixed parameter A = 4, see Fig. 9.

•

Global view on logistic equation behavior for various parameter of A ∈ [3.4 − 4] and numerical precision ∈ [1, 20], logistic equation has been iterated for 1000, 10 000, 100 000 (see Fig. 8) iterations.

•

Analyze and vizualize why and when is n periodic behavior generated, see Fig. 2 - 3.

•

Detail analysis (the most important one) of logistic equation behavior for parameter A = 4, numerical precision ∈ [1, 20], initial conditions xstart ∈ [0.01, 0.99] (x was incremented periodically by = 0.01) and 1 000 000 iterations for each combination of this parameters, see Fig. 8 - 10 and Table II.

Impact of the precision on the dynamics of logistic equation is depicted on Fig. 2 (compare with Fig. 3). The original

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B. Determinism or Randomness This part of research uses results from the previous one. Our experiments have been set so that periodical deterministic time series based on deterministic chaos generators (see for example Fig. 1 and Eq. 1), were used instead of PRNGs. Based on the fact that numerical precision has impact on existence of periodicity in deterministic chaos, we have selected logistic equation, Eq. 1, and data series generated by this equation for numerical precisions from interval [1, 13] with setting A = 4, see Tab. II which shows minimal and maximal period for current setting. Algorithms selected for our experiments were variants of differential evolution: DERand1Bin and DELocalToBest, [22]. Setting of all versions of DE algorithm is in Tab. I. Based on this setting and algorithm architecture it is easy to calculate how many times data series of pseudochaotic numbers (PCHNs) generated by DCHS have been used in EAs. Total cost function evaluation was done 130 000 times for both DE versions. Tab. II summarizes how many times were PCHNs repeatedly used in DE. All experiments were done in Mathematica 9, on MacBook Pro, 2.8 GHz Intel Core 2 Duo. Test function used in this experiment was 10th dimensional Schwefel’s function (Eq. 2) and each experiment was repeated 100 times for each numerical precision set. The aim was to find global extreme (10 × −418.983 at position 420.969, 420.969, ..., 420.969) of this function as precise as possible. So 3900 evolutionary experiments were done in total. In each experiment (Case 1 and 2) was PRNGs used on the start of Eq. 1 to set initial condition xstart ∈ [0, 1]. Remaining use of Eq. 1 was PRNGs free, i.e. PRNGs was not further in use. There were in fact three case studies of our experiments. n ∑

√ −xi sin( |xi |)

(2)

i=1

Case 1. In this case study we were focused on use of PCHNs instead of PRNG only for mutations, so classical PRNG was used in DE to select individuals from population as well as in special error-correction procedures (for example, when individual leave searched space and has to be returned back).

NP Dimensions (individual length) Generations F CR

TABLE I.

20 10 500 0.9 0.5

A LGORITHMS SETTING . F OR BOTH VERSIONS OF DE USED THE SAME SETTING .

Numerical Precision 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Minimal Period 4 2 10 15 67 143 421 1030 2277 2948 9668 65837 518694 75594 1 1 264446 18491 46854 70767

Maximal Period 4 10 29 36 170 481 758 4514 11227 35200 57639 489154 518694 316645 0 0 264336 18491 46854 922744

WAS

Repeated in DE 32500 13000 4482 3611 764 270 171 28 11 3 2 0 0 0 0 0 0 0 0 0

Fig. 2. Cobweb diagram: impact of low numerical precision = 1 (one decimal behind zero) on shape of the mapping function. Four periodic orbit can be easily observed (blue line), see Fig. 4

P ERIODICITY DEPENDANCE OF E Q . 1 ON VARIOUS TABLE II. NUMERICAL PRECISION ( UP TO PRECISION 20). TABLE ALSO SHOWS HOW MANY TIMES WAS USED n PERIODICAL SERIES BY DE ALGORITHM ( UP TO PRECISION 13).

Case 2. In the second case study all PRNG numbers were fully replaced by PCHNs (with different numerical precision and thus period length). Case 3. The last part of our experiments was focused on use of the same algorithms (DE) in its canonical versions with classical PRNGs just to compare performance with both Cases 1 and 2. III.

