Gene Targeting Differential Evolution A Simple and Efficient Method for Large-Scale Optimization

—Large-scale optimization problems (LSOPs) are challenging because the algorithm is difﬁcult in balancing too many dimensions and in escaping from trapped bottleneck dimensions. To improve solutions, this article introduces targeted modiﬁcation to the certain values in the bottleneck dimensions. Analogous to gene targeting (GT) in biotechnology, we experiment on targeting the speciﬁc genes in the candidate solution to improve its trait in differential evolution (DE). We propose a simple and efﬁcient method, called GT-based DE (GTDE), to solve LSOPs. In the algorithm design, a simple GT-based modiﬁ-cation is developed to perform on the best individual, comprising probabilistically targeting the location of bottleneck dimensions, constructing a homologous targeting vector, and inserting the targeting vector into the best individual. In this way, all the bottleneck dimensions of the best individual can be probabilistically targeted and modiﬁed to break the bottleneck and to provide global guidance for more optimal evolution. Note that the GT is only performed on the globally best individual and is just carried out as a simple operator that is added to the standard DE. Experimental studies compare the GTDE with some other state-of-the-art large-scale optimization algorithms, including the winners of CEC2010, CEC2012, CEC2013, and CEC2018 competitions on large-scale optimization. The results show that the GTDE is efﬁcient and performs better than or at least comparable to the others in solving LSOPs.

Gene Targeting Differential Evolution: A Simple and Efficient Method for Large-Scale Optimization Zi-Jia Wang , Member, IEEE, Jun-Rong Jian , Student Member, IEEE, Zhi-Hui Zhan , Senior Member, IEEE, Yun Li , Fellow, IEEE, Sam Kwong , Fellow, IEEE, and Jun Zhang , Fellow, IEEE Abstract-Large-scale optimization problems (LSOPs) are challenging because the algorithm is difficult in balancing too many dimensions and in escaping from trapped bottleneck dimensions.To improve solutions, this article introduces targeted modification to the certain values in the bottleneck dimensions.Analogous to gene targeting (GT) in biotechnology, we experiment on targeting the specific genes in the candidate solution to improve its trait in differential evolution (DE).We propose a simple and efficient method, called GT-based DE (GTDE), to solve LSOPs.In the algorithm design, a simple GT-based modification is developed to perform on the best individual, comprising probabilistically targeting the location of bottleneck dimensions, constructing a homologous targeting vector, and inserting the targeting vector into the best individual.In this way, all the bottleneck dimensions of the best individual can be probabilistically targeted and modified to break the bottleneck and to provide global guidance for more optimal evolution.Note that the GT is only performed on the globally best individual and is just carried out as a simple operator that is added to the standard DE.Experimental studies compare the GTDE with some other state-of-the-art large-scale optimization algorithms, including the winners of CEC2010, CEC2012, CEC2013, and CEC2018 competitions on large-scale optimization.The results show that the GTDE is efficient and performs better than or at least comparable to the others in solving LSOPs.
Currently, many DE variants have also been widely studied for solving large-scale optimization problems (LSOPs) [26].However, their performance is still restricted by the phenomenon of "curse of dimensionality" as the number of dimensions increases [27]- [31].Since an LSOP often involves various characteristics of different variables, the evolution in different dimensions (i.e., variables) may become imbalanced.That is, a solution may have outstanding values on some dimensions, but very poor values on some other dimensions, which are regarded as the "bottleneck dimensions."These bottleneck dimensions will limit the algorithm's ability to find globally optimal solutions.Therefore, it is crucial for a large-scale optimization algorithm to tackle the bottleneck dimensions efficiently, so as to balance the evolution among all the dimensions.If we can introduce targeted modification to the values of bottleneck dimensions to facilitate the evolution of algorithm, the solutions could be much improved.
In biotechnology, gene targeting (GT) is an experimental method to target the specific gene of living organisms for modification (i.e., targeting modification the specific gene), so as to improve the trait of living organisms, which has obtained a great success [32].After getting the location of causative gene, the GT first constructs a homologous targeting vector using the deoxyribonucleic acid (DNA) homologous recombination technique.Then, the homologous targeting vector is inserted into the embryonic stem cell to form a mutated embryonic stem cell.This mutated embryonic stem cell can express a much better trait than the original embryonic stem This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/cell since the causative gene has been targeted for modification.In 2007, Capecchi, Smithies, and Evans shared the Nobel Prize in Physiology or Medicine in recognition of their contributions to GT [32].In 2018, half of the Nobel Prize in Chemistry was awarded to Arnold for her contributions in targeting evolution of enzyme [33].In 2020, Doudna and Charpentier won the Nobel Prize in Chemistry for their contributions in clustered regularly interspaced short palindromic repeats (CRISPR/Cas9) gene editing [34].The effects of GT include that some causative genes can be targeted and modified for better performance.The need of targeted modification to bottleneck dimensions in solving LSOPs is very similar to the GT.Therefore, in this article, motivated and inspired by the GT technique, we propose a simple and efficient method, called gene targeting DE (GTDE), to solve the LSOPs.
In the algorithm design, a simple GT-based modification inspired by the GT technique is proposed to perform on the best individual to balance the evolution among all the dimensions.This includes probabilistically targeting the location of causative genes (i.e., bottleneck dimensions), constructing the homologous targeting vector, and inserting the homologous targeting vector into the embryonic stem cell (the best individual).In this way, all the bottleneck dimensions of the best individual can be probabilistically targeted and modified, so that the property of the best individual can be greatly improved.The improved best individual will, in turn, provide a proper guidance to the evolution and push the algorithm to find the global optimum gradually.Note that the GT is only performed on the globally best individual and is just carried out as a simple operator that is added to the standard DE.Therefore, compared with other large-scale optimization algorithms [35]- [39], GTDE not only maintains the simple algorithm structure of standard DE but also avoids the sensitive decomposition strategy and the complicated learning strategy or local search technique, which is simpler to understand and easier to implement.
In the experimental results, GTDE is better than, or at least comparable to other state-of-the-art large-scale optimization algorithms, including the winners of CEC2010, CEC2012, CEC2013, and CEC2018 competitions on largescale optimization, on both CEC2010 and CEC2013 test suites.The experimental results further show the efficiency of GTDE.
The remainder of this article is organized as follows.Section II describes the basic DE algorithm and some related methods for LSOPs.Section III presents the GTDE algorithm in detail.The discussion and analysis of experimental results are shown in Section IV.Finally, the conclusion is drawn in Section V.

