Optimized Fixed-Time Synergetic Controller via a modified Salp Swarm Algorithm for Acute and Chronic HBV Transmission System

MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX ISSN: 1803-3814 (Printed), 2571-3701 (Online) https://doi.org/10.13164/mendel.2023.k.162 A New Integral Function Algorithm for Global Optimization and Its Application to the Data Clustering Problem Ridwan Pandiya1,  , Atina Ahdika2, Siti Khomsah3, Rima Dias Ramadhani3 1Department of Informatics, Institut Teknologi Telkom Purwokerto, Indonesia 2Department of Statistics, Universitas Islam Indonesia, Indonesia 3Department of Data Sciences, Institut Teknologi Telkom Purwokerto, Indonesia [email protected]  , [email protected], [email protected], [email protected] Abstract The filled function method is an approach to finding global minimum points of multidimensional unconstrained global optimization problems. The conventional parametric filled functions have computational weaknesses when they are employed in some benchmark optimization functions. This paper proposes a new integral function algorithm based on the auxiliary function approach. The proposed method can successfully be used to find the global minimum point of a function of several variables. Some testing global optimization problems have been used to show the ability of this recommended method. The integral function algorithm is then implemented to solve the center-based data clustering problem. The results show that the proposed algorithm can solve the problem successfully. Keywords: Global Optimization, Global Minimum Point, Integral Function Algorithm, Data Clustering. Received: 23 August 2023 Accepted: 02 October 2023 Online: 02 October 2023 Published: 20 December 2023 1 Introduction Many problems in real life require optimal solutions. It encourages scholars working in numerical optimization to create the most efficient method. Bio-inspired algorithms and surrogate-assisted methods (see [20,14, 10,18,12,27]) can solve the black-box optimization problem. Flexibility is the main characteristic of these methods. However, the convergence is based on probability. Therefore, those algorithms are expensive in CPU time. Deterministic approaches promise guaranteed convergence, although the problems solved are usually less extensive than those that stochastic approaches can solve. The filled function method [7,6,8] and the DIRECT method [11,31] are examples of methods that fall into the deterministic approach. This research is focused on the filled function method. There has been a recent growth in parametric-filled function methods in global optimization problems. This is due to the development of nonlinear function optimization problems, which have many local minimum points in the search domains. These problems are widespread in applications such as engineering, finance, social science, biology, etc. The methods such as Newton’s steepest descent, quasi-Newton, trust region, and conjugate gradient methods are well executed for obtaining a local solution and globally determined when solving convex optimization problems [25]. Nevertheless, if the problems contain a non-convexity structure, those methods fail to find a global solution. Consequently, the global optimization problem is a challenging computational job for researchers [32]. The parametric filled function method is an auxiliary function method that uses a point in an objective function, usually a minimum point, to construct a function that satisfies the definition of the filled function. The parametric filled function, which was first presented by Ge [6,7], gives us an excellent idea of using the local minimum point to find the global minimum point. Furthermore, the basic outline of the function is explained as follows. a. Minimize the objective function by employing a local minimization method; a minimum point of the objective function is attained. b. The filled function is constructed, and the point close to the present minimum point is used as an initial point to further minimize the filled function. c. The minimum point of the filled function obtained in b is used as an initial