R ESULTS

The results of experiment copy the structure of experiment design and they are divided into 2 parts. The first one reports about results of dependance of periodicity generated by deterministic chaos systems on numerical precision. The second one discusses what impact n periodical time series generated by chaotic systems (with given numerical precision) have inside EAs on its performance and it is compared with EAs powered by PRGNs. Both parts are reported here. A. Chaos or Periodicity

Fig. 3. Cobweb diagram: chaotic dynamics with of logistic equation with high numerical precision. Compare with Fig. 2, the same initial conditions are there.

caused different periodic data series as well as initial value xstart has impact on appearance of n periodic series, compare Fig. 4 - Fig. 7, then Fig. 8, Fig. 9 and Fig. 10. In this part it has been numerically demonstrated that different numerical precision causes different n periodic series and reported in Tab. II. B. Determinism or Randomness

As described in the previous section, dynamics of the logistic equation have been investigated and discussed when, how and what n periodical orbits can be observed in this dynamics and at the end of this paper there is also discussed whether random number generator is really so important for evolutionary algorithms. Here a few simple demonstrations have been made on logistic equation. Chaotic behavior of logistic equation, with numerical precision 1 and xstart = 0.45, (Fig. 4 ), xstart = 0.8 (Fig. 5 ) has been generated for numerical precision = 2, for xstart = 0.6, numerical precision = 3 (Fig. 6) and for xstart = 0.02, numerical precision = 4 (Fig. 7). As clearly visible, different numerical precision

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Results based on all three experiments are reported in Tab. III - VII and Fig. 11 - 13. Tables and figures are organized sequentially according to case examples. Results of the Case 1 are recorded in Fig. 11, further in Tab. III and IV. Results of the Case 2 are recorded in Fig. 12, further in Tab. V and VI. The last one, Case 3, is in Fig. 13 and Tab. VII. For Case 1 and Case 2 has been used numerical precision in DCHS to generate PCHNs, see Tab. II. In this table there is also recorded how many times was n periodic series repeatedly used in each algorithm. It is visible that in the frame of our experiments was enough to set numerical precision to 5 (Case

Fig. 4. Periodic behavior of logistic equation with numerical precision 1 and xstart = 0.45.

Fig. 8. Periodicity and chaos. Logistic equation has been iterated for 100 000 iterations, xstart = 0.02. Black squares represent settings of logistic equation producing periodic behavior. This black-white configuration depend only on certain xstart , precision and number of iterations Fig. 5. Periodic behavior of logistic equation with numerical precision 2 and xstart = 0.8.

Fig. 6. Periodic behavior of logistic equation with numerical precision 3 and xstart = 0.6.

Fig. 7. Periodic behavior of logistic equation with numerical precision 4 and xstart = 0.02.

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Fig. 9. Global view on behavior of logistic equation with dependance on numerical precision and iterations for xstart = 0.02. Periodic patterns are visible there.

Fig. 10. Zoom view on behavior of logistic equation with dependance on numerical precision and iterations for xstart = 0.45. Periodic patterns at each line (i.e. precision case) are visible there.

Numerical Precision 1 2 3 4 5 6 7 8 9 10 11 12 13

TABLE III.

Numerical Precision 1 2 3 4 5 6 7 8 9 10 11 12 13

TABLE IV.

Max -1518.23 -2113.45 -4189.83 -4071.39 -4071.39 -4071.39 -4071.39 -4189.81 -4189.75 -4071.39 -4189.73 -4071.39 -4071.39

75% -1844.57 -2676.34 -4189.83 -4189.83 -4189.83 -4189.83 -4189.82 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

Median -2030.3 -2851.81 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

25% -2267.56 -3071.15 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

Min -2850.71 -3401.61 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

C ASE 1. VALUES OF THE COST FUNCTION FOR DER AND 1B IN

Max -1339.95 -834.557 -971.065 -2004.78 -4052.35 -3886.55 -4071.28 -3980.9 -4071.39 -4071.39 -4071.38 -3895.89 -4015.18

75% -1872.95 -2592.75 -3429.78 -4172.27 -4189.76 -4186.71 -4189.77 -4189.76 -4189.82 -4189.82 -4189.81 -4189.8 -4189.8

Median -2060.63 -2767.6 -4189.83 -4188.8 -4189.82 -4189.71 -4189.82 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

25% -2292.03 -2918.88 -4189.83 -4189.68 -4189.83 -4189.82 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

Numerical Precision 1 2 3 4 5 6 7 8 9 10 11 12 13

TABLE VI.

