Local Binary Pattern-Based Adaptive Differential Evolution for Multimodal Optimization Problems

The multimodal optimization problem (MMOP) requires the algorithm to find multiple global optima of the problem simultaneously. In order to solve MMOP efficiently, a novel differential evolution (DE) algorithm based on the local binary pattern (LBP) is proposed in this paper. The LBP makes use of the neighbors’ information for extracting relevant pattern information, so as to identify the multiple regions of interests, which is similar to finding multiple peaks in MMOP. Inspired by the principle of LBP, this paper proposes an LBP-based adaptive DE (LBPADE) algorithm. It enables the LBP operator to form multiple niches, and further to locate multiple peak regions in MMOP. Moreover, based on the LBP niching information, we develop a niching and global interaction (NGI) mutation strategy and an adaptive parameter strategy (APS) to fully search the niching areas and maintain multiple peak regions. The proposed NGI mutation strategy incorporates information from both the niching and the global areas for effective exploration, while APS adjusts the parameters of each individual based on its own LBP information and guides the individual to the promising direction. The proposed LBPADE algorithm is evaluated on the extensive MMOPs test functions. The experimental results show that LBPADE outperforms or at least remains competitive with some state-of-the-art algorithms.


Local Binary Pattern-Based Adaptive Differential Evolution for Multimodal Optimization Problems
Hong Zhao, Student Member, IEEE, Zhi-Hui Zhan , Senior Member, IEEE, Ying Lin , Member, IEEE, Xiaofeng Chen , Xiao-Nan Luo , Jie Zhang, Sam Kwong , Fellow, IEEE, and Jun Zhang , Fellow, IEEE D IFFERENTIAL evolution (DE) is a kind of evolutionary algorithm (EAs) proposed by Storn and Price in 1995 [1].Like other EAs, DE is a class of heuristic optimization algorithms that includes mutation, crossover, and selection operators.The DE algorithm is efficient for solving many real-world optimization problems [2]- [5].
With the increasing complexity of the real-world problems, many problems have multiple optimal solutions, namely, multimodal optimization problems (MMOPs).Over the past decades, MMOPs have drawn considerable attention [6].Many researchers try to use EAs [7] and swarm intelligence algorithms [8] to solve MMOPs, such as the genetic algorithm [9], ant colony optimization [10], estimation of distribution algorithm [11], particle swarm optimization (PSO) [12], and DE [13].Although tremendous efforts have been put into utilizing the above algorithms to solve MMOPs, there are still many drawbacks as discussed in [14].First, how to efficiently form niches to locate as many peaks as possible is still a challenge.Second, the parameters of the algorithm are still difficult to set for balancing the exploration and exploitation in solving MMOPs.Third, it is still difficult to maintain the found optima until the end of the evolution.
When dealing with the above difficulties, the existing methods have the following limits.First, some of the existing methods try to divide the population into several separate niches.However, it is difficult to determine the niche parameters, such as the number of niches and their sizes.Second, if we use only the local information from the niche in the evolutionary operator to guide the individuals, the algorithms may get trapped in local optima.This problem would be more serious especially by the influence of the structural bias of the algorithms [15], [16].Third, some of the existing methods adopt fixed parameters during the entire evolution, which is not efficient to balance the exploration and exploitation abilities in solving MMOPs.Although self-adaptive parameters may sometimes tend to have weak exploration ability and are 2168-2267 c 2019 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.
more prone to premature convergence [17], they can be efficient if the diversity is well addressed by cooperating with the niche information.Therefore, there is a great need to design an efficient niching method that can form niches without sensitive parameters to find as many peaks as possible.Moreover, an efficient evolutionary operator and an adaptive parameter control strategy that cooperate with the niche strategy are in great need to balance the exploration and exploitation abilities to refine the found peaks and to maintain them during the entire evolution process.
To these aims, this paper borrows the idea of the local binary pattern (LBP) operator from image processing to the optimization domain for solving MMOPs efficiently.Similar to the multiple peaks detection in MMOPs, researchers also want to find all of the regions of interest in a picture in the image processing.LBP is a simple yet efficient multiresolution approach used in the image processing [18], which can process the grayscale and rotation-invariant texture classification based on the nonparametric discrimination of prototype distributions.Meanwhile, LBP uses the local pattern information to identify multiple textures in order to identify multiple objects, which can be analogous to locate multiple peaks in MMOPs.Therefore, by borrowing this idea, a novel LBP-based niching strategy is proposed, which also uses the local information of each individual to help the individual construct niche.In this way, more peak regions can be detected because all of the individuals can form their own LBP niches.Moreover, the current individual can be guided by both the local information from niche and the global information from the population, in order to prevent individuals from being trapped into local optima.Furthermore, we can adaptively control the relevant parameters according to the neighbors' information from the LBP niche to reduce the limitation of fixed parameters.
Inspired by the aforementioned motivations, we incorporate the LBP-based niching strategy in DE and propose an LBPbased adaptive DE (LBPADE) in this paper.The proposed LBPADE algorithm has the following three advantages.
1) A novel niching strategy based on LBP is designed to identify as many peaks as possible.This strategy avoids the sensitive niching parameters, such as the number of niches and their sizes.2) A new mutation strategy called the niching and global interaction (NGI) mutation strategy is developed to guide the individual to a more promising position by using both the local information of the LBP niche and the global information of the entire population.3) Instead of using fixed parameters in DE, this paper proposes a new adaptive parameter strategy (APS) for each individual according to its fitness and distribution information about the LBP niche, to relieve the sensitivity of parameters like scaling factor F and crossover rate CR.The LBP-based niching strategy can help the LBPADE to locate as many peaks as possible, while the NGI and APS based on the LBP niche can help the LBPADE to balance the exploration and exploitation abilities efficiently.The numerical experiments are conducted on all 20 widely used multimodal benchmark functions in CEC'2013.The experimental results show that the LBPADE is superior to other algorithms in comparison.
The remainder of this paper is structured as follows.Section II describes the basic DE algorithm and the related works on MMOPs.Section III presents the details of the proposed LBPADE.Next, Section IV shows the extensive experimental studies.Finally, Section V draws the conclusion of this paper.