point for minimizing the objective function, and the better minimum point of the objective function will be obtained. d. The global minimum point will be found if phases b and c are repeated until the stopping criterion is satisfied. The filled function proposed in [7] has some unexpected attributes. The intended method strongly depends on two parameters that are difficult to adjust to satisfy several conditions. Because of this problem, Ge and Qin in [8] proposed filled function that has only one parameter and is twice continuously differentiable everywhere. However, the filled function proposed in [8] 162 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX and its derivative are still influenced by the exponential term, which will be very large when the iterative points are far from the current local minimum point. Subsequently, several two-parameter and oneparameter filled functions have been proposed, as presented in [4,19,3,21,23,22], to solve global optimization problems. However, many filled function methods proposed by various authors in the literature have some drawbacks since their filled functions still include exponential terms, logarithmic terms, weight factors, and norms. In addition, selecting an appropriate method for searching minimum points, particularly for filled functions, involves numerous variables because they must contain the Hessian matrix and the inverse of a matrix. Several methods do not include inverse or Hessian matrix, but it is still believed that the gradient-based methods are more effective in determining the minimum point of the function. In 2010, [26] introduced the parameter-free filled function. However, the function is discontinuous at some points. Besides, the filled function proposed by [26] has the arctan term that makes this function very flat if the points are far from the incumbent local minimum point. This nature is the same as Ge’s filled function [7]. Several continuously differentiable filled functions were proposed in [24,28,1]. The local minimum point of those filled functions does, however, occasionally exist. According to this fact, this study introduces an integral function algorithm. Our new integral function method is made to eliminate the limitation of previous methods because our study has successfully developed such an auxiliary function that can solve multidimensional optimization problems. The advantage of our filled function lies in the non-involvement of parameters, exponential, logarithmic, and weight factors. This paper is organized as follows: In section 2, our new integral function, including its definitions and theorems, is presented. Section 3 describes the algorithm that includes an integral function in its phase. Numerical results are presented in Section 4. Application of the integral function to the data clustering problem is provided in Section 5, and the conclusion will be presented in Section 6. 2 A New Integral Function and its Properties This study solves the following unconstrained global optimization problems. (min ω(t) t∈Rn.(1) Throughout this paper, we require the following assumptions. Assumption 1: ω(t) is continuously differentiable. Assumption 2: ω(t)→ ∞ as ∥t∥→∞. Assumption 3: The set G, defined by G={ω(t)|t∈ F} , is finite, with Frepresenting the set of all local minimum points of Problem (1) Since the number of local minimum points can be infinite, Assumption 3 requires that the number of local minimum values of Problem (1) is finite. Assumption 2 implies the existence of a compact set ξ∈Rnwhose interior contains all minimum points of ω(t). We assume that the value of ω(t) for ton the boundary of ξ is greater than the value of ω(t) for any tin the interior of ξ. Then, Problem (1) is equivalent to Problem (2). (min ω(t) t∈ξ(2) In order to develop an auxiliary function approach that can address the deficiencies experienced by the filled function method, we offer the following integral function. ψi(t, t∗ m) =            −Zti t∗ m(i) (ω(t)−ω(t∗ m)) dtiti≥t∗ m(i) −Zt∗ m(i) ti (ω(t)−ω(t∗ m)) dtiti< t∗ m(i) , (3) where t∗ m=t∗ m(1), . . . , t∗ m(n) is a local minimum point of ω(t), i= 1, . . . , n, with n is the number of variables of ω(t), and ω(t)−ω(t∗ m) = ωt∗ m(1), . . . , tm(i), . . . , t∗ m(n) −ωt∗ m(1), . . . , t∗ m(n),(4) for m= 1, . . . , q, with qis the number of local minimum point of ω(t). Based on the form shown in Eq. (3), it is clear that ψi(t, t∗ m) is a one-dimensional function. It is known that all filled functions given in the literature still need to select the directions ei(i= 1, . . . , m0) and m0≥2n. In contrast, we do not need to select the direction in this paper since every function in Eq.