Taguchi-Enhanced Binary Differential Evolution ...

Taguchi-Enhanced Binary Differential Evolution Algorithm Cheng-Hong Yang Dept. of Electronic Eng., National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan. [email protected]

Yu-Da Lin* Dept. of Electronic Eng., National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan [email protected]

Senior Member, IEEE

Member, IEEE

Li-Yeh Chuang* Dept. of Chemical Eng., I-Shou University Kaohsiung, Taiwan [email protected]

Abstract This paper proposed an improved binary differential evolution (BDE) algorithm based on a Taguchi approach (Taguchi-BDE) for optimization problems. An estimation operator with efficient probability was adopted to preserve the diversity of populations and enhance the global search ability in Taguchi-BDE, and the Taguchi approach was used to optimize solutions after the crossover operations in Taguchi-BDE. The performance of the Taguchi-BDE algorithm was evaluated on a set of ten numerical benchmarks, and the results showed that our algorithm achieved better mean values over 40 repeated tests than did the other BDE-based algorithms. Taguchi-BDE was also applied to a real-world problem in statistical data mining, namely identification of interactions between single nucleotide polymorphisms. A total of 1,500 data sets under different settings were used to evaluate the efficiency of Taguchi-BDE, and the results showed that the proposed algorithm can achieve a 92.07% success rate. In conclusion, Taguchi-BDE revealed efficient convergence toward promising search regions, and achieved satisfactory search results.

been made to combine evolutionary algorithms with advantageous approaches, including PSO [2], HS [5], and the interior point algorith m [12], to deliver superior performance in numerica l optimization problems. So me of these hybrid algorithms, such as discrete binary DE (DBDE) [13], DE based on a binary bit-string framework (MBDE) [14], and DE based on a probability estimat ion operator (NM BDE) [15], have outperformed the original algorithms.

Keywords binary differential evolution, genome-wide association study, global numerical optimization, Taguchi approach

A. Review of Differential Evolution DE is a practical approach to global numerical optimization problems [16]. Co mpared with PSO, DE is easily adjustable because DE only uses two parameters. DE has been evaluated with several numerica l benchmarks in global optimization problems, and has been successfully applied to solve some real-world mathematical problems [17, 18]. This indicates that it may be more suited to such real-world problems than either PSO or GA [19]. The DE process consists of initialization, mutation, crossover, and selection operations . The population includes three vectors, namely target vector xi , mutant vector u i , and trial vector vi . Each target vector represents an potential solution in the search space. During init ialization, all target vectors are randomly distributed in the search space, and all of them are within the upper and lower limits on all dimensions. Subsequent mutation can expand the search region because a third vector is formed fro m the weighted difference of the two vectors within a mutant vector. New target vectors are then obtained by combining mutant and trial vectors in a crossover operation. The algorithm continues to execute mutation, crossover, and selection operations until the assigned number of iterations is reached. The target vector with the optimal value in the population is returned as the best solution when the algorithm terminates.

I.

INT RODUCT ION

Nonlinear optimizations often return faulty results due to mu ltiple local optima, and thus substantial challenges remain to finding global optima. Nu me rous algorithms make attempts to mimic some characteristics of natural phenomena or social behavior, such as particle swarm optimization (PSO) [1, 2], genetic algorithms (GAs) [3], differential evolution (DE) [4], and harmony search (HS) [5]. These algorithms have been applied to various real-world problems, including bioinformatics [6-8]. The performance characteristics of algorithms have been evaluated, and have been determined to have several problems [9-11]. Recently, some studies have sought to improve the performance of global numerical optimization algorithms on real-world problems. In evolutionary algorithm studies, both intensification and diversification are the essential aspects of metaheuristic methods. The intensification leads to population exploitation around the appreciable solutions and selects beneficial candidates to generate more competitive solutions. Diversification can improve populations and enable them to exp lore more efficient search spaces. Various attempts have

This paper proposes an improved DE method (TaguchiBDE) that combines NMBDE with a Taguchi approach to solve numerica l optimization problems. The Taguchi approach can find the optimal mating for additional crossover operations in NM BDE. According to our experiments, Taguchi-BDE showed increased population diversity and an appreciable convergence toward promising search regions. II.

TAGUCHI BINARY DIFFERENT IAL EVOLUT ION

TABLE I.