A. DE
The basic idea of DE is to generate new individuals through the difference between individuals and select the better individual to enter the next generation.The standard DE process includes the following steps.
1) Initialization: The initial population is randomly generated within a given boundary domain as where i = 1, 2, . . ., NP and j = 1, 2, . . ., D. Herein, NP represents the population size, and D is the problem dimension.rand(0, 1) is a random number uniformly distributed in the interval of (0, 1), and L j and U j denote the lower and upper bounds of the jth dimension, respectively.
2) Mutation Operator: At each generation, a mutation vector v i is obtained based on the difference between individuals.Here, we list some typical mutation strategies as follows. DE/rand/1: DE/best/1: DE/current-to-best/1: DE/current-to-rand/1: DE/current-to-pbest/1 (without archive): DE/current-to-pbest/1 (with archive): where , . . ., NP}, and F is the scaling factor.x best is the best individual that has the best fitness value in the population.x p best is randomly chosen as one of the top 100p% individuals in the current population with p ∈ (0, 1).x * r2 is randomly chosen from the union of P and A, where P is the set of current population and A is the set of archived inferior solutions [19].
3) Crossover Operator: After the mutation operation, the crossover operator is performed on the individuals v i and x i to form a trial vector u i = [u i,1 , u i,2 , . . ., u i,D ].Herein, we describe two typical crossover operators, namely, the binomial crossover and exponential crossover.The binomial crossover is used in this paper.
In binomial crossover, each dimension of u i is separately determined to come from v i or x i by a crossover rate CR as where rand(0, 1) returns a random number between 0 and 1, while the jrand is a random index in {1, 2, . . ., D} to ensure that at least one dimension of u i comes from v i .
In exponential crossover, L consecutive dimensions come from v i as where dimension k is randomly selected from {1, 2, . . ., D}.
Then, all of the following L dimensions come from v i .Note that <•> D means modulo D and returns D if the result is 0.Moreover, L is the length of the sequence, which is no longer than D and is determined by the crossover rate CR as shown in Algorithm S.I in the supplementary material.

4) Selection Operator:
The selection operator is conducted by comparing the objective values of the original individual x i and the trial vector u i using (10) for a maximization problem, where the better one is selected for the next generation The DE repeats the above mutation, crossover, and selection operators until it meets the terminal conditions.The pseudocode of a traditional DE variant called DE/rand/1/bin is given in Algorithm S.II in the supplementary material.