(3) has included the directions. In other words, each direction has a different function. Based on Eq.(3), ψ1(t, t∗ m) and ψ2(t, t∗ m) are integral functions for searching the global minimum point of ω(t) in t1-axis (right side and left side of tirespectively), while ψ3(t, t∗ m) and ψ4(t, t∗ m) are integral functions for searching the global minimum point of ω(t) in t2-axis, and so forth. Therefore, we will have 2nof integral function where each function indicates the directions. Theorems 1-5show that ψi(t, t∗ m) satisfies the specification to be qualified as a filled function, or in other 163 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Pandiya-g2igHX,gA New Integral Function Algorithm for Global Optimization and Its Application to the Data Clustering... words, Eq.(3) satisfies three conditions demanded by the Definition 1. Definition 1. [33] A function ψ(t, t∗ m) is categorized as a filled function constructed at t∗ mif ψ(t, t∗ m) meets the following three conditions. •t∗ mis a strict local maximum point of ψ(t, t∗ m); •ψ(t, t∗ m) does not own local minimum and saddle points in ˆ ξ; •If ¯ ξ=∅, then ¯ ξcontains local minimum point of ψ(t, t∗ m), where ˆ ξ={t∈ξ|ω(t)> ω (t∗ m)} \ {t∗ m} and ¯ ξ={t∈ξ|ω(t)≤ω(t∗ m)}. For the simplicity, integral function (3) can be written as ψ(t, t∗ m) = −Zt t∗ m ω(t)−ω(t∗ m)dt, because either −Zti t∗ m(i) ω(t)−ω(t∗ m)dtiti≥t∗ m(i) or −Zt∗ m(i) ti ω(t)−ω(t∗ m)dtiti< t∗ m(i) has the same behavior. Theorem 1. If (1) t∗ m∈ F, (2) tmis a point such that tm=t∗ m, where ω(tm)≥ω(t∗ m). Then, ψ(t, t∗ m)≤0 = ψ(t∗ m, t∗ m). Proof. From Eq. (3), we have ψ(tm, t∗ m) = −Ztm t∗ m ω(tm)−ω(t∗ m)dtm. It follows from the assumption of the theorem, ω(tm)≥ω(t∗ m). This implies ω(tm)−ω(t∗ m)≥0. Since tm=t∗ m, we have Ztm t∗ m ω(tm)−ω(t∗ m)dtm>0. Therefore, ψ(tm, t∗ m)<0 = ψ(t∗ m, t∗ m). Theorem 2. If t∗ m∈ F, then t∗ mis a strict local maximum point of ψ(t, t∗ m). Proof. t∗ mis the element of F. This implies there exists N(t∗ m, ζ), which is the neighborhood of F, such that ω(t)≥ω(t∗ m), ∀t∈ N (t∗ m, ζ). It follows from Theorem 1, for all t∈ N (t∗ m, ζ), t=t∗ m, ψ(t, t∗ m)<0 = ψ(t∗ m, t∗ m). Thus, t∗ mis a strict local maximum point of ψ(t, t∗ m). Theorem 3. If (1) t∗ m∈ F, (2) t1and t2are two points such that ∥t1−t∗ m∥<∥t2−t∗ m∥and ω(t∗ m)< ω (t1)< ω (t2). Then, ψ(t2, t∗ m)< ψ (t1, t∗ m)<0 = ψ(t∗ m, t∗ m). Proof. Consider ψ(t2, t∗ m)−ψ(t1, t∗ m) = −Zt2 t∗ m (ω(t2)−ω(t∗ m)) dt2 +Zt1 t∗ m (ω(t1)−ω(t∗ m)) dt1. Since ω(t∗ m)< ω (t1)< ω (t2), then ω(t2)−ω(t∗ m)> ω (t1)−ω(t∗ m)>0. Therefore, Zt1 t∗ m (ω(t1)−ω(t∗ m)) dt1−Zt2 t∗ m (ω(t2)−ω(t∗ m)) dt2<0. Consequently, ψ(t2, t∗ m)< ψ (t1, t∗ m)<0 = ψ(t∗ m, t∗ m). Theorem 4. If (1) t∗ m∈ F, (2) tis a point such that ω(t)> ω (t∗ m). Then tis not a stationary point of ψ(t, t∗ m). Proof. For a given tmwith ω(tm)> ω (t∗ m), we have ∇ψ(tm, t∗ m) = Dtm −Ztm t∗ m (ω(tm)−ω(t∗ m)) dtm!. From the fundamental theorem of Calculus, ∇ψ(tm, t∗ m) = −(ω(tm)−ω(t∗ m)) . Since ω(tm)> ω (t∗ m), then ∇ψ(tm, t∗ m)= 0 164 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Theorem 5. If t∗ m∈ F, then any local minimum point or saddle point of ψ(t, t∗ m)must belong to the set ¯ ξ={t∈Rn|ω(t)≤ω(t∗ m)}. Proof. Assume that Theorem 5is not true. Then, there is a local minimum point or saddle point of ψ(t, t∗ m), denoted by ˜ t, such that ˜ t /∈¯ ξand ω˜ t> ω (t∗ m). It follows Theorem 2t∗ mis a strict local maximum point of ψ(t, t∗ m). Since ˜ tis a local minimum or saddle point of ψ(t, t∗ m), thus t∗ m=˜ t. If ˜ tis a local minimum point of ψ(t, t∗ m), it contradicts Theorem 3, while if ˜ tis a saddle