A. Basic DE
Assume there is a population with N individuals.DE generates new individuals according to the difference between individuals, using three operations called mutation, crossover, and selection.These operations in each generation are described as follows.
Mutation: In each generation g, each individual x i,g will create its mutant vector v i,g by the mutation operation.Three mutation strategies frequently used are listed as follows: 1) DE/rand/1 2) DE/best/1 3) DE/current-to-best/1 where r1, r2, and r3 are different integers randomly selected from {1, 2, . . ., N}, which are all different from i.The amplification factor F is used to amplify the differential vectors.
x best,g is the best individual in generation g.
It should be noted that some dimensions of v i,g may exceed the boundary constraints.In this case, v i,j,g is randomly generated according to a uniform distribution as where L j and U j are the lower and upper bounds of the jth dimension, respectively, and rand is a random number in the range of [0, 1].
Crossover: After the mutation, a trial vector u i,g is generated through a binomial crossover operation on x i,g and v i,g u i,j,g = v i,j,g , if rand ≤ CR or j = j rand x i,j,g , otherwise where j rand is an integer randomly selected from {1, 2, . . ., D} to ensure that u i,g has at least one dimension different from x i,g , and D is the dimension of problem.The crossover rate CR determines the fraction of u i,g inherited from v i,g .

Selection:
The selection operator is conducted by comparing u i,g with x i,g .The better one is selected into the next generation.For instance, for a minimization problem, the individual with a smaller fitness value survives into the next generation, as where f (x) is the fitness function.

B. Methods for Solving LSOPs
To solve the LSOPs efficiently, two kinds of methods are commonly used: 1) introducing cooperative coevolution (CC) framework in EAs, termed CCEAs and 2) designing new search strategies for EAs to improve the population diversity and avoid the local optima.
Van den Bergh and Engelbrecht [42] first introduced CC in PSO and propose CCPSO-S K .It first divides the whole problem into K equal subproblems, then PSO is used to optimize each subproblem.Based on CCPSO-S K , CCPSO-H K is further proposed [42], which combines PSO with CCPSO-S K , where PSO and CCPSO-S K are optimized alternately.In order to relieve the sensitivity of the subproblem size K, Li and Yao developed CCPSO2 [44], where the subproblem size K is randomly selected from a predefined size pool.
In CCEAs, we wish to place the interdependent dimensions into the same group and categorize the independent dimensions into different groups.Therefore, the decomposition strategy is the most important and hot research for CCEAs.Yang et al. [47] proposed a random grouping scheme and apply this decomposition scheme into the DE (DECC-G).Following the DECC-G, Yang et al. [48] further propose a multilevel CC framework for large-scale optimization (MLCC), where several problem decomposers are constructed based on the random grouping strategy with different group sizes.Differential grouping (DG) [49] uncovers the underlying interaction structure of the decision dimensions and forms subcomponents more accurately, resulting in the DECC-DG.To improve the decomposition strategy, Tang [50] used a ring connection and design a new mutation strategy for DE.Meselhi et al. [51] proposed a novel decomposition strategy to minimize the number of common variables between subproblems.With a CC framework and a hybrid mutation strategy, Deng et al. [52] combined quantum computing with DECC for LSOPs.Different from the decomposition in the original decision space, Ren et al. [53] used singularvalue decomposition in the CC framework and propose an eigenspace divide-and-conquer approach.
CCEAs provide a potential approach and perspective to LSOPs.However, they also bring some limitations and drawbacks.First, CCEAs are very sensitive to the decomposition strategy.Second, many decomposition strategies need extra FEs to detect the correlation among dimensions.Third, CCEAs are only effective to the separable problems, when dealing with the partially separable problems or nonseparable problems, the performance of CCEAs degrades severely.
2) Novel Learning and Updating Strategies for EAs: Due to the above limitations and drawbacks of CCEAs, researchers also consider the nondecomposition method which regards to all the dimensions as a whole.Specifically, they propose some novel search strategies to enhance the population diversity for fully exploration and avoiding local optima.
Cheng and Jin [54] proposed social learning PSO (SL-PSO), where each particle randomly learns from other better particles.Besides, they also present a competitive swarm optimizer (CSO) [55], where two particles are selected for competing, then the loser particle learns from the winner particle.Jian et al. [56] proposed a novel region encoding scheme to help evolutionary computation algorithm evolve fast and use it into SL-PSO to solve LSOP.Li et al. [57] enhanced the CSO by embedding the covariance matrix adaptation evolution strategy.He et al. [58] provided an efficient model to generate mutation vectors from a mixture model for solving LSOPs.Yang et al. [59] proposed a dynamic segment-based predominant learning swarm optimizer (DSPLSO).The loser particle learns from several winner particles using the segmentbased predominant learning strategy, and different winner particles guide the different learning components of the loser particle.Following DSPLSO, Yang et al. [60] developed dynamic level-based learning swarm optimizer (DLLSO).The dynamic multiswarm PSO (DMS-L-PSO) proposed by Zhao et al. [61], generated several subswarms dynamically and frequently to improve the population diversity, also utilizes a Quasi-Newton method named the BFGS local search strategy to further refine the solutions.Wang et al. [62] developed a dynamic grouping learning strategy for distributed PSO, where the size of each group is dynamically changed to balance the diversity and convergence.After that, they further propose adaptive granularity learning distributed PSO [63], where the size of each subpopulation is adaptively changed based on the estimation of the evolutionary state.Lan et al. [64] designed a two-phase learning swarm optimizer (TPLSO), which involves two learning strategies called "mass learning" and "elite learning," respectively.Li et al. [65] also used the two-strategy mechanism and propose adaptive PSO with decoupled exploration and exploitation (APSO-DEE).Yildiz and Topal [66] solved LSOPs using a small population size, which propose micro DE with directional local search (μDSDE).Schoen and Tigli [67] applied a cluster method in DE to determine the time of local search.Incorporated by the cloud computing model, He et al. [68] proposed a Spark-based DE with the group topology model.Guo et al. [69] utilized a rankbased mutation strategy to improve the DE performance in LSOPs.Now, many new swarm intelligence algorithms have been proposed and also applied to LSOPs, such as the sine-cosine algorithm (SCA), whale optimization algorithm (WOA), and social spider algorithm.Li et al. [70] improved SCA by using the nonlinear random convergence parameter and dynamic inertia weight strategy (termed DSCA).Chakraborty et al. [71] enhanced WOA with adaptive parameter mechanism for LSOPs (termed eWOA).Baş and Ulker [72] designed two techniques called "spider blasting" and "explorer spider memory" to improve social spider algorithm in LSOPs.
3) Top Algorithms in CEC Large-Scale Competition: Besides the above algorithms, some top algorithms which join up the large-scale competition are described as follows.Molina et al. [73], [74] proposed a memetic algorithm based on local search chains (MA-SW-Chains), which combines a steady-state GA with a local search method.Such an algorithm later became the winner of the CEC2010 competition [75].In CEC2012 competition (also using the CEC2010 test suite), many advanced algorithms were proposed.LaTorre et al. [76] developed a multiple offspring sampling (MOS) framework, which combines different algorithms to solve LSOPs and is the winner of CEC2012.Self-adaptive DE with adaptive population size (jDElsgo) is proposed by Brest et al. [77], which is ranked the second in CEC2012.CC framework with global search (CCGS) is reported in [78], which is ranked the third in CEC2012.In CEC2013 competition, a new test suite is proposed [79].Subsequently, LaTorre et al. [80] used the MOS framework to solve the CEC2013 test suite, which becomes the

A. GT-Based Modification
The GT technique is first proposed by Capecchi [32] to target modification to the genetic information of living organisms.Specifically, after getting the location of the causative gene, GT modifies the causative gene by constructing the homologous targeting vector and inserting the homologous targeting vector into the embryonic stem cell to form a mutated embryonic stem cell.Since the causative gene is targeted for modification, the mutated embryonic stem cell can express the much better trait.Fig. 1 displays an example of GT.
Inspired by the GT, the GT-based modification is proposed for effectively solving LSOPs by probabilistically targeting and modifying the bottleneck dimensions of individuals.Since the best individual x best,g is much more likely close to the global optimum and we wish to accelerate the approaching speed to the best solution, our GT-based modification is only performed on the x best,g .Specifically, x best,g can be regarded as the embryonic stem cell, whereas the dimension of x best,g can be perceived as the exon gene of the embryonic stem cell.The process of the GT-based modification can be shown as the following three steps.
1) Probabilistically Targeting the Location of Causative Genes (i.e., the Bottleneck Dimensions): Herein, the bottleneck dimensions which limit the individual to find the global optimal solution can be regarded as the causative genes.A Monte Carlo-based detection method is adopted here to check whether the dimension j is a bottleneck dimension with the probability P j .
Monte Carlo method is a class of computational algorithms that rely on repeated random sampling to obtain numerical results [83].Herein, we use the Monte Carlo method to random sample the dimensions to determine the bottleneck Algorithm 1 Target the Bottleneck Dimensions Begin 1.For j = 1 to D 2.
If rand < P j 4.
The j th dimension is targeted as the bottleneck dimension; 5.
End If 6. End For End dimensions.The sample probability P j is drawn from a univariate Gaussian distribution with 0.01 as the mean and 0.01 as the standard deviation for all the dimensions.Specifically, for each dimension, we generate a random value in [0, 1].If the value is smaller than the probability P j , the current dimension can be targeted as the bottleneck dimension.Otherwise, the current dimension is regarded as a normal dimension.
Although this method is simple and easy to use, it is very effective.Moreover, using the Monte Carlo method, all the dimensions have the same chance to be targeted as the bottleneck dimensions.In this way, all the bottleneck dimensions of the best individual can be probabilistically targeted and modified.Therefore, the evolution of different dimensions will be balanced, pushing the global convergence on all dimensions.
The overall procedure of determining the bottleneck dimension is shown in Algorithm 1.We have investigated the efficiency of this Monte Carlo-based bottleneck dimensions detection method and the sensitivity of the probability P j in Section IV-H.
2) Construct the Homologous Targeting Vector (v best,g ): In the GT technique, the homologous targeting vector is constructed using the DNA homologous recombination technique.Since the GT-based modification is only performed on the x best,g , herein, we choose the DE/best/1 mutation strategy in (2) as the homologous recombination to construct the homologous targeting vector v best,g .As for the amplification factor F, we directly adopt the setting as 0.5 since it is suggested in many literature [84], [85].However, to enhance the parameter diversity, the independent parameter is utilized here.Therefore, we use 0.5 as the mean and 0.1 as the standard deviation for a Gaussian distribution to generate the amplification factor F.
Moreover, a new mutation strategy DE/r-best/1 is proposed to extend the DE/best/1 mutation strategy by inserting a new randomly generated individual, shown as DE/r-best/1 is a variant of DE/best/1, the only difference between them is that the new randomly generated individual x rand replaces the randomly selected x r2,g in DE/best/1.
As we can see, the DE/best/1 mutation strategy can keep the evolutionary information of the population since both of the individuals x r1,g and x r2,g are from the population.While the DE/r-best/1 mutation strategy can bring new information of the search space and provide more diversity since it uses the new randomly generated individual x rand out of the population.In our method, the information of the homologous targeting vector (i.e., v best,g ) is constructed by these two mutation strategies according to a probability P m .Specifically, for each dimension, we first generate a random value in [0, 1].
If rand < P m 5.
Randomly generate a new individual x rand ; 6.
Generate the j th dimension of homologous targeting vector v best,j,g using the DE/r-best/1 mutation strategy in (7) and F; 7. Else 8.
Generate the j th dimension of homologous targeting vector v best,j,g using the DE/best/1 mutation strategy in (2) and F; 9.
Fix homologous targeting vector v best,g using (4).11.End For End Algorithm 3 Insert the Targeting Vector Into the x best,g If the j th dimension is the bottleneck dimension 3.
End If Target the bottleneck dimensions using Algorithm 1; 3.
Construct the homologous targeting vector using Algorithm 2; 4.
Insert the homologous targeting vector into the x best,g and generate the new individual x best,g using Algorithm 3; 5.

End For End
If this value is smaller than the probability P m , then the current dimension of x best,g will select the DE/r-best/1 mutation strategy to generate the information of this dimension of the v best,g .Otherwise, the DE/best/1 mutation strategy is chosen.To preserve the evolutionary information of the population, we should pay more attention to DE/best/1 mutation strategy, so that P m is set relatively small as 0.01.
Using this method, we have constructed the homologous targeting vector (v best,g ) using two mutation strategies to complete the second step of GT.As we can see, the whole evolutionary information of the population can be preserved to a great degree since we use the DE/best/1 mutation strategy with a relatively large probability, which can keep the properties of x best,g in exploitation.Moreover, the new randomly generated individual x rand is used and the new mutation strategy DE/r-best/1 is proposed, which can further provide new information of search space and more diversity.
The overall procedure of constructing the homologous targeting vector is shown in Algorithm 2. We have investigated the efficiency of this DE/r-best/1 mutation strategy and the sensitivity of the probability P m in Section IV-H.
3) Insert the Homologous Targeting Vector Into the Embryonic Stem Cell (x best,g ): After generating the homologous targeting vector, it is inserted into the embryonic stem cell.In other words, the bottleneck dimensions in x best,g should be changed by the corresponding dimensions in homologous targeting vector v best,g , while the other dimensions in x best,g will be preserved.The mutated embryonic stem cell, which means the newly generated individual, is termed x best,g .The overall procedure of inserting the homologous targeting vector into x best,g is shown in Algorithm 3.
After these three steps, we have finished a complete process of the GT-based modification and generated a new individual x best,g .If the fitness of x best,g is better than x best,g , x best,g will replace x best,g ; otherwise, x best,g is ignored.The GT-based modification is executed N GT times.The whole framework of the GT-based modification is shown in Algorithm 4.
Using this GT-based modification method, the bottleneck dimensions of the best individual are probabilistically targeted and modified, which can provide a proper guidance to the evolution and push the population toward the global optimum gradually.Moreover, all the dimensions have the same chance to be targeted as the bottleneck dimensions, so that the evolution of different dimensions will be balanced.

B. Evolutionary Process
Since the best individual x best,g can update itself constantly using the GT-based modification, the whole population should also learn from the x best,g to converge to the global optimum gradually.However, due to the massive local optima when dealing with the LSOPs, we should improve the population diversity to avoid being trapped in the local optima at the same time.To this aim, the mutation strategy DE/current-to-best/1 is selected in GTDE, as shown in (3).
As we can see, the second term of (3) is F × (x best,gx i,g ), which means each individual will learn from x best,g to some degree.Meanwhile, it is also disturbed by the third term F × (x r1,g − x r2,g ), which helps maintain and improve the diversity to discover the global optimum sufficiently.
Besides, the performance of DE is very sensitive to the parameter settings (F and CR).Many adaptive parameter mechanisms have been proposed with significant successes, such as adaptive DE with optional external archive [86], self-adapting control parameters in DE [87], and successhistory-based parameter adaptation for DE [88].Here, we adopt a simple independent parameter mechanism rather than the adaptive mechanism to reduce the parameter sensitivity and to enhance parameter diversity.F and CR are generated using Gaussian distribution with μ F and μ CR as the mean, and σ F and σ CR as the standard deviation, respectively.In this way, each individual has its own independent parameters, further increasing the parameter diversity and having many different parameter combinations.The parameter diversity will further achieve the enhancement of search diversity.

C. Complete GTDE Algorithm
Based on the GT-based modification, the pseudocode of the complete GTDE algorithm is outlined in Algorithm 5.
Find out the best individual x best,g ; 7.
For i = 1 to N 8.
If x i,g is the best individual x best,g 9.
Execute GT-based modification on x i,g using Algorithm 4; 10.
Generate mutant vector v i,g using the DE/current-to-best/1 mutation strategy in (3) and F i ; 14.
Generate trial vector u i,g using ( 5) and CR i ; 16.
If f (u i,g ) ≤ f (x i,g ) 17.

End While End
In every generation, we judge whether the current individual x i,g is the best individual x best,g .If x i,g is the best individual x best,g , then x i,g will carry out the GT-based modification using Algorithm 4. Otherwise, x i,g will be updated with the DE/current-to-best/1 mutation strategy and its corresponding independent parameters (F and CR).The procedure is repeated until the maximum number of FEs (MaxFEs) is met.
Compared with other CCEAs, GTDE avoids the sensitive decomposition strategy and the disadvantages of CCEAs, such as the extra FEs in dimension correlation detection and poor performance on partially separable or nonseparable problems.Compared with other non-CCEA large-scale optimization algorithms, the whole GTDE algorithm avoids the complicated learning strategy [54]- [60] (such as the dynamic segmentbased predominant learning in DSPLSO [59]), extra local search techniques [61], [73], [82] (such as the MTS and L-BFGS-B local search techniques in SHADE-ILS [82]), and algorithm component selection [76], [80] (such as the MOS framework, [76], [80]).Compared with the traditional DE, GTDE has almost the same simple algorithm structure, with only adding the GT-based modification as shown in lines 8-10 in Algorithm 5, which is easy and straightforward to understand and implement, showing its simplicity.Moreover, GTDE can probabilistically target and modify the bottleneck dimensions of the best individual and make the evolution of different dimensions balanced, which obtains satisfying experimental results, showing its efficiency.

A. Experimental Setup
In this section, 35 commonly used large-scale optimization benchmark functions in CEC2010 (f 1 −f 20 ) [75] and CEC2013 (F 1 −F 15 ) [79] are used to test the performance of GTDE.For more details about the test functions, refer to [75] and [79], respectively.
The MaxFEs is set as 3 000 000 for all competitors and all functions, the population size N is set as 100 in GTDE, while the frequency of the GT-based modification N GT is set as 400 in every generation in GTDE.The values of μ F and σ F in GTDE are set as 0.7 and 0.5, respectively, while the values of μ CR and σ CR in GTDE are both set as 0.5.For a fair and convincing comparison, the parameters used in the compared algorithms are set the same as their original papers because they have been fine tuned for these large-scale optimization benchmark functions.All the algorithms are run 30 times and the mean results are reported.For clarity, the best results are marked in boldface.Moreover, Wilcoxon's rank sum test at α = 0.05 is used to evaluate whether GTDE is significantly better than (+), similarly to (≈), or significantly worse than (-) the compared algorithm [89], [90].

B. Comparison Results on the CEC2010 Test Suite
The detailed comparison results on the CEC2010 test suite are listed in Table I.From Table I, we can see the following.
For the first three separable functions f 1 − f 3 , GTDE generally outperforms other algorithms, especially on f 1 .
For the next 15 partially separable functions f 4 − f 18 , GTDE achieves the best performance on nine functions (f 4 , f 7 − f 9 , f 12 − f 14 , f 17 , and f 18 ).Conversely, all the competitors cannot achieve the best performance on no more than four functions and cannot outperform GTDE on more than six functions.
For the last two nonseparable functions f 19 and f 20 , the performance of GTDE is still much better than other algorithms.
Overall, GTDE performs better than the competitors on at least 13 functions.Conversely, no competitor can surpass GTDE on more than seven functions.
Next, we draw the convergence curves of different algorithms to study their evolutionary behaviors on all the 20 CEC2010 test functions.The convergence curves of GTDE and compared algorithms on all the 20 benchmark functions are shown in Fig. S.1 in the supplementary material.Herein, for a fair and convincing comparison, we select several benchmark functions from all the three groups, including separable function f 1 , partially separable functions f 4 , f 8 , f 9 , and f 13 , and nonseparable function f 20 , as the representative instances for main discussion.(m), we find that GTDE seems stagnation in the middle of the evolutionary process, but obtains the more satisfying results at last, on partially separable functions f 8 and f 13 .On nonseparable function f 20 shown in Fig. S.1(t), most algorithms get the similar performance.However, GTDE is still more accurate than the other algorithms.
Overall, in the CEC2010 test suite, GTDE achieves the best performance and generally outperforms other large-scale optimization algorithms.

C. Comparison Results on the CEC2013 Test Suite
The detailed comparison results on the CEC2013 test suite are listed in Table II.From Table II, we can see the following.
For the first three separable functions F 1 − F 3 , GTDE can achieve the best performance on F 1 and F 3 .In particularly, on F 1 , only GTDE can find the global optimal solution in all the runs, while other algorithms converge slowly.
For the next eight partially separable functions F 4 − F 11 , GTDE performs the best on five functions (F 4 , F 6 − F 8 , and F 11 ).Conversely, all the competitors cannot achieve the best performance on no more than 3 functions and cannot outperform GTDE on more than three functions.
For the next three overlapping functions F 12 − F 14 , GTDE still outperforms all the other algorithms on all the three functions.
For the last nonseparable function F 15 , GTDE achieves the best performance significantly again.
Overall, GTDE performs better than the competitors on at least 11 functions.Conversely, no competitor can surpass GTDE on more than three functions.
We also draw the convergence curves of different algorithms to study their evolutionary behaviors on all the 15 CEC2013 test functions.The convergence curves of GTDE and the compared algorithms on all the 15 benchmark functions are shown in Fig. S Overall, in the CEC2013 test suite, GTDE achieves the best performance and generally outperforms other large-scale optimization algorithms.

D. Comparison With the Winners of the CEC2010 and CEC2012 Competitions
GTDE is further compared with the winner of the CEC2010, MA-SW-Chains [73], and the top three algorithms of the CEC2012 competition, to illustrate the efficiency and superiority of GTDE.The top three algorithms of the CEC2012 competition are MOS [76], jDElsgo [77], and CCGS [78].For a fair and convincing comparison, we directly cite the final results from their original papers [73], [76]- [78].The detailed comparison results between GTDE and the compared algorithms are listed in Table III.Since we only have the final results of the compared algorithms [73], [76]- [78], whether GTDE performs better than (+), similarly to (≈), or worse than (-) the compared algorithms is just measured by the mean results.From Table III  functions, GTDE performs better than all these three algorithms on all the two nonseparable functions.Overall, we see that GTDE is better than the top algorithms in the CEC2012 competition.

E. Comparison With the Winners of the CEC2013 and CEC2018 Competitions
In this section, we further compare GTDE with the winner of the CEC2013 competition, MOS [80], and the top three algorithms of the CEC2018 competition, to further evaluate GTDE.The top three algorithms of the CEC2018 competition are SHADE-ILS [82], MLSHADE-SPA [81], and MOS [80].Similarly, for a fair comparison, we directly cite the final results of the compared algorithms from the original paper [80]- [82].Whether GTDE performs better than (+), similarly to (≈), or worse than (-) these three algorithms is just measured by the mean results.The detailed comparison results between GTDE and the compared algorithms are listed in Table IV.From Table IV, we find the following.
1) When compared with MOS, GTDE still keeps its promising performance.GTDE has similar or even better performance than MOS on separable functions, such as F 1 and F 2 .MOS may have the superiority and performs better than GTDE on the partially separable functions.However, GTDE performs significantly better than MOS on all the overlapping functions (F 12 − F 14 ) and nonseparable function (F 15 ).Overall, GTDE dominates MOS on eight functions, while only dominated by MOS on six functions.Moreover, GTDE is a simple enhanced version of DE, being much simpler and easier to understand and implement, when compared with MOS.MOS is a framework that combines several different algorithms to generate offspring.For different large-scale benchmark function sets, MOS needs to seek and find the suitable algorithms to work together and the proper parameter settings for each adopted algorithm to optimize a specific problem or a specific problem set.Compared with MOS, GTDE is much easier for implementation and not sensitive to the parameter configuration, fully facilitating its practical application.Overall, GTDE can achieve better performance than the winner of the CEC2013 competition.Moreover, GTDE is much simpler and easier to understand and implement, which is another advantage over MOS.
2) When compared with SHADE-ILS and MLSHADE-SPA, GTDE also keeps its promising performance.GTDE performs much better than these two algorithms on separable functions, especially on F 1 and F 2 .SHADE-ILS cannot outperform GTDE on any separable function.As for the partially separable functions, GTDE achieves the similar performance compared with MLSHADE-SPA, where GTDE dominates MLSHADE-SPA on four functions and is dominated by MLSHADE-SPA on four functions.SHADE-ILS may have the superiority and performs better than GTDE on the partially separable functions, however, GTDE performs significantly better than these two algorithms on the nonseparable function (F 15 ).When dealing with overlapping functions, GTDE performs better than MLSHADE-SPA on F 13 and F 14 , performs better than SHADE-ILS on F 14 .Overall, GTDE dominates MLSHADE-SPA and SHADE-ILS on nine and six functions, respectively, while dominated by MLSHADE-SPA and SHADE-ILS on six and eight functions, respectively.Although GTDE performs a little worse than SHADE-ILS, we can see that GTDE performs much better than SHADE-ILS on separable functions and nonseparable function.In particularly, on F 1 , GTDE can find the global optimum 0.00E + 00 in all the runs, while SHADE-ILS cannot.On F 2 , GTDE achieves the mean result with 7.54E + 01, which is much better than the mean result in SHADE-ILS with 1.00E + 03.Even on the functions where SHADE-ILS dominates GTDE, GTDE still obtains the close performance to SHADE-ILS.For example, on F 10 −F 13 , GTDE achieves the mean results with 9.26E + 07, 5.74E + 05, 1.19E + 02, and 1.58E + 05, respectively, which is very similar to the mean results in SHADE-ILS with 9.18E + 07, 5.11E + 05, 6.18E + 01, and 1.00E + 05, respectively.Overall, we can see that GTDE is competitive or even better than the top algorithms of the CEC2018 competition.

F. Scalability of GTDE on 2000-D Problems
We further conduct the experiments on the CEC2010 test functions with dimension increasing to 2000 to investigate the scalability of GTDE.Herein, the MaxFEs is set as 6 000 000 for all algorithms.The population size N is set as 200 in GTDE, while the frequency of the GT-based modification N GT is set as 800 in every generation in GTDE.The detailed experimental results can be seen in Table S.I in the supplementary material.
From Table S.I, we can see that with the dimension increases, the performance of many algorithms degrades severely, except GTDE.Moreover, GTDE still keeps its promising performance and superiority on the 2000-D problems.GTDE outperforms the competitors on at least 13 functions.Conversely, no competitor can surpass GTDE on more than seven functions.These results fully demonstrate the efficiency and scalability of GTDE.

G. Effects of the GTDE Components
The main component of GTDE is the GT-based modification.Herein, we will discuss the property and effect of the GT-based modification.
To investigate the effect of GT-based modification, we compare GTDE with its variant without GT-based modification.Herein, we denote the GTDE variant without GTbased modification as GTDE-noGT.The GTDE-noGT is just removing lines 8-10 in Algorithm 5, which is similar to the traditional DE.Moreover, as some kinds of local search techniques used in μDSDE [66], DMS-L-PSO [61], and SHADE-ILS [82] also deal with the modification of dimensions when solving LSOPs, we further compare the GTbased modification and the local search-based modification herein.Specifically, in μDSDE [66], the local search technique is conducted by using a directional vector, where the individual will constantly update itself by adding this directional vector until the directional vector cannot improve the individual.In DMS-L-PSO [61] and SHADE-ILS [82] (the winner of the CEC2018 large-scale competition), they both use the BFGS local search technique.In some multimodal optimization algorithms [17], [18], the local search technique is conducted by sampling some points around the best individual using Gaussian distribution.Herein, we compare GTDE with its variants using different local search techniques to verify the efficiency of the GT-based modification.We replace the GT-based modification by the local search techniques, such as using directional vector, Gaussian distribution, and BFGS, and term them as GTDE-directional, GTDE-Gaussian, and GTDE-BFGS, respectively.The detailed comparison results among GTDE, GTDE-noGT, and GTDE with its variants using different local search techniques on the CEC2010 test suite are listed in Table S.II in the supplementary material.From Table S.II, we find that GTDE is much better than GTDE-noGT, since GTDE outperforms GTDE-noGT on 15 functions, while only dominated by GTDE-noGT on five functions.The experimental results also show that the GT-based modification is more suitable than other local search techniques.GTDE outperforms GTDE-directional, GTDE-Gaussian, and GTDE-BFGS on 15, 15, and 14 functions, respectively, while only performs worse than these variants on five, five, and six functions, respectively.Therefore, the GT-based modification is efficient and more suitable than the other local search techniques.
Next, GTDE is further tested to see how it can reduce the number of bottleneck dimensions.We investigate the number of bottleneck dimensions during the evolutionary process in GTDE.In order to determine the bottleneck dimension, herein, we first denote the region, where the optimum as the center, 5% of the search space, as the quasi-optimal region.The dimension lies in the quasi-optimal region is regarded as the feasible dimension; otherwise, the dimension is regarded as the bottleneck dimension.For example, assume that the range of the jth dimension is [−100, 100] and its optimum is 10.Then, if the jth dimension of x best,g lies in the range [10 − 5% × 200, 10 + 5% × 200], i.e., [0,20], the jth dimension is regarded as the feasible dimension, otherwise, the jth dimension is regarded as the bottleneck dimension.We adopt this bottleneck dimension measurement to measure the distance between a solution and the optimum, since the bottleneck dimensions in this article refer to the dimensions which have not been optimized well yet.We analyze the evolution behavior of the average number of bottleneck dimensions over 30 times in both GTDE and GTDE-noGT.In Fig. S.3 in the supplementary material, we plot the variation of the average number of bottleneck dimensions on all the 20 functions as the algorithm progresses.Herein, we select separable function f 3 , partially separable function f 8 , and nonseparable function f 20 as the representative instances for main discussion.
As we can see, in f 3 and f 8 shown in Fig. S.3(c) and (h), GTDE can target the bottleneck dimensions for modification and the number of bottleneck dimensions decreases gradually and finally becomes to 0. However, in GTDE-noGT, the number of bottleneck dimensions decreases slowly and finally stagnates at about 600 and 100, respectively.In f 20 shown in Fig. S.3(t), the number of bottleneck dimensions finally decreases to 0 in both GTDE and GTDE-noGT.However, GTDE achieves a much faster convergence speed than GTDE-noGT.Therefore, the GT-based modification in GTDE is very helpful and useful to different problems.

H. Sensitivity of the GTDE Parameters
We first investigate the sensitivity of probability P j .We compare GTDE with its variants on different values of P j .In GTDE, P j is set as the Gaussian distribution with 0.01 as the mean and 0.01 as the standard deviation.Herein, P j is tested by Gaussian distribution with 0.01 as the standard deviation and 11 values as the mean, i.e., 0.001, 0.003, 0.005, 0.009, 0.03, 0.05, 0.09, 0.1, 0.3, 0.5, and 0.9.We denote the GTDE variant with a different P j = Gaussian(a, 0.01) as GTDE-j(a).The detailed comparison results between GTDE and its variants on different P j settings on the CEC2010 test suite are listed in Table S.III in the supplementary material.
From Table S.III, we see that GTDE is still much better than its variants.It appears that the smaller P j setting has a better performance than larger P j setting, especially on f 1 − f 4 .However, the larger P j setting also achieves the satisfying results on f 5 , f 6 , f 15 , and f 16 .Even so, GTDE with P j as Gaussian(0.01,0.01) still outperforms GTDE-j(0.03),GTDEj(0.05),GTDE-j(0.09),GTDE-j(0.1),GTDE-j(0.3),GTDEj(0.5), and GTDE-j(0.9)on at least 13 functions, respectively, while these variants cannot outperform GTDE on more than seven functions.Note that the P j with mean as 0.001, 0.003, 0.005, 0.009, and 0.01 offer similar performance, which indicates that smaller P j settings are more suitable for LSOPs and GTDE is not very sensitive to the parameter P j with smaller settings.Therefore, P j with a relatively smaller setting [such as P j = Gaussian(0.01,0.01)] is a suitable setting for GTDE.
We further investigate the sensitivity of probability P m .We compare GTDE with its variants on different values of P m .In GTDE, P m is set to 0.01 and is tested against 13 values, i.e., 0, 0.001, 0.003, 0.005, 0.009, 0.03, 0.05, 0.09, 0.1, 0.3, 0.5, 0.9 and 1. Herein, we denote the GTDE variant with a different P m = b as GTDE-m(b).Note that the GTDE-m(0) means only mutation strategy DE/best/1 is used, GTDE-m(1) means only mutation strategy DE/r-best/1 is used.The detailed comparison results between GTDE and its variants with different P m settings on the CEC2010 test suite are listed in Table S.IV in the supplementary material.
From Table S.IV, we find that generating v best,g using only one strategy DE/best/1 [i.e., GTDE-m(0)] can still obtain satisfying results, while using two strategies DE/best/1 and DE/r-best/1 can bring better results, since GTDE outperforms GTDE-m(0) on ten functions, especially on f 2 , f 3 , and f 8 .That is, because the DE/r-best/1 mutation strategy can bring new information of the search space and will further provide more diversity.However, using DE/r-best/1 too frequently with larger P m cannot preserve the evolutionary information of the population, which results in the poor performance of GTDE-m(0.1),GTDE-m(0.3),GTDE-m(0.5),GTDE-m(0.9),and GTDE-m(1).Smaller P m seems to have a better performance than larger P m , especially on f 2 , f 6 , f 12 , and f 13 , however, GTDE with P m = 0.01 still performs better than GTDE-m(0), GTDE-m(0.03),GTDE-m(0.05),GTDEm(0.09),GTDE-m(0.1),GTDE-m(0.3),GTDE-m(0.5),GTDEm(0.9), and GTDE-m(1) on at least ten functions, respectively, while no variant can outperform GTDE on more than 7 functions.Note that the P m with 0.001, 0.003, 0.005, 0.009, and 0.01 offer similar performance, which indicates that smaller P m settings are more suitable for LSOPs and GTDE is not very sensitive to the parameter P m with smaller settings.Therefore, P m with a relatively smaller setting (such as P m = 0.01) is suitable for GTDE.
Besides, there are some other parameters in the whole GTDE algorithm, such as the population size N, the frequency of the GT-based modification N GT , and the mean values of μ F and μ CR .Here, we investigate the sensitivities of these four parameters.
1) Sensitivity of Population Size N: A larger N provides a higher diversity, while a smaller N saves the number of FEs to prolong the evolution iterations.We compare GTDE with its variants on different values of N. In GTDE, N is set to 100 and is tested against three values, i.e., 50, 200, and 300.We denote the GTDE variant with a different N = c as GTDE-N(c).The detailed comparison results between GTDE and its variants on three different N settings on the CEC2010 test suite are listed in Table S.V in the supplementary material.
From Table S.V, we see that GTDE is still much better than its variants.It appears that a smaller N setting delivers better performance than a larger N does, especially on f 18 − f 20 .However, too small an N, such as N=50 in GTDE-N(50), may result in the lack of diversity, which cannot perform well on f 5 and f 6 .Even so, GTDE with N=100 still outperforms GTDE-N(50), GTDE-N(200), and GTDE-N(300) on 10, 11, and 13 functions, respectively, while no variant can outperform GTDE on more than eight functions.Therefore, N = 100 is the most suitable setting for GTDE to perform well in this situation.
2) Sensitivity of Frequency N GT : We compare GTDE with its variants on different values of N GT .A larger N GT can further refine the accuracy of the best individual, but consumes more FEs.In GTDE, N GT is set to 400 and is tested against three values, i.e., 200, 300, and 500.We denote the GTDE variant with a different N GT = d as GTDE-T(d).The detailed comparison results between GTDE and its variants with different N GT settings on the CEC2010 test suite are listed in Table S.VI in the supplementary material.
From Table S.VI, we see that a larger N GT delivers better performance than a smaller N GT does, especially on f 12 , f 13 and f 17 .However, a smaller N GT saves the number of FEs, which still offers satisfying results on f 4 .GTDE with N GT = 400 performs better than GTDE-T(200), GTDE-T(300), and GTDE-T(500) on 12, 10, and 8 functions, respectively, while worse than these variants only on four, five, and six functions, respectively.Therefore, N GT = 400 is the most suitable setting for GTDE to achieve the more accuracy results and require fewer FEs at the same time.
3) Sensitivity of the Mean Value μ F : Since the performance of DE is sensitive to parameters F and CR, we assess the sensitivities of mean values μ F and μ CR .We first compare GTDE with its variants with different values of μ F .In GTDE, the μ F is set to 0.7 and is tested against four values, i.e., 0.1, 0.3, 0.5, and 0.9.We denote the GTDE variant with a different μ F = e as GTDE-F(e).The detailed comparison results between GTDE and its variants with different μ F settings on the CEC2010 test suite are listed in Table S.VII in the supplementary material.
4) Sensitivity of the Mean Value μ CR : We now compare GTDE with its variants with different values of μ CR .In GTDE, the μ CR is set to 0.5 and is tested against four values, i.e., 0.1, 0.3, 0.7, and 0.9.We denote the GTDE variant with a different μ CR = f as GTDE-CR(f).The detailed comparison results between GTDE and its variants with different μ CR settings on the CEC2010 test suite are listed in Table S.VIII in the supplementary material.
From Table S.VIII, we see that a larger μ CR generally performs better than a smaller μ CR , especially on f 18 − f 20 .Smaller μ CR settings may be suitable for f 5 and f 6 .GTDE with μ CR = 0.5 performs better than GTDE-CR(0.1)and GTDE-CR(0.3)on 14 and 13 functions, respectively, while worse than these variants only on four and five functions, respectively.Note that the μ CR with 0.5, 0.7, and 0.9 offer similar performance, which indicates that larger CR settings are more suitable for LSOPs and GTDE is not very sensitive to parameter μ CR with larger settings.Therefore, μ CR with a relatively larger setting (such as μ CR = 0.5) is appropriate for GTDE.

V. CONCLUSION
Inspired by the technique of GT, this article developed a simple and efficient GTDE in solving LSOPs.
In the algorithm design, the GT-based modification was developed to perform on the best individual to balance the evolution among all the dimensions, including probabilistically targeting the location of bottleneck dimensions, constructing the homologous targeting vector, and inserting the homologous targeting vector into the best individual.In this way, all the bottleneck dimensions of the best individual were probabilistically targeted and modified, so that the property of the best individual can be greatly improved.The improved best individual will, in turn, provide a proper guidance to the evolution and pushing the population to find the global optimum gradually.Compared with other large-scale optimization algorithms, GTDE not only maintains the simple algorithm structure of standard DE but also avoids the sensitive decomposition strategy and the complicated learning strategy or local search technique, which is much simpler to understand and more accessible to implement.
In the experimental results, GTDE generally outperformed the state-of-the-art large-scale optimization algorithms, including the winners of CEC2010, CEC2012, CEC2013, and CEC2018 competitions on large-scale optimization.The comparison results fully showed the efficiency of GTDE.
In summary, GTDE is a simpler and more efficient method to solve LSOPs.Future work will hence focus on its realworld applications.In addition, we will also extend methods for measuring bottleneck dimensions to benefit more largescale optimizers.

Fig. S. 1 (
a) shows that only GTDE and MLCC have a faster convergence speed and find the global optimum finally on separable function f 1 , while other algorithms evolve slower.However, GTDE can find the global optimum more quickly than MLCC.On partially separable functions f 4 and f 9 , shown in Fig. S.1(d) and (i), GTDE still keeps a faster convergence speed and gets more accurate results.From Fig. S.1(h) and .2 in the supplementary material.Similarly, we select several benchmark functions from all the four groups, including the separable function F 1 , partially separable functions F 4 , F 7 , and F 8 , overlapping function F 14 , and nonseparable function F 15 , as the representative instances for main discussion.From Fig. S.2(a), we can see that only GTDE has a faster convergence speed than other algorithms and finally finds the global optimum on separable function F 1 .On partially separable functions F 4 and F 8 , shown in Fig. S.2(d) and (h), GTDE still keeps a faster convergence speed and gets more accurate results.From Fig. S.2(g), we find that GTDE converges to the result with 1E + 00, which is much more satisfying than other algorithms with results from 1E + 07 to 1E + 13, on partially separable functions F 7 .On overlapping function F 14 in Fig. S.2(n), GTDE shows its great advantage at the late state of the evolutionary process, which gets much better results finally.From Fig. S.2(o), it can be seen that all the algorithms get the similar performance on nonseparable function F 15 , but GTDE still obtains more accurate results.

TABLE III EXPERIMENTAL
RESULTS BETWEEN THE GTDE AND TOP ALGORITHMS IN CEC2010 AND CEC2012 COMPETITIONS TABLE IV EXPERIMENTAL RESULTS BETWEEN THE GTDE AND TOP ALGORITHMS IN CEC2013 AND CEC2018 COMPETITIONS