B. Related Works on MMOPs
The algorithm for solving MMOPs is required to maintain the diversity of population to find as many peaks as possible.Moreover, due to the limited budget on the fitness evaluations (FEs), the algorithm is also required to converge fast in each peak region.To better review the related works on these efforts to solve MMOPs, we attempt to describe them in three aspects.
1) Niching Strategies: In order to localize as many peaks as possible, a fruitful research line is based on the usage of niching schemes [20], [21].The two most famous niching methods are the crowding method [20] and the speciation method [21].The crowding DE (CDE) compares the fitness of an offspring individual with the nearest parental individual of the crowd formed by some parental individuals.The offspring will replace its nearest parental individual if it has a better fitness value.Otherwise, the offspring will be ignored.The speciation DE (SDE) solves MMOPs by evolving multiple species, each of which evolves independently around a peak region to locate more global optima.However, both of the two niching methods introduce additional parameters, that is, the crowding size in crowding and the species radius in speciation, which are problem-dependent and highly sensitive to the algorithm performance.
To reduce the influence of parameters in niching methods, some improved niching methods have been proposed.Li [22] proposed using parameter-free ring topology according to the index of the individual for niching, resulting in R2PSO and R3PSO.Gao et al. [23] introduced a selfadaptive cluster-based DE (Self-CCDE) for MMOPs.It adopted the multipopulation strategy to locate different optima, and employed the self-adaptive parameter control to enhance searchability Therefore, designing an efficient niching method that can form niches without sensitive parameters is in great need.To this aim, this paper proposes an LBP-based niching method where the neighbors that form the niche can be determined similar to the idea of the LBP operator in image processing.Moreover, our proposed APS based on the LBP information can be adaptive control of the parameters in DE.The proposed LBPADE algorithm will be compared with the above algorithms with different niching strategies.
2) New Evolutionary Operators: Since MMOP requires the algorithm to find all of the peaks and refine their accuracy, the population diversity and convergence ability are both very important.Therefore, many new evolutionary operators have been proposed to combine with the niching strategy.Using the clustering methods to initialize population can increase the diversity [24] and may be good in dealing with MMOPs.Qu et al. [25] proposed a neighborhood mutation strategy to design the neighborhood CDE (NCDE) and neighborhood SDE (NSDE) algorithms.These algorithms are promising in maintaining the multiple optima found during evolution.Qu et al. [26] also proposed a distance-based locally informed particle swarm (LIPS) algorithm, which utilized several local bests to guide the evolution of particles.Biswas et al. [27] introduced an improved parent-centric normalized neighborhood mutation operator for DE (PNPCDE), which contained a niching scheme by combining the parent-centric mutation operator with the crowding replacement rule.They also introduced an information-sharing mechanism among the individuals to enhance the niching behavior and proposed the LoICDE algorithm [28].
Therefore, how to design an efficient evolutionary operator to balance the population diversity and convergence ability is significant in solving MMOP.To this aim, this paper proposes the NGI mutation strategy that can combine the niche information and global information to balance the exploration (diversity) and exploitation (convergence) abilities of the algorithm.The proposed LBPADE algorithm will be compared with the above algorithms with different evolutionary operators.
3) Multiobjective Techniques: In recent years, some researchers also transformed MMOPs into multiobjective optimization problems [29]- [31].Generally, the first objective can be the multimodal function itself and the second objective is a specially designed function.Cheng et al. [32] proposed an estimation of the potential optimal areas method, which utilized an adaptive diversity indicator as the second optimization objective.Basak et al. [33] defined mean distance-based selection as the second objective.Differently, a very efficient way was proposed by Wang et al. [34] where the MMOP was transformed into a multiple objective optimization problem whose two objectives are designed according to the definition of multiobjective optimization, so that the two objectives conflict.This is the MOMMOP algorithm and shows general better performance than other multiobjective-based methods.Therefore, the MOMMOP algorithm will be adopted to compare with our proposed LBPADE algorithm.

III. LBPADE
In this section, we first introduce the detailed process of the niching strategy based on LBP.Next, we describe the NGI mutation strategy, the new APS based on LBP, and a boundary constraint strategy (BCS).Finally, the complete LBPADE algorithm is derived.

A. LBP Operator in Image Processing
LBP is widely used to extract the local texture feature of the image in image processing [18].More specifically, LBP marks the difference between the center-point pixel and its neighborhood pixel by a threshold value, where the neighborhood is defined as a window of 3 × 3. Therefore, a local area of LBP has nine pixels.LBP uses the gray value of the window center as a threshold, and compares the gray value of the center with the other eight adjacent pixels.As shown in Fig. 1, if the gray value of the surrounding pixel is not smaller than that of the center pixel, the pixel is marked as 1; otherwise 0. The LBP in the window is used to reflect the texture information of this local area.Using the LBP operator, we can find the edge of the image in the local area, so we can identify different objects in an image.
Similar to the idea of LBP in image processing, we can also describe the information of the neighbors around each individual in DE when solving MMOPs.That is, we regard each individual as a pixel.Each individual (pixel), together with its similar M individuals (M nearest surrounding pixels), forms a niche (local area).Herein, the similarity is measured by the Euclidean distance to simulate the surrounding information, and M can be set as 8 due to the original LBP being within a 3 × 3 window with 8 surrounding pixels.We also regard the fitness value of each individual as the gray value of each pixel.Then, we can compare the fitness value of each individual in the current niche with the fitness value of the current (center) individual.For example, for maximization MMOPs, if a neighboring individual has a fitness value larger than or equal to the current individual (i.e., the neighboring individual has a better fitness value), it is marked with 1. Otherwise, it is marked with 0. Then, we introduce an external archive set S to store all of the individuals marked with 1 in the current niching.Note that each individual forms its own LBP-based niche and has its own set S to store the individuals in the current niche that are equal or better than itself.In this way, the current individual can learn from the better individuals in S via a novel-designed NGI mutation strategy, which will be discussed in the following section.