point of ψ(t, t∗ m), it contradicts Theorem 4. As a result, Theorem 5is true. Theorem 6. If t∗ mis a global minimum point of ω(t), then ψ(t, t∗ m)<0,∀t∈ξ. Proof. Since t∗ mis a global minimum point of ω(t), ω(t)≥ω(t∗ m) for all t∈ξ. Thus, it follows from Theorem 1,ψ(t, t∗ m)<0 for all t∈ξ. 3 The Algorithm Derived from the theoretical properties of ψi(t, t∗ m), in this section, we describe our integral function algorithm, which applies to an unconstrained global optimization problem. The general iterative of this integral function is as follows: assume that t∗ 1is not a global minimum point of ω(t), then we can create an integral function ψ(t, t∗ m) at t∗ 1to acquire such a point that can be applied as a new initial point in order to bring the minimization of ω(t) to the better minimum point than t∗ 1. The process is conducted repeatedly until the stopping criterion is met. Input:t0, the initial point Output:t∗ I, global minimum point of ω(t) Initialization: a. Choose a constant υ, e.g., set υ:= 0.1 b. Set i= 1, . . . , i0, where i0= 2n c. Set i:= 1. Main step 1. Minimize ω(t) by using t0to obtain t∗ I. 2. Built Eq.(3) 3. if i>i0 then go to 7 else go to the next step 4. Find ¯ ti, the stationary point of ψi(t, t∗ I) 5. Minimize ω(t) by using ¯ tito obtain t∗ II 6. if ω(t∗ II )< ω (t∗ I) and t∗ II ∈ξ then set i:= 1, t∗ I=t∗ II , and go to 2 else set i:= i+ 1 and go to 3 7. Algorithm stops and t∗ Iis taken as a global minimum. There are two phases in the algorithm. First, minimizing the objective function ω(t) using a local search procedure (we implement the steepest descent method). In this phase, t∗ Iis obtained. In the second phase, integral function ψi(t, t∗ I) is constructed. Integral function ψi(t, t∗ I) is minimized, then we will yield ¯ ti, and phase 2 ends. The algorithm will reenter phase 1, with ¯ tias the starting point to find t∗ II , a new minimum point of ω(t) if such one exists. The iteration process is repeated until a stopping criterion is met. The last local minimum point to be found is assumed to be the global minimum point of ω(t). The explanation of our integral function in the algorithm is given in the following paragraphs. A set of 2nintegral functions is chosen in step 2, where each integral function represents the direction. For example, assume that we solve the two-variable function ω(t), with t∗ I= (t∗ 1, t∗ 2) as the first local minimum point of ω(t), which is gained in step 1. Thus, we have the following four integral functions. ψ1(t, t∗ I) = −Zt1 t∗ 1 (ω(t1, t∗ 2)−ω(t∗ 1, t∗ 2))dt1, for (t1≥t∗ 1, t2=t∗ 2). ψ2(t, t∗ I) = −Zt∗ 1 t1 (ω(t1, t∗ 2)−ω(t∗ 1, t∗ 2))dt1, for (t1< t∗ 1, t2=t∗ 2). ψ3(t, t∗ I) = −Zt2 t∗ 2 (ω(t∗ 1, t2)−ω(t∗ 1, t∗ 2))dt2, for (t2≥t∗ 2, t1=t∗ 1). ψ4(t, t∗ I) = −Zt∗ 2 t2 (ω(t∗ 1, t2)−ω(t∗ 1, t∗ 2))dt2, for (t2< t∗ 2, t1=t∗ 1). It can be seen clearly that ψ1-ψ4are one-variable functions. ψ1is one-variable function for which t2 is fixed for the positive t1-axis direction, ψ2is onevariable function for which t2is fixed, for the negative t1-axis direction, ψ3is one-variable function which t1 is fixed for the positive t2-axis direction, and ψ4is onevariable function for which t1is fixed for the negative t2-axis direction. Each stationary point of ψ(t, t∗ m) will be paired with the number that has been fixed. For example, if we obtain the local minimum point or inflection point ¯ t1 of ψ1or ψ2, then (¯ t1, t∗ 2) will become the initial point for minimizing ω(t) in step 5. Otherwise, if the stationary point ¯ t2is obtained from ψ3or ψ4, (t∗ 1,¯ t2) will be taken as the initial point. 165 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Pandiya-g2igHX,gA New Integral Function Algorithm for Global Optimization and Its Application to the Data Clustering... 