Experiment number 1 2 3 4 5 6 7 8

A 1 1 1 1 1 2 2 2 2

ORTHOGONAL ARRAY OF L8 (2 7 ) Factors B C D E Column number 2 3 4 5 1 1 1 1 1 1 2 2 2 2 1 1 2 2 2 2 1 2 1 2 1 2 2 1 2 1 1 2 2 1 2 1

F

G

6 1 2 2 1 1 2 2 1

7 1 2 2 1 2 1 1 2

B. Taguchi Approach The proposed Taguchi approach uses a signal-to-noise ratio (SNR) and an orthogonal array (OA) to control the parameters of the experiments [20-22]. Previously, Taguchi approaches have been applied to optimize the parameters in a GA [23] and a DE [24]. However, a search of the literature revealed no instances of Taguchi approaches being applied to improve BDE. In this study, we used an two-level OA represented as Ln (2n 1 ) where n = 2k, k is a positive integer (k > 1), the number 2 represents the levels for each factor, and n 1 is the number of columns in the OA. Table I shows the OA L8 (27 ). Notably, the number to the left of each row is the experiment nu mber, and runs from 1 to 8. SNR( ) is the adaptation of Taguchi approach design parameters, and the present experiments were fulfilled on the basis of the-smaller-the-better fitness values. In individual targets, the SNR value can be increased when parameter design is improved. SNR is determined by (1); in (2), the Taguchi approach establishes an orthogonal table by calculating each experiment. The i can be used to modify SNR, which thus increases mating [25]. A smaller i value indicates a better result. The fitness value of each result is denoted by yi where i is the experiment number, i = 1 ... n, and n is the number of experiments. In (3), Efl is the effect of factors , where f is the factor name and l is the level nu mber. The s mallest Ef value can be determined as the optimal dimension. (1) (2) (3) C. Taguchi Binary Differential Evolution In this study, Taguchi-BDE comb ines NMBDE [15] and Taguchi methods to improve search stability. The Taguchi approach stabilizes the convergence of cross -selection operations and prevents the population from falling into local optima. In this research, Taguchi methods are used to optimize the solutions after cross selection and to select the preferable cross-dimension in terms of the mutation probability (i.e., mutation rate). Taguchi-BDE uses a binary encoding differential and probability estimation operator to control the mutation calculation pxi = pxi1 , pxi2 , ..., pxij , where i=1 ... NP

ALGORITHM I. T AGUCHI-BDE D 1 Define: Let f( be the fitness function. Let V be vectors. N is D the number of vectors in the swarm, each having a vector vi in the search-space where D is the number of dimension. U(b low , b up ) is a uniformly distributed random vector, b low and bup are the lower and upper boundaries in the search space. 2 Create orthogonal arrays 3 Calculation each experiment of orthogonal arrays 4 Get crossover dimensions 5 Randomly initialize population 6 while (the termination criterion is not met) 7 Evaluate fitness 8 for i 1 to N 9 for j 1 to D 10 if U(0, 1) CR or j U(0, i) then 11 Create a mutant vector 12 if U(0, 1) P(vij) then 13 pvij 1 14 else 15 pvij 0 16 end 17 else 18 pvij pxij 19 e nd 20 end 21 Using pvi and pxi crossover by T aguchi solution 22 if f(pvi) < f(pxi) then 23 pxi pvi 24 if f(pvi) < f(p g ) 25 p g pvi 26 end 27 e nd 28 end 29 end

and j=1 ... N. Notably, i is the ith target vector, NP is the population size, j is the number of the dimensions, pxi is a possible target vector in the binary search space, N is the total number of dimensions, and the value of pxij is either 0 or 1. As indicated in Algorith m 1, the processes of Taguchi-BDE include initialization, mutation, crossover, the Taguchi operation, and selection. The target vector that includes the optimal objective function value in the population is considered to be the best solution when the routine terminates. Taguchi-BDE solves binary problems by increasing the probability estimation to calculate the fitness values. In the mutation operation, Taguchi-BDE uses the machine operator (MO) formu la [(4), (5)] to provide a binary code mutation formula. In (4), b is the bandwidth factor, and is an appropriate value to increase the search efficiency. In (5), F is the weight scale factor of the weight difference algorithm. MO values can be used to obtain a probability value, and then a result (pvi ) of zero or one is produced as determined by (6). The rand() is a random number within [0, 1].