Max 823.851 -58.9429 220.071 -1700.31 -1457.35 -2451.4 -2567.36 -3335.31 -4162.43 -3109.22 -4132.83 -4071.39 -3985.53

75% 687.668 -58.9429 220.071 -2360.66 -2878.78 -3426.59 -3694.01 -4189.57 -4189.81 -4189.83 -4189.79 -4189.81 -4189.8

Median 687.668 -58.9429 -2390.42 -2587.63 -3613.31 -3601.66 -4079.63 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

25% 123.225 -848.729 -2647.82 -2836.82 -4056.6 -3850.79 -4189.81 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

Min -12.9572 -2146.33 -3341.34 -3402.46 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

C ASE 2. VALUES OF THE COST FUNCTION FOR DEL OCALT O B EST

Min -3371.71 -3428.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

C ASE 1. VALUES OF THE COST FUNCTION FOR DEL OCALT O B EST

1, Fig. 11, Tab. III and Tab. IV) and 8 (Case 2, Fig. 12, Tab. V and Tab. VI). All results from both cases can be compared with Fig. 13 and Tab. VII from Case 3 that was using only PRNG. When compared with Tab. VII according to the Median value, then it is visible that in Case 1 it is comparable with numerical precision equal to 3 (DERand1Bin and DELocalToBest). For Case 2 it is enough to have numerical precision to 7 (DERand1Bin) and 8 (DELocalToBest). From Tab. II is visible how many times was n periodic PCHNs used for those numerical precisions. IV.

Fig. 11. precision.

Case 1. Dependance of algorithm performance on numerical

Fig. 12. precision.

Case 2. Dependance of algorithm performance on numerical

C ONCLUSION

The main motivation of the research reported in this paper is to check whether it is possible to replace random number generators by deterministic processes originated in systems of deterministic chaos. For different numerical precisions there Numerical Precision 1 2 3 4 5 6 7 8 9 10 11 12 13

TABLE V.

Max 823.851 687.668 220.071 -1138.12 -1656.53 -2602. -2417.19 -2591.58 -4189.65 -4071.39 -4189.76 -4188.91 -4071.39

75% 687.668 -58.9429 220.071 -1936.64 -2347.12 -3712.17 -4189.32 -4189.82 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

Median 687.668 -58.9429 -1615.84 -2183.85 -2625.79 -3975.73 -4189.81 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

25% 123.225 -58.9429 -2050.05 -2477.27 -3036.31 -4189.82 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

Min -12.9572 -58.9429 -2635.02 -3026.53 -4159.5 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83 -4189.83

C ASE 2. VALUES OF THE COST FUNCTION FOR DER AND 1B IN

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Algorithm DERand1Bin DELocalToBest

TABLE VII.

Max -4071.39 -3738.01

75% -4189.83 -4160.41

Median -4189.83 -4185.35

25% -4189.83 -4189.32

C ASE 3. VALUES OF THE COST FUNCTION FOR VERSION WITH PRNG.

Min -4189.83 -4189.83 BOTH

DE

[2] Caponetto R., Fortuna L., Fazzino S., Xibilia M., Chaotic sequences to improve the performance of evolutionary algorithms, IEEE Trans. Evol. Comput. 2003, 7:3, 289-304. [3] Pluhacek, M., Senkerik, R., Davendra, D., Kominkova Oplatkova, Z. On the Behaviour and Performance of Chaos Driven PSO Algorithm with Inertia Weight, Computers and Mathematics with Applications, In print, ISSN 0898-1221 [4] Pluhacek M., Budikova V., Senkerik R., Oplatkova Z., Zelinka I. Extended Initial Study on the Performance of Enhanced PSO Algorithm with Lozi Chaotic Map, In Proceedings of Nostradamus 2012: International conference on prediction, modeling and analysis of complex systems, Springer Series: Advances in Intelligent Systems and Computing, Vol. 192, 2012, pp. 167 178, ISBN: 978-3-642-33226-5.