4 Numerical Results In this section, we apply our integral function algorithm to some standard testing problems derived from the literature [26]. The proposed algorithm, and the algorithm given in [26] for comparison purposes, is programmed by Microsoft Visual C++ 6.0 for working on the Windows 10 system with Intel(R) Core(TM) i3 Processor 2.3G CPU and 4G RAM. Numerical results illustrate that the method is competent. The following notations are used in Tables 1-3 1. nis the number of variables of a testing functions, 2. Ftis the total number of function evaluations of ω(t) and ψi(t, t∗ m) before termination by our algorithm, 3. Gtis the total number of gradient evaluations of ω(t) before termination by our algorithm, 4. Ftl is the total number of function evaluations of ω(t) and ψi(t, t∗ m) before termination by the algorithm proposed by [26], 5. Gtl is the total number of gradient evaluations of ω(t) before termination by the algorithm proposed by [26] 6. υis a small positive number used as an addition to the minimum point when it will be used to minimize the integral function. Table 1: General computational results by our algorithm. Problem t0υ n FtFtl 1 (-1.6,0.9) 0.1 2 90 40 2 (-2 , -1) 0.1 2 304 145 3 (2 , 2) 0.1 2 159 74 4 (-1 , -1) 0.1 2 707 349 5 (10 , 10) 0.1 2 425 137 6 (1 , -1) 0.1 2 797 398 7 (1 , 1) 0.1 2 221 94 8 (3 , 3) 0.1 2 788 147 (0 , 0) 0.1 2 1432 706 (10 , -10) 0.1 2 2111 1045 9 (2 , 2 , 2) 0.1 3 4447 2208 (2 , ..., 2) 0.1 5 5087 2525 (2 , ..., 2) 0.1 7 4354 2156 (2 , ..., 2) 0.1 10 4692 2316 Table 1provides the general computational result of the integral function algorithm (Algorithm 3). It indicates that Algorithm 3successfully obtained the global minimum point of the given examples. Table 2compares Algorithm 3and the algorithm offered in [26]. We use the same initial point for our algorithm and algorithm in [26]. In the framework that the comparison is fair enough, we use the same method in minimizing the objective function and integral function (filled Table 2: Comparison of the results between our algorithm and the algorithm proposed in [26]. Problem t0υ n FtFtl Ftl Gtl 1 (-1.6,0.9) 0.1 2 90 40 1013 110 2 (-2 , -1) 0.1 2 304 145 1525 168 3 (2 , 2) 0.1 2 159 74 1173 91 4 (-1 , -1) 0.1 2 707 349 1425 775 5 (10 , 10) 0.1 2 425 137 1363 186 6 (1 , -1) 0.1 2 797 398 1775 95 7 (1 , 1) 0.1 2 221 94 1972 231 8 (3 , 3) 0.1 2 788 147 2143 279 (0 , 0) 0.1 2 1432 706 1723 327 (10 , -10) 0.1 2 2111 1045 3439 127 9 (2 , 2 , 2) 0.1 3 4447 2208 2467 227 (2 , ..., 2) 0.1 5 5087 2525 4497 1053 (2 , ..., 2) 0.1 7 4354 2156 9735 1683 (2 , ..., 2) 0.1 10 4692 2316 13091 2167 function for the method proposed in [26]). We found that in minimizing the filled function in [26], we could not use its gradient for the search direction. Therefore, another search direction stated in [26] was used. It can be seen in Table 3 that our integral function is more effective. All function evaluations are less than those yielded by the algorithm proposed by [26]. The exception is for the n-dimensional function (n= 3). Function evaluation obtained by Algorithm 3is 4447, while function evaluation gained by the algorithm in [26] is 2467. So as for the aspect gradient evaluation, Algorithm 3performs better than that algorithm in [26]. Except for gradient evaluations, the Rastrigin and two dimensional functions (c= 0.5) have more gradient evaluations. However, it can be understood since our integral function can obtain the entire extreme points of a function. However, in this paper, we are only concerned with obtaining a global minimum point. However, this kind of nature of our integral function has affected function evaluation, especially for functions with many extreme points. 5 Application to Data Clustering Problem Grouping data sets into some clusters is one of the unsupervised pattern classifications. It has been broadly implemented in various real problems such as pattern recognition, machine learning, text classification, and other problems (see [2,30]). Some approaches can be employed in clustering data. One of the most popular methods is the K-means algorithm. The algorithm obtains the minimum distance of the data into the center for each partition. The K-means algorithm originally used the Euclidean distance to operate its algorithm. However, according to [29], that distance sometimes is unsuitable. Since clustering is the minimization problem, K-means challenges non-convex and nonlinear optimization. This fact brings many scholars to solve the