T ABLE II. BENCHMARK FUNCTIONS.

Name

Function formulas

Search range ±2.048

Multimodal?

Separable?

Scalable?

N Y

N N

Y N

Griewank

Y

N

Y

±10

Schaffer-F6

Y

N

Y

±10

Dixon Ack ley 1 Sphere Sum squares Powell Sum

f17 ( x)

2 n i 1 xi

N Y Y N N

N N Y Y Y

Y Y Y Y Y

±10 ±32 ±10 ±5.12 ±1

Schwefel 2.22

f 20 ( x)

n i 1

N

N

Y

±10

Rosenbrock Freudenstein-Roth

xi

n i 1

xi

(4) (5) (6)

In the crossover operation, the target and trial vectors obtained fro m mutation operations are used to find better solutions in the search space. Each dimension is determined by using the target or mutant vectors pu i as shown in (7).

(7)

III.

IMPLEMENT AT ION

A. Benchmark Numerical Experiments Typical studies on global numerical optimizat ion problems use numerical benchmarks to evaluate algorithm performance [26]. To analyze the effectiveness and performance of TaguchiBDE for nu merica l optimization proble ms, the ten most widely used representative benchmark functions were emp loyed in our study. The formulations of these benchmark functions are presented in Table II. B. Initialization We applied Taguchi-BDE to the global numerical optimization problems listed in Table II. Each dimension of the target vector had 20-b it binary encoding to represent the decimal. The conversion formula is shown in (9).

In this study, the selection operation applies elite selection. A comparative test decides whether an individual must be saved for the next generation. When pxi is better than pui , pxi replaces pu i as (8).

(8)

Next, an OA is generated according to the number of dimensions , followed by an orthogonal table according to the fitness value of each orthogonal. The results of the OA are used to obtain the best crossover. For the obtained orthogonal dimensions pvi and pxi , the crossover operation is executed before NMBDE selects the operation for each vector.

±10

(9)

C. Parameter Settings The population size was set to 40, the maximu m number of generations was set to 250 iterations, and b was set to 6 [15]. A total of 40 imple mentations were used to test the algorithms under each setting with completely different initial conditions , so as to produce meaningful statistics. The results that were obtained were almost independent of the initial guess es. The statistical measures , such as mean values and their standard deviations (Std. Dev.), were used to evaluate the performance of Taguchi-BDE. IV.

SIMULAT ION RESULT S

A. Tuning Learning Parameters CR, b, and F The optimal para meters of both DE and NMBDE algorith ms have been reported elsewhere [15]. Ho wever, these

TABLE III. COMP ARISON OF T AGUCHI- BDE WITH FOUR DIFFERENT ALGORITHMS FOR TEN BENCHMARK FUNCTIONS .

Name Rosenbrock Freudenstein-Roth Griewank Schaffer-F6 Dixon Ack ley 1 Sphere Sum squares Powell Sum Schwefel 2.22

BPSO 1.70E-04 8.37E-04 9.11E-09 6.00E-05 1.56E-05 3.65E-02 2.77E-05 1.15E-05 1.43E-08 5.69E-03

DBDE 2.02E-04 4.69E-01 4.56E-08 2.41E-03 1.57E-03 6.35E-02 7.61E-05 4.20E-05 7.78E-09 8.14E-03

parameters were not suitable for Taguchi-BDE; thus, optimization of the parameters was required to achieve optimal search results. Therefore, we evaluated two parameters, CR and F, under various settings. The crossover rate CR affects the individual contribution through mutation, and the scaling factor F determines the ratio of MO calculations. Two well-known functions (Rosenbrock and schaffer-F6) were used as numerical benchmarks to optimize the CR and F parameters of Taguchi-BDE. CR was set to 0.1, 0.3, and 0.6, and F was set to 0.2, 0.6, 1.0, 5.0, and 10.0. Each implementation was processed using 20-b it encoding and 3000 generations per experiment; each experiment was run 40 t imes. The best mean values were 3.77E-10 and 1.01E-11, and the optimal values of CR and F are 0.5 and 5.0, respectively. B. Comparison Study In the ten benchmarks , the population size was set to 40, with 3000 iterations per experiment, and each experiment was repeated 40 times. The results are shown in Table III, and the values are the average of fitness values over 40 runs. TaguchiBDE obtained the optimal solutions (i.e., 0.000) for five benchmarks, namely Griewank , Sphere, Sum squares, Powell Sum, and Schwefel 2.22, whereas BPSO, DBDE, MBDE, and NMBDE d id not achieve the optimal solution. For other benchmarks, Taguchi-BDE showed solutions that were superior to those of other methods; one such benchmark was Ack ley 1 (Taguchi-BDE: 4.44E-16; BPSO: 3.65E-02; DBDE: 6.35E-02; M BDE: 9.49E-03; NM BDE: 3.73E-02). Overa ll, the results indicate that Taguchi-BDE provides a better performance than do BPSO, DBDE, M BDE, and NM BDE. Moreover, Taguchi-BDE demonstrated more appreciable improvement than NM BDE, indicat ing that the proposed Taguchi approach substantially imp roves NMBDE in terms of both intensification and diversification. V. BIOLOGICAL BENCHMARK A. Identification of Single-Nucleotide Polymorphism (SNP) SNP Interaction Problem Identificat ion of the relat ionship between a disease and related factors could help disease investigators understand the association between the genotype and phenotype for that

MBDE 5.35E-05 4.97E-02 5.59E-10 2.76E-06 2.16E-04 9.49E-03 1.50E-06 7.01E-07 5.52E-10 1.36E-03

NMBDE 8.89E-05 6.14E-02 1.37E-08 6.57E-04 1.69E-04 3.73E-02 3.36E-05 1.68E-05 1.88E-09 4.59E-03

Taguchi-BDE 5.00E-10 1.00E-06 0.000 1.02E-10 1.01E-10 4.44E-16 0.000 0.000 0.000 0.000

disease. In a genome's sequence, single-nucleotide polymorphis ms (SNPs) are vital variat ions that can be located in reg ions between coding and noncoding genes [27], and may cause changes to a mino acid and mRNA transcript stability [28]. Nu merous disease-related investigations have indicated that SNPs are associated with diseases or cancers and some particular SNPs may have interactions associated with diseases or cancers. Moreover, the interactions of SNPs display an appreciable impact on risk association [28-31]. A SNP involves three genotypes: a homozygous reference genotype, heterozygous genotype, and homozygous variant genotype. A SNP genotype may associate with other SNP genotypes to produce an interaction between SNPs that influences their biological functions in terms of genes. The statistical data mining fields assume that SNP SNP interactions can display the phenomenon with significant differences between any two groups [32]. However, identification of SNP SNP interactions is a staggering computational challenge for statistical data mining due to the complex co mbinations of data of the genotypes related to various SNPs [28]. Determining the significant SNP SNP interactions requires calculation of the possible combinations of genotypes of SNPs using [C(S, G) 3G ] N, where N is the number of subsets of crossover validation, S is the total number of SNPs, and G is the number of genotypes. Consequently, statistical data mining based on exhaustive searching may fail to comp letely identify the SNP SNP interactions when both total numbers of SNPs and samples continue to be raised. Some algorithms, such as DE, have been applied to this challenge [8]. However, the identification of SNP SNP interaction must be improved. B. Encoding and Initialization A binary encoding is used to formalize the target vector, donor vector, and trial vector. Let D be L(O, S ) = O log 2 (S) , where O is the order of SNP SNP interaction and S is the total number of SNPs. The O-order SNP SNP interaction identified fro m S SNPs can be coded by Ddimensional bits Vi = {v1 vD | vi {0, 1}}. During initialization, all target vectors are randomly assigned to the entire parameter space, and each target vector is called a model.

TABLE IV. COMPARISON OF THE SUCCESSFUL DETECTION RATES OF THE MODELS UNDER DIFFERENT GENETIC CONSTRAINTS.

50-SNPs 100-SNPs 150-SNPs NM T D.I. NM T D.I. NM T h2 BDE BDE BDE BDE BDE BDE 0.005 84 15 81 17 69 99 98 79 0.05 83 17 81 14 77 100 95 86 0.1 85 15 77 16 72 100 93 79 0.3 84 16 69 18 68 100 97 82 0.4 85 14 78 15 71 99 93 81 2 h : heritability; TBDE: Taguchi-BDE algorithm; D.I.: Degree improvement with T BDE over than NMBDE; bold type is the better.

D.I. 10 9 7 14 10 of

C. Fitness Evaluation Multifactor-d imensionality reduction (MDR) is a statistical data min ing method based on exhaustive searching that can be used to identify SNP SNP interaction [33]. The advantages of MDR are that a representation of a data space can be transformed so that contingency table measures can be easily used [34], and that MDR uses cross-validation to prevent over-fitting. Here, we used an MDR operation (Algorithm II) to design the fitness function. Let L be a set of genotypes in a model, also called a multifactor class (i.e., L = {l 1 l O }), where O is the order of SNP SNP interaction. For that multifactor class, the ratio of the number o f cases (patients) to the number o f controls (normal) within each multifactor class is computed by using (10).

ALGORITHM II. FITNESS FUNCTION BASED ON MDR 1 2 3 4 5 6 7 8 9 10

11 12 13 14

Define: N is the number of fold cross-validation. C is the total number of possible-genotypes combination between SNPs. divide data into N subsets for n 1 to N T est_data n-th sub-data T raining_data other sub-data for c = 1 to C //training data determine the H/L groups at c-th combination e nd for c = 1 to C //test data assign the H/L groups at c-th combination //the H/L is assigned according to determined result in training data end Compute TP, FP, FN, and TN Compute balanced classification error rate as fitness value end

balanced classification error is used to evaluate the model as given in (12). (12) The aforementioned procedure is repeated until all of the testing data sets have been used. The average classification error is the fitness value. D. Parameter Settings and Datasets

(10)

where (11) where the jth sample of cases is labeled Pj ; the jth samp le of controls is labeled N j ; P* and N * are the total numbers of samples of cases and controls in the training set, respectively; and u( ) is a function to count samples whose all genotypes are matched with L. When all possible mu ltifactor c lasses have been computed, each mult ifactor c lass is determined as if the ratio 1; otherwise it is determined as . Next , the testing data is used to evaluate the trained model. True positive (TP) is the cardinality of the set of samp les of cases that are predicted as belonging to the group; otherwise these samples of cases belong to the false negative (FN) set. However, true negative (TN ) is the card inality of the set of samples of controls that are predicted as belonging to the L group; otherwise these samples of controls belong to the false positive (FP) set. Thus, the four frequencies can be generated in a 2-way contingency table [TP, FP, TN, and FN]. The

According to our previous experiment, CR was set to 0.5, F was set to 5, and b was set to 6. The population size was the same as the number of SNPs , and the maximu m generation was set to 1000. Data sets with 2-loci, strict, and pure genetic models were generated by GAM ETES software [35] under d ifferent genetic constraints, including heritability of 0.005, 0.05, 0.1, 0.3, and 0.4 and minor allele frequency of 0.2. For each genetic constraint, 100 replicate data sets were generated with a sample size of 2,000 for balanced cases and controls , and 50, 100, and 150 were adopted as the total number of SNPs for various runs. Thus, a total of 1,500 [5 (heritability) × 100 (replicates) × 3 (number of SNPs)] data sets were generated to evaluate the algorith ms. In each data set, a pair of highly interactive SNPs (M0P0 and M1P1) became the target; the rest of the SNPs were named Ni. E. Comparison Study Each of the 15 genetic constraints had 100 rep licates and was generated by GAM ETES to evaluate the algorith ms for the ability to identify SNP-SNP interaction. We used success rate as a useful criterion for evaluating the performance of algorithms, which was defined as:

Sr

100

N successful N all

(13)

where N all was the total number o f trials and Nsuccessful was the number of trials in wh ich the solution (i.e., M0P0 and M1P1) was successfully identified. The success rates were calculated for each set of 100 rep licates (Tab le IV). The results showed Taguchi-BDE was able to successfully identify the SNP SNP interactions, and both of the two algorithms showed a down trend for the increased dimension of data sets. However, in all genetic constraints, Taguchi-BDE outperformed NM BDE. In particular, Taguchiaverage success rate for all 15 genetic constraints was 92.07, whereas NMBDE had an average success rate of 77.80. VI.

DISCUSSION AND CONCLUSIONS

In global numerical optimizat ion problems, intensification and diversification are essential to preserve exploitation and exp loration until termination [36]. To preserve the diversity of the population, DE uses mutation to constantly generate new positions by referencing two vectors. However, the degree of diversification can slowly decline when the population converges to intensive areas; this is called the local optima problem in evolutionary algorithms . Taguchi approaches can profit fro m the optimal para meters of a process that can be scientifically analyzed [37]. The experimental design must determine the focused factors and the levels of their fitness functions before starting the experiment. Consequently, a Taguchi approach does not reduce the number of experiments to a minimu m. The OA table is able to immediately collocate the parameter disposition table for the focused factors and their levels. These advantages enable Taguchi approaches to preserve population diversity. The ten numerical benchmarks in this study included unimodal and multimodal types to test the superior search capabilities of Taguchi-BDE. Taguchi-BDE delivers satisfactory performance for the identification of real-world SNP SNP interactions under various genetic constraints. In typical investigations of SNP SNP interactions, the total number of SNPs is roughly 100, and they are selected methodically fro m d isease-related genes. However, increasing the number of SNPs used to identify a significant SNP SNP interaction enables investigation of the associations between SNPs and diseases (particularly cancers). Evolutionary algorithms can facilitate such investigations by increasing the total number of SNPs , and thus finding more associations among SNPs. In the present study, the same initialized population was used for both NMBDE and TaguchiBDE in each data set to ensure a fair comparison of search ability without chance. Although NMBDE showed notably favorable results, Taguchi-BDE outperformed NMBDE for the identification of SNP SNP interactions, especially for the 150SNP case. The results indicate that Taguchi approaches could improve the search ability of NMBDE in co mplex real-world problems.

Future studies should conduct convergence analys es by observing the fluctuation of fitness values in all generations. Taguchi-BDE could also be tested with more numeric benchmarks because different parameters (CR and F values) may result in different convergence rates. Moreover, the application of Taguchi-BDE could be extended to solve more complex binary problems. A CKNOWLEDGMENT This study was partly supported by the Ministry of Science and Technology of Taiwan for Grant 103-2221-E-151 -029 MY3, 104-2221-E-214 -035 -MY2 and 105-2811-E-151 -002 -. REFERENCES [1] [2]

J. Kennedy, and R. C. Eberhart, "A discrete binary version of the particle swarm algorith m." pp. 41044108, 1997. J. Kennedy, "Particle swarm optimization," Encyclopedia of machine learning, pp. 760-766: Springer, 2011.

[3]

[4]

fast and elit ist mult iobjective genetic algorithm: NSGA Evolutionary Computation, IEEE Transactions on, vol. 6, no. 2, pp. 182-197, 2002. R. Stor a simp le and effic ient heuristic for global optimizat ion Journal of global optimization, vol. 11, no. 4, pp. 341-359, 1997.

[5] [6]

[7]

[8]

[9]

[10]

[11]

heuristic opt Simulation, vol. 76, no. 2, pp. 60-68, 2001. O. Y. Fu, Y. D. Lin, L. Y. Chuang, H. W. Chang, C. polymorphis ms of ORAI1 gene are associated with preventive models o f breast cancer in the Biomed Research International, vol. 2015, no. 2015, pp. 281263, 2015. C. H. Yang, Y. D. Lin, Y. C. Chiang, and L. Y. hybrid approach for CpG island detection in the human g PLoS One, vol. 11, no. 1, pp. e0144748, Jan 4, 2016. C. H. Yang, Y. D. Lin, L. Y. Chuang, and H. W. -order SNP barcodes in mitochondrial D-loop for chronic d ialysis Journal of Biomedical Informatics, vol. 63, pp. 112-119, Oct, 2016. M. A. A l-Betar, and A. search algorithm for university course timetabling, Annals of Operations Research, vol. 194, no. 1, pp. 3-31, Apr, 2012. -based diffe rential evolution algorithm for permutation flow shop s Advances in Engineering Software, vol. 55, pp. 10-31, Jan, 2013. L. Y. Chuang, S. H. Moi, Y. D. Lin, and C. H. Yang,

optimizations

[12] [13] [14]

[15]

[16]

[17]

[18]

[19]

[20] [21] [22]

for

detecting single nucleotide Artificial Intelligence in Medicine, vol. 73, pp. 23-33, Oct, 2016. Y. Nesterov, and A. Nemirovskii, Interior-point polynomial algorithms in convex programming: Siam, 1994. C. Peng, L. Jian, and L. Zhiming, "So lving 0-1 knapsack problems by a discrete binary version of differential evolution." pp. 513-516, 2008. C.-Y. Wu, and K.of structures using modified binary diffe rential Structural and Multidisciplinary Optimization, vol. 42, no. 6, pp. 939-953, 2010. L. Wang, X. Fu, Y. Mao, M . I. Menhas, and M. Fei,

[27]

Neurocomputing, vol. 98, pp. 55-75, 2012. K. Price, R. M . Storn, and J. A. Lamp inen, Differential evolution: a practical approach to global optimization: Springer Science & Business Media, 2006. J. Ilonen, J.-K. Kamarainen, and J. Lampinen, Neural Processing Letters, vol. 17, no. 1, pp. 93-105, 2003. J. Vesterstrom, and R. Tho msen, "A comparat ive study of differential evolution, particle swarm optimization, and evolutionary algorith ms on numerical benchmark problems." pp. 1980-1987, 2004. S. Paterlin i, and

[30]

Computational statistics & data analysis, vol. 50, no. 5, pp. 1220-1247, 2006. R. K. Roy, A primer on the Taguchi method: Society of Manufacturing Engineers , 2010. M. S. Phadke, Quality engineering using robust design: Prentice Hall PTR, 1995. D. C. Montgomery, Design and analysis of experiments: John Wiley & Sons, 2008.

[23]

[28]

[29]

[31]

[32] [33]

[34]

[35] crossover operators for o Computers & Operations Research, vol. 22, no. 1, pp. 135-147, 1995.

[25]

[26]

mach ine learning and statistical approaches for detecting SNP interactions in high-dimensional IEEE/ACM transactions on computational biology and bioinformatics, no. 99, 2016. L. Y. Chuang, H. Y. Lane, Y. D. Lin, M. T. Lin, C. H. barcode biomarkers for genes associated with fac ial emotion perception using particle swarm Annals of General Psychiatry, vol. 13, pp. 15, May 21, 2014. C. H. Yang, Y. D. Lin, S. J. Wu, L. Y. Chuang, and o rder gene-gene interactions in eight single nucleotide poly morphis ms of reninangiotensin system genes for hypertension association s Biomed Research International, pp. Article ID 454091, 2015. C.-H. Yang, L.-Y. Chuang, and Y.based differential evolution identifies the epistatic interaction in genomeBioinformatics, in press, 2017. -gene interactions that Nature Reviews Genetics, vol. 10, no. 6, pp. 392-404, Jun, 2009. M. D. Ritchie , L. W. Hahn, N. Roodi, L. R. Bailey, W. D. Dupont, F. F. Parl, and J. H. Moore, -dimensionality reduction reveals highorder interactions among estrogen-metabolis m genes in spo American Journal of Human Genetics, vol. 69, no. 1, pp. 138-147, Ju l, 2001. W. S. Bush, T. L. Ed wards, S. M. Dudek, B. A. contingency table measures improve the power and detection of mu l Bmc Bioinformatics, vol. 9, May 16, 2008. R. J. Urbanowicz, J. Kiralis, N. A. Sinnott-Armstrong, T. Heberling, J. M. Fisher, and J. H. Moore, pure,

[24] algorith m fo r the selection of optimal machin ing Applied Soft Computing, vol. 13, no. 3, pp. 1561-1566, Mar, 2013. J.-T. Tsai, T.-K. Liu, and J.Taguchi-genetic algorithm for global nu merical Evolutionary Computation, IEEE Transactions on, vol. 8, no. 4, pp. 365-377, 2004. S. Rahnamayan, H. R. Tizhoosh, and M. M. Salama, IEEE Transactions on Evolutionary computation, vol. 12, no. 1, pp. 64-79, 2008.

-wide PLoS Comput Biol, vol. 8, no. 12, pp. e1002822, 2012.

[36]

[37]

strict,

epistatic models with random Biodata Mining, vol. 5, Oct 1, 2012. M. Lo zano, and C. Garc iametaheuristics with evolutionary algorith ms specializing in intensification and diversification: Computers & Operations Research, vol. 37, no. 3, pp. 481-497, Mar, 2010. R. E. Walpole, R. H. Myers, S. L. Myers, and K. Ye, Probability and statistics for engineers and scientists: Macmillan New York, 1993.