Fig. 13.

[5] Pluhacek M., Senkerik R., Zelinka I. Impact of Various Chaotic Maps on the Performance of Chaos Enhanced PSO Algorithm with Inertia Weight an Initial Study, In Proceedings of Nostradamus 2012: International conference on prediction, modeling and analysis of complex systems, Springer Series: Advances in Intelligent Systems and Computing, Vol. 192, 2012, pp. 153 166, ISBN: 978-3-642-33226-5.

Case 3. Results of EAs powered by PRNG, see Tab. VII

were generated periodic series, see Tab. II, that were used instead of number (series) generated by PRGNs. Based on reached results it can be stated that at least, in our case studies, all experiments exhibit fact that random number generators can be replaced by deterministic processes with small period (29 4514), repeated use in evolutionary algorithms with quite big frequency (5956 - 28 times). The advantage of proposed use of deterministic processes is the fact that in such case it is possible to fully repeat the run of given algorithm, to analyze its behavior deterministically, including its full path on searched fitness landscape. We also believe that mathematical proofs can be constructed more easily for such class of evolutionary algorithms.

[6] Pluhacek M., Senkerik R., Davendra D., Zelinka I. PID Controller Design For 4th Order system By Means Of Enhanced PSO algorithm With Lozi Chaotic Map, In: Proceedings of 18th International Conference on Soft Computing - MENDEL 2012, 2012, pp. 35 - 39, 2012, ISBN 978-80214-4540-6. [7] Pluhacek M., Budikova V., Senkerik R., Oplatkova Z., Zelinka I. On The Performance Of Enhanced PSO algorithm With Lozi Chaotic Map An Initial Study, In: Proceedings of 18th International Conference on Soft Computing - MENDEL 2012, 2012, pp. 40 - 45, 2012, ISBN 978-80214-4540-6. [8] Persohn K.J., Povinelli R.J., Analyzing logistic map pseudorandom number generators for periodicity induced by finite precision floatingpoint representation, Chaos, Solitons and Fractals, 45, 2012, 238-245. [9] Drutarovsky M.. Galajda, P., A robust chaos-based true random number generator embedded in reconfigurable switched-capacitor hardware, 17th International Conference Radioelektronika, Vols 1 and 2 Pages: 29-34, Brno, CZECH REPUBLIC Date: APR 24-25, 2007

Despite the widely presumed fact that pseudo-random number generators for evolutionary algorithms use have to have as big period as possible (for example Mersenne twister with 219937 −1) and such as the 232 common in many software packages, we show here that deterministic periodical processes with period 29 - 4514 are enough for our experiments.

[10] Bucolo M., Caponetto, R., Fortuna, L., Frasca M., Rizzo A,. Does chaos work better than noise?, Circuits and Systems Magazine, IEEE, On page(s): 4 - 19 Volume: 2, Issue: 3, Third Quarter 2002

Our further research is focused on more extensive and intensive testing of our ideas proposed here. Wider class of different algorithms, test functions and chaotic systems (to generate deterministic processes) will be selected for future experiments to prove and specify the domain of validity of our ideas proposed here.

[12] Pluchino A., Rapisarda A., Tsallis C., Noise, synchrony, and correlations at the edge of chaos, PHYSICAL REVIEW E, Volume: 87, Issue: 2, 2013, DOI: 10.1103/PhysRevE.87.022910

ACKNOWLEDGMENT The following grants are acknowledged for the financial support provided for this research: Grant Agency of the Czech Republic - GACR P103/13/08195S, by the Development of human resources in research and development of latest soft computing methods and their application in practice project, reg. no. CZ.1.07/2.3.00/20.0072 funded by Operational Programme Education for Competitiveness, co-financed by ESF and state budget of the Czech Republic, further was supported by European Regional Development Fund under the project CEBIA-Tech No. CZ.1.05/2.1.00/03.0089 R EFERENCES [1] Zelinka I, Celikovsky S, Richter H and Chen G., (2010) Evolutionary Algorithms and Chaotic Systems, (Eds), Springer, Germany, 550s, 2010.

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[11] Hu HP., Liu LF., Ding ND., Pseudorandom sequence generator based on the Chen chaotic system, COMPUTER PHYSICS COMMUNICATIONS, Volume: 184, Issue: 3, Pages: 765-768, 2013, DOI: 10.1016/j.cpc.2012.11.017

[13] Davendra D., Zelinka I., Senkerik R., Chaos driven evolutionary algorithms for the task of PID control, Computers and Mathematics with Applications, Vol. 60, Issue 4, 2010, pp 1088-1104, ISSN 0898-1221. [14] Senkerik R., Davendra D., Zelinka I., Oplatkova Z., Pluhacek M., Optimization of the Batch Reactor by Means of Chaos Driven Differential Evolution, In Proceedings of SOCO 2012: Soft Computing Models in Industrial and Environmental Applications, Springer Series: Advances in Intelligent Systems and Computing, Vol. 188, pp. 93 102, ISBN: 978-3-642-32922-7. [15] R. Lozi, Emergence Of Randomness From Chaos, International Journal of Bifurcation and Chaos, Vol. 22, No. 2 (2012) 1250021, World Scientific Publishing Company, DOI: 10.1142/S0218127412500216 [16] Xing-yuan Wang, Xue Qin, A new pseudo-random number generator based on CML and chaotic iteration, Nonlinear Dynamics An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, ISSN 0924-090X Volume 70 Number 2, Nonlinear Dyn. (2012) 70: pp 1589-1592 DOI 10.1007/s11071-012-0558-0 [17] Narendra K Pareek, Vinod Patidar, and Krishan K Sud, A Random Bit Generator Using Chaotic Maps, International Journal of Network Security, Vol.10, No.1, pp 3238, Jan. 2010 [18] Wang Xing-Yuan, Yang Lei, Design Of Pseudo-Random Bit Generator Based On Chaotic Maps, International Journal of Modern Physics B Vol. 26, No. 32 (2012) 1250208 (9 pages) World Scientific Publishing Company DOI: 10.1142/S0217979212502086

[19] Sun Yang Zhang, Lingbo Gu Xingsheng, A hybrid co-evolutionary cultural algorithm based on particle swarm optimization for solving global optimization problems, International Conference on Life System Modeling and Simulation / International Conference on Intelligent Computing for Sustainable Energy and Environment (LSMS-ICSEE) Location: Wuxi, PEOPLES R CHINA Date: SEP 17-20, 2010 [20] Hong Wei-Chiang, Dong Yucheng, Zhang, Wen Yu, Chen Li-Yueh, Panigrahi, B. K., Cyclic electric load forecasting by seasonal SVR with chaotic genetic algorithm, International Journal of Electrical Power and Energy Sysytems Volume: 44 Issue: 1 pp 604-614 DOI: 10.1016/j.ijepes.2012.08.010 [21] Zelinka I. SOMA – Self Organizing Migrating Algorithm, in New Optimization Techniques in Engineering, Eds.: Babu B. V., Onwubolu G. (Springer-Verlag, New York), pp 167-218, 2004 [22] Price K. An Introduction to Differential Evolution, New Ideas in Optimization, Ed.: Corne D., Dorigo M. Glover F. (McGraw-Hill, London, UK), pp 79-108, 1999 [23] Glover F., Laguna M. a Mart R., Scatter Search in Ghosh A. a Tsutsui S. (Eds.), Advances in Evolutionary Computation: Theory and Applications, Springer-Verlag, New York, pp 519-537, 2003 [24] Beyer H. G., Theory of Evolution Strategies, (SpringerVerlag, New York), 2001 [25] Holland J. H., Genetic Algorithms, Scientific American, July, pp 44 50, 1992 [26] Clerc M., Particle Swarm Optimization, ISTE Publishing Company, ISBN 1905209045, 2006

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