B. NGI Mutation Strategy
In classic DE, the differential vector(s) in the mutation operator are generated by individuals randomly selected from the entire population, without concerning the distance among them.Even individuals far from each other can be used to produce the differential vector(s).Such a mutation operator might not be suitable for MMOPs, because it may slow down the convergence toward the global optima in each peak region, despite not increasing the overall population diversity.In many DE algorithms specifically designed for MMOPs, the mutation strategy is modified to make use of the niching information [20], [21], that is, the differential vector(s) in mutation are generated by individuals from the same niche.These methods are helpful in locating more peaks, but may also increase the risk of getting trapped into local optima due to the potential loss of population diversity.
In order to accurately locate the regions with optimal solutions and maintain global searchability, an NGI mutation strategy is proposed, which combines the local information of the niche and the global information of the entire population.This strategy not only increases the diversity of the population but also enhances the convergence of the population.That is, for an individual x i , the NGI mutation strategy is as where M is the set that stores the M neighbors, N is the set that stores the entire population, and |N| = NP.The set S stores the individuals that have an equal or better fitness value than the value of f r 1 and r 2 are randomly selected from the set M, while g_r 1 and g_r 2 are randomly selected from the set N, and x nbest is the best individual in the current niche.
In our proposed NGI mutation strategy, there are two situations.One situation is that |S| ≥ 1, meaning that better individual(s) exit in the niche, which can guide the current individual x i .In this case, x nbest is used to guide x i , so that the individual x i can locate the potential optima quickly.Meanwhile, in order to prevent individuals from being trapped into local optima, the global disturbance is also added in the NGI mutation strategy, that is, g_r 1 and g_r 2 are randomly selected from the entire population.This situation is shown as the if statement of (11).The other situation is that |S| = 0, meaning that the current individual x i is the best in the LBPbased niche.In this case, x i may be close to the optimum.Therefore, we randomly select two individuals from the set M to generate local exploitation information for x i .This way, we can improve the convergence speed to the current potential optimum.

C. LBP-Based APS
In this section, to balance the exploration and exploitation abilities, a method of APS is proposed.APS can adjust the parameters based on the information of the fitness and the LBP niche of each individual.
1) Adaptive Parameter F: According to the LBP-based niche, we utilize the neighbor's information of each individual to find the promising direction of evolution in MMOPs.Generally speaking, when |S| is larger, the number of better individuals (the fitness is equal to or better than current individual x i ) is large, and x i needs to learn more from these better individuals, so F i should be larger.On the contrary, if |S| is smaller, the fitness of the current individual is promising, and the number of better individuals is less, so x i does not need too much learning, and F i should be smaller.Therefore, the value of |S|/|M| can be used to guide the learning scale (i.e., the parameter F i ) of the current individual.
Besides, it should be noticed that in the early stage of the evolution, most of the individuals are in exploration.Therefore, F i should be larger to search more globally optimal regions.However, in the later stage of the evolution, most of the individuals are in exploitation.In order to accelerate the convergence speed, it is better to reduce F i to avoid oscillation.Based on the above two considerations, F i is set as where a and b are the lower and upper boundaries of F i , respectively; fe is the current number of FEs; MaxFEs is the maximum number of FEs; and λ is a parameter that indicates the evolutionary stage of the algorithm.As the range of F is usually within [0.1, 0.9], a and b are simply set as 0.1 and 0.9, respectively, whose influences are also investigated in Table S.I in the supplementary material.Moreover, through the empirical studies, we can find that most of the individuals finish the exploration stage after 80% of the evolutionary process and start the exploitation stage in the last 20% process.Therefore, we set the value of λ as 0.8, which is investigated in Section IV-G.It should be noted that in the later stage of the evolution, most of the individuals are in the exploitation.
In order to accelerate the convergence speed, it is better to reduce F i to avoid oscillation.Therefore, we set F i to shrink to 0.001 times in the later stage of the evolution, which helps to balance the exploration and exploitation of the population.
2) Adaptive Parameter CR: To better balance the diversity and convergence, the parameter CR i of x i is adaptively controlled based on the location distribution information of the individuals in its LBP niche.The detailed process of adaptively controlling CR i is given in the following steps.
Step 1: According to the location of each individual x j in the LBP niche, the center position of the |M| + 1 individuals is denoted as center, whose kth dimension center k is where k = 1, 2, . . ., D, and x j,k is the kth dimension of the jth individual in the current niching.
Step 2: Calculate the Euclidian distance d j between the jth individual and the center as where j = 1, 2, . . ., |M| + 1.The d j reflects the distribution information of the individual x i in the niche.
Step 3: Calculate the average distance d as The rationale of d is to reflect the uniformity of the individuals distribution in the current niche.As illustrated in Fig. 2(a), if all d i have similar values, the individuals are likely to concentrate and distribute uniformly around the peak.Otherwise, the distribution of individuals in the current niche is likely to be scattered and uneven as in Fig. 2(b).In order to measure the uniformity (UN) of the niche, step 4 is designed.
Step 4: Calculate the UN of x i in the current niche as According to (16), the concentrated and uniform niche like Fig. 2(a) will have a small UN i .Conversely, the UN i will be large if the individuals of the current niche are more scattered and uneven as in Fig. 2(b).
Step 5: Adaptively control the value of CR i according to UN i as where m and n are similar as a and b in (12) to clamp the range of CR.Since the range of CR in DE is usually within [0.1, 0.9], the values of m and n are simply assigned to 0.1 and 0.9, respectively, whose influences are also investigated in Table S.II in the supplementary material.Fig. S.I in the supplementary material shows the rationale and detailed changes of CR i with different UN i .It can be found that CR i becomes larger as UN i increases until CR i reaches the maximum value of n, that is, 0.9.For a larger value of UN i , the distribution of current niche is more uneven, indicating that the current individual has not reached the convergence stage, and needs to increase the search diversity.Therefore, a larger CR i value is needed to obtain more information from the mutated vectors, which is helpful for increasing the diversity.On the contrary, when UN i is smaller, the individuals' distribution of the niche is more concentrated and uniform.This indicates that the niche may have reached a convergence stage and, therefore, the value of CR i should be smaller to make the individual keep more information from the current position, in order to increase the solution accuracy.