clustering problem through the optimization point 166 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX of view. In the optimization approaches, the data is modeled into the objective function and then solved by implementing minimization procedures. Therefore, the appropriate method chosen is one of the key successes in obtaining the optimal cluster. Since the integral function algorithm in this paper is a global minimization algorithm, in this section, we are willing to examine the proposed algorithm’s reliability to solve the clustering problem. The purpose of this section is the implementation of Algorithm 3. For this purpose, we use the optimization model initiated by [29], which has the following form. min τ1,...,τk∈R −ε m X p=1 ln k X q=1 exp −∥τq−tp∥2 ε!!,(5) where τ1, . . . , τk∈Rdenotes the centers for each kdisjoint partitions of the data set A={tp∈Rn|p= 1, . . . , m} ⊂ Rn. Eq.(5) will be solved by Algorithm 3. We use the example in [13]. Example 1. Let Table 3is a data set where the elements denoted by tp,p= 1, . . . , 20. The data will be partitioned into two clusters. In the first phase, all tp are substituted to Eq.(5). The next step is solving the optimization model using Algorithm 3to obtain the optimal center for each cluster. Table 3: Data set t1t2t3t4t5 0.456535 0.658425 0.868230 0.086283 0.704274 t6t7t8t9t10 0.063848 0.795001 0.684515 0.040520 0.824774 t11 t12 t13 t14 t15 0.957827 0.486618 0.008372 0.081007 0.251257 t16 t17 t18 t19 t20 0.907372 0.014313 0.783009 0.743946 0.066294 For the data set given in Table 3, Algorithm 3success in obtaining centers (τ∗ 1, τ∗ 2) = (0.0765,0.7392) , with the number of iterations and function evaluations are 40 and 1829, respectively. From the results, it can be shown that Algorithm 3can be used as one of the alternatives when solving center-based data clustering problems. 6 Conclusion and Future work The more problems in the real world can be modeled into the optimization model, the higher the demand for minimization methods with efficient computational performance. In this regard, many methods have been offered with various advantages. This paper offers a new deterministic approach to solve global optimization problems: the integral function algorithm. Uniquely, even though the optimization problems solved are multi-dimensional, the approach used in this study is a one-variable function approach. With this property, the computation stage becomes simpler. In addition, the integral function initiated in this study does not involve parameters usually included by similar methods. The reliability and efficiency of the proposed method are demonstrated through numerical simulations and numerical comparisons with other similar methods available in the literature. The proposed algorithm is also implemented in the final phase to solve the center-based data clustering problem. The implementation results show that the proposed algorithm can obtain the optimal center of the given data set. The numerical simulations conducted in this research are limited to several benchmark objective functions. In order to confirm that the proposed method is more reliable in solving global optimization problems, further research will be carried out on non-trivial optimization problems, such as black-box optimization problem [9,15], GKLS generator [5,16], and zigzagbased benchmark functions [13,17]. Acknowledgment This research is supported by the Ministry of Education, Culture, Research and Technology, Indonesia, through the fundamental research grant 2023. References [1] Ahmed, A. A new filled function for global minimization and system of nonlinear equations. Optimization 71 (2022), 4083–4106. [2] Dhillon, I., Mallela, S., and Kumar, R. A divisive information-theoretic feature clustering algorithm for text classification. Journal of Machine Learning Research 3 (2003), 1265–1287. [3] El-Gindy, T., Salim, M., and Ahmed, A. A new filled function method applied to unconstrained global optimization. Applied Mathematics and Computation 273 (2016), 1246–1256. [4] Gao, Y., Yang, Y., and You, M. 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