D. BCS
During evolution, the individual may exceed the search range due to the exploration in the mutation operator.A simple handle method is to directly set the exceeded dimension of the individual to the corresponding boundary value.This way may be helpful to increase the search diversity of the population in the early stage of the evolutionary process to find more peaks.However, in the later stage of the evolutionary process, most of the individuals have arrived convergence around the best individual of the niche.In this case, if the range-exceeded individuals are still set to the boundary, the convergence may be destroyed.Therefore, in order to balance the diversity and convergence, a novel BCS is proposed in this paper.The BCS deals with the range-exceeded individual according to different stages during the evolutionary process.In the early stage of the evolutionary process, the BCS resets the range-exceeded individual directly to the boundary (L j or U j ).However, in the later stage of the evolutionary process, the BCS adopts another method to reset the range-exceeded individuals by using information of the best individual (x nbest ) in the current niching to accelerate the population convergence.In conclusion, when the solution v i,j is range exceeded, v i,j can be reset as where μ is a parameter that controls the way of dealing with boundary, and μ = 1E-04, which is investigated in Section IV-F.x nbest is the best individual in the current niche.

E. Complete LBPADE Algorithm
This section provides the complete LBPADE algorithm.The initialization and the crossover operators are inherited from the traditional DE, and the NGI mutation operators have been introduced as above.Therefore, only the selection operator and the termination condition are described here.
1) Selection Operator: The selection operator is to find the nearest individual in the parental population to the current individual, and compare their fitness.Then, the individual with better fitness is selected for the next generation.For maximization MMOP, after forming the trial vector u i by the crossover operator, the selection operator needs to find the nearest individual x p to u i in the parental population.For i = 1 to NP 4:

Algorithm 1 LBPADE
Find the most nearest (measured by Euclidean distance) |M| individuals of the current individual x i to form a niche; 5: Compute the mutation scaling factor F i by Eq. ( 12) ; 6: Produce a v i using NGI mutation strategy by Eq. ( 11); 7: Use Eq. ( 18) to reset v i if the v i is range-exceeded; 8: Compute the crossover rate CR i by Eq. ( 17); 9: Produce a u i using crossover operator by Eq. ( 8); 10: End For 11: For each trial vector u i 12: Evaluate the fitness values of u i ; 13: Find the individual x p that is nearest to u i in parents population; 14: Select x i by comparing the fitness values of u i and x p as Eq. ( 19); 15: End For 16: fe = fe + NP; 17: End While End

TABLE I BASIC INFORMATION OF THE TEST FUNCTION
The selection operator is represented as 2) Termination Condition: The termination condition is defined by the maximum number of FEs (MaxFEs).If the termination criterion is satisfied, the algorithm stops and returns the fitness value of the final population as the result.Moreover, if the algorithm has found all known global peaks, it also stops.
Overall, the pseudocode of the complete LBPADE algorithm is outlined in Algorithm 1.

A. Test Functions and Experimental Settings
The CEC'2013 benchmark set includes 20 multimodal test functions.All test functions are acquainted as maximization problems.The property of these multimodal test functions has been introduced in [35].Basic information of the test functions is also summarized in Table I.
In order to validate the effectiveness of the proposed LBPADE algorithm, experimental tests on the benchmark functions are undertaken in this section.The performance of the proposed LBPADE algorithm will be compared with some state-of-the-art multimodal optimization algorithms.The related parameters used in this paper and the compared algorithms are as follows: the test function is the set of CEC'2013, and the other parameters are set as the original paper except for population size and MaxFEs.
Our proposed algorithm is compared with 11 state-of-theart multimodal algorithms.They are CDE [20], SDE [21], R2PSO [22], R3PSO [22], Self_CCDE [23], NCDE [25], NSDE [25], LIPS [26], PNPCDE [27], LoICDE [28], and MOMMOP [34].The reason for choosing these 11 algorithms is that they are representative algorithms in the three categories of related works on MMOP, as reviewed in Section II-B.Therefore, all 11 algorithms are adopted to be compared with LBPADE, where the first five algorithms (i.e., CDE, SDE, R2PSO, R3PSO, and Self-CCDE) are in the niching strategies category, the second five algorithms (i.e., NCDE, NSDE, LIPS, PNPCDE, and LoICDE) are in the new evolutionary operators category, and the last MOMMOP is the typical and well-performed algorithm in the multiobjective techniques category.All of these algorithms use the parameter settings recommended by their source literature, except for the population size (NP) and the termination criterion (MaxFEs).As listed in Table I, different settings of the population size and the termination criterion are adopted for different functions depending on their degrees of complexity, but the settings are set the same across all algorithms, and each algorithm is tested for 51 independent runs.
With the accuracy level ε set at 1E-04, the peak ratio (PR), the success rate (SR), and the average FEs (AveFEs) are calculated to evaluate the algorithmic performance.PR is defined as the average percentage of the global optima found in multiple runs where HFP i is the number of have found peaks (HFP) in the end of the ith run, HKP is the number of have known peaks, and TR is the number of total runs.SR is defined as the percentage of successful runs out of total runs where NSR denotes the number of successful runs.A successful run means that all known peaks have been found, that is, HFP = HKP.The AveFEs over multiple runs be calculated as where FE i denotes the number of FEs used to find all peaks in the ith run.
In addition, we implement LBPADE using C++ language, and all experiments are executed on a PC with 4 Intel Core i5-3470 3.20-GHz CPUs, 4-GB memory, and a Ubuntu 12.04 LTS 64-bit system.

B. Comparisons With State-of-the-Art Algorithms
For all test functions, each algorithm terminates if all known peaks have been found or the termination condition is met.The above three metrics-PR, SR, and AveFEs-are used to describe their results.
The results of F1-F20 are shown in Table II with ε = 1.0E-04.The boldface represents the best results in all of the compared algorithms.Meanwhile, the Wilcoxon's rank-sum test [36] (α = 0.05) with respect to PR between LBPADE and other algorithms is used to examine the significance of the difference.
Table II compares the results of PR and SR to different multimodal algorithms.The results show that LBPADE performs significantly better than the other algorithms on most test functions.For example, LBPADE obtained the best PR results on 15 out of all 20 test functions, among all 11 compared algorithms, as indicated by the boldface.It should be noticed that if the algorithm has the same PR, the AveFEs can be used to measure the performance of the algorithm.Therefore, we compare the AveFEs results of LBPADE with CDE, Self_CCDE, NCDE, LIPS, PNPCDE, and MOMMOP on F1-F5.These algorithms are chosen because they are well-performing algorithms in their corresponding categories according to the results in Table II, that is, CDE and Self-CCDE in the niching strategies category, NCDE, LIPS, and PNPCDE in the new evolutionary operators category, and MOMMOP in the multiobjective techniques category.Moreover, F1-F5 are tested because it is not necessary to measure the other complicated functions by the AveFEs since the results on SR of these algorithms are almost 0 when the algorithms terminate.Herein, we test the AveFEs with three different ε levels (i.e., 1.0E-02, 1.0E-03, and 1.0E-04).The detailed comparison results are shown in Table III, and the best results are marked as boldface.
1) For the First Five Simple and Not Scalable Functions F1-F5: From the observation of Table II, it can be seen that LBPADE has the best results on F1-F5, and the results are the same as CDE, Self_CCDE, NCDE, PNPCDE, and MOMMOP.Moreover, these algorithms can find all global optima in each run (i.e., SR is 1.000).Since LBPADE can form a stable niche based on LBP, each individual is guided by the information of the current niche, which helps LBPADE to deal with MMOPs.Meanwhile, LBPADE performs significantly better than SDE, R2PSO, R3PSO, NSDE, LIPS, and LoICDE on F1-F5.
From Table III, we can find that LBPADE converges faster than all compared algorithms on F1-F5 when ε = 1.0E-02.Meanwhile, LBPADE obtains the best AveFEs values on F1, F2, and F4 when ε = 1.0E-03, and on F1, F2, and F5 when ε = 1.0E-04.The significant tests also show that LBPADE generally outperforms all compared algorithms with significantly small AveFEs values.These indicate the fast convergence ability of LBPADE.
Therefore, we can conclude that LBPADE generally performs better than the other state-of-the-art multimodal algorithms, on the performance metrics of PR, SR, and AveFEs.
2) For the Scalable Multimodal Functions F6-F10: F6-F10 are the MMOPs that have many global optima, so that many algorithms cannot find all global optima.However, LBPADE has generally better performance on these difficult problems than most of the compared algorithms.Table II shows that only LBPADE, CDE, LoICDE, and MOMMOP can find all global optima on F6 (i.e., PR and SR are 1.000).For F7, LBPADE obtains the PR value of 0.889 and performs better than all others except for MOMMOP.For F8 and F9, LBPADE performs better than many others like CDE, SDE, R2PSO, R3PSO, NCDE, NSDE, LIPS, PNPCDE, and LoICDE.Significantly, LBPADE also obtains the best PR and SR values 1.000 on F10.All of these indicate that LBPADE has a good ability to globally search and can find all or most of the global optima in these complex functions.
3) For the Composition Functions F11-F20: F11-F15 are the composite functions.Moreover, they have many local optima.F16-F20 are also complicated functions.Moreover, F18-F20 are the high-dimensional functions, which are 10D, 10D, and 20D, respectively.It is difficult to measure the performance of each algorithm with SR, because the SR values of F11-F20 are equal to 0. But the PR values of them can reflect the difference of the algorithm performances.Therefore, only the PR values of F11-F20 are analyzed in the following contents.
For F11, LBPADE performs better than CDE, SDE, R3PSO, NSDE, and LoICDE.LBPADE performs the best with F12 except for MOMMOP.For F13 and F14, the result of PR on LBPADE is the same with NCDE and MOMMOP, while is better than CDE, SDE, R2PSO, R3PSO, Self_CCDE, NSDE, PNPCDE, and LoICDE.For F15, it is worth noticing that only LBPADE and MOMMOP can obtain PR results larger than 0.5 (i.e., find more than half of the global optima).Moreover, LBPADE is also the winner, whose PR value is 0.654, which is significantly better than that of MOMMOP.
When dealing with F16-F20, most of the algorithms show poor performance, and some algorithms even cannot deal with these test functions.LBPADE still performs better than CDE, SDE, R2PSO, R3PSO, Self_CCDE, NCDE, NSDE, LIPS, PNPCDE, LoICDE, and MOMMOP on F16-F20.Especially, for F18 to F20, LBPADE obtains the best results among all compared algorithms.It indicates that LBPADE has a strong ability to globally search when dealing with the high-dimensional functions.In general, LBPADE performs better than other algorithms on most test functions.We can also conclude that CDE, SDE, R2PSO, R3PSO, NSDE, and LIPS perform better on simple and low-dimensional functions, but their abilities of dealing with complicated problems are poor.Self_CCDE, PNPCDE, LoICDE, and MOMMOP also perform poorly when dealing with high-dimensional functions.In contrast, LBPADE always better in both low-and high-dimensional functions.Overall, the is feasible and promising in solving most of the tested MMOPs.
To further investigate the LBPADE, the final solution distributions of some functions (i.e., F1, F2, F3, F4, F6, F7, F10, F11, F12, and F13) are shown in Fig. 3, which is formed when the algorithm achieved the MaxFEs of test functions or all of the global optima have been found in the evolutionary process.
From Fig. 3, we find that LBPADE can find most of the global optima on these functions that have only a few numbers of global optima, such as Fig. 3(a)-(d).More important, for the functions with a large number of global optima, LBPADE also can find all of the global optima, such as Fig. 3(e)-(g).Moreover, for some complicated functions, LBPADE still can find most of the global optima, such as Fig. 3(h)-(j).Therefore, LBPADE has a good ability to locate global optima in MMOPs.
Moreover, we compared the runtime of each algorithm on F1-F20 at accuracy 1.0E-04.The results are presented in Table S.III in the supplementary material.From Table S.III in the supplementary material, we find that the of LBPADE is promising compared with the other algorithms.Generally speaking, although LBPADE spends slightly more time than some algorithms like CDE, SDE, R2PSO, R3PSO, and Self_CCDE, the runtime differences are acceptable when combined with the comparisons on solution quality.More important, LBPADE spends less time than other algorithms like NCDE, NSDE, LIPS, PNPCDE, LoICDE, and MOMMOP, almost on all of the functions.Therefore, the proposed LBPADE has a very promising runtime in handling MMOPs.

C. Effects of the NGI Mutation Strategy
In this section, we investigate the effect of the NGI mutation strategy by comparing it with the results derived from the usage of different mutation strategies on LBPADE, such as DE/rand/1 and DE/best/1, which are denoted as LBPADE-rand and LBPADE-best, respectively.Meanwhile, we also compare the LBPADE with better DE mutation strategy, such as DE/current-to-pbest/1 with or without archive and DE/currentto-rand/1, which are denoted as LBPADE-pbest(A), LBPADEpbest, and LBPADE-crand, respectively.The PR and SR results of different LBPADE variants on ε = 1.0E-04 are listed in Table IV.
From Table IV, we find that LBPADE significantly outperforms all compared mutation strategies on most test functions.For F1 to F5, all LBPADE variants have promising results.LBPADE locates all of the global optima in each run with F6, which performs best among all LBPADE variants.For F7, all LBPADE variants obtain a promising result except LBPADEbest.This may be because the DE/best/1mutation strategy is somehow greedy to make some individuals trapped into local optima.But for F8, LBPADE obtains slightly worse PR values than LBPADE-rand due to F8 having many unevenly distributed global optima.It should be noted that LBPADE performs better than other LBPADE variants as the function dimension increases.This phenomenon can be seen from the PR values on F11-F20.That is, LBPADE obtains higher PR results than LBPADE-best, LBPADE-rand, LBPADEpbest(A), LBPADE-pbest, and LBPADE-crand on these functions.It indicates that LBPADE has better global searchability when dealing with complicated or high-dimensional functions.In conclusion, our NGI mutation strategy is a promising method by combining the niching information from the LBP niche and the global information from the entire population, which helps LBPADE to solve the MMOPs more efficiently.

D. Advantage of APS
To validate the advantage of APS, five different variants of LBPADE are designed as follows.The first variant, namely, LBPADE-CR, fixes F and lets CR be controlled by APS.The second variant, namely, LBPADE-F lets F be controlled by APS but fixes CR.The third variant, namely, LBPDE, fixes F and CR at 0.5 and 0.1, respectively.Besides, some other adaptive parameter mechanisms are also used to compare with our APS, such as the adapt mechanisms introduced by JADE [19] and LSHADE [37], which are denoted as LBP-JADE and LBP-LSHADE, respectively.Besides, for a more comprehensive analysis of LBPADE, we further investigated the contributions of the two components (i.e., APS and NGI) in LBPADE.In fact, the contribution of APS can be obtained from the LBPADE variant without NGI, namely, the LBPADE-rand, the LBPADE-best, and the other LBPADE variant in Table IV.Similarly, the contribution of NGI can be obtained from the LBPADE variant without APS, such as LBPADE-F, LBP-JADE, and others in Table V.
To avoid the potential bias caused by selecting inappropriate counterparts, we adopt a novel way herein to estimate the contributions made by each component.First, according to Table IV, we calculate the value of "number of +" minus "number of −" for each compared variant.For example, for the LBPADE-best, the value is 13 − 0 = 13, and the value for LBPADE-crand is 15 − 0 = 15.Then, the average of these values is calculated and the result is (13 + 12 + 10 + 12 + 15)/5 = 12.4.We can regard that this value can reflect the contribution that NGI brings to LBPADE, the larger the value, the more contributions it brings.Similarly, we can obtain this value from Table V as (9+5+9+15+10)/5 = 9.6 to indicate the contribution that APS brings to LBPADE.From the above analyses and the result (12.4 > 9.6), we can conclude that NGI generally contributes more than APS to the performance of LBPADE.

E. Maintaining the Identified Optima
During evolution, the MMOPs require that the algorithm can locate many global optima simultaneously.The algorithm with good performance not only can maintain the global optima that have been identified (found) but also can continue to search for the other global optima that have not been found.To investigate the performance of LBPADE on maintaining the identified optima, the solution distribution of some functions (i.e., F2, F4, and F10) is presented on some specific generations.Fig. 4 shows the solution distribution of F2 with different generations (i.e., the generations are 1, 10, 20, and the final generation, respectively).We can find that LBPADE   has obtained all of the global optima when the generation is 20 on F2 [i.e., Fig. 4(c)].It indicates that LBPADE can locate the global optima quickly, namely, LBPADE has a good ability of global search.Fig. 5 shows the solution distribution of F4 with different generations (i.e., 1, 30, 50, and the final generation, respectively).From Fig. 5, we can find that the population nearly converges completely when the generation is 30, but some solutions still do not achieve the convergence state until the generation is 50.Similarly, the solution distributions of F10 with different generations are presented in Fig. 6.With the increasing generation, all solutions gradually achieve the convergence state.All global optima are found until the generation is 100.
Overall, it is clear that the found solutions are not dispersed with the increasing evolution in LBPADE.Namely, our LBPADE can maintain the global optima until the end of evolution.

F. Impacts of Parameter Settings
In this section, we mainly investigate the impacts of parameters λ and μ on our LBPADE.The best PR are highlighted in boldface, and the last row of the table called "#Best" counts the number of the best PR each algorithm obtains on the total 20 functions, namely, the number of the bolded PR.
1) Effect of Parameter λ: The parameter λ is used in (12) to control the state of exploration and exploitation during the evolution process.Table S.IV in the supplementary material shows the PR results with different values of λ.
From Table S.IV in the supplementary material, we have two observations.It is found that a small λ value will cause most of the FEs to be exhausted during exploitation.This may lead to the ineffectiveness in exploring the search space, and it may degrade the diversity of solutions.If the λ value is large, most FEs are exhausted during exploration, which may lead to a poor ability of global search.Therefore, selecting a suitable λ value is important to balance the state of exploitation and exploration during evolution.We find that the #Best is 20 when λ is 0.8 in Table S.IV in the supplementary material, which means that LBPADE performs best with different λ.For F1 to F5, there is not much difference as the value of λ varies.The reason is that F1 to F5 have achieved convergence in the early stage, so the value of λ has less of an effect on F1 to F5.For F6 to F20, LBPADE also performs the best when λ is 0.8.It indicates that λ = 0.8 is a reasonable setting.
2) Effect of Parameter μ: As introduced in Section III-E, μ is a parameter of BCS, which controls the way to deal with boundary during the evolution process.To investigate the effect of μ on LBPADE's performance, two groups of possible μ values are tested.In group 1, μ is set as 0, 1E-05, 1E-04, 1E-03, and 1E-02, respectively.In group 2, μ is set as 0.1, 0.2, 0.4, 0.5, 0.6, 0.8, and 1, respectively.It should be noticed that with μ being equal to 1, the range-exceeded solution is reset to the boundary (L j or U j ).With μ being equal to 0, the range-exceeded solution is reset to the best solution (x nbest ) in the current niching.The result of PR under different settings of μ is shown in Table S.V in the supplementary material.
From Table S.V in the supplementary material, it is clear that the μ has a significant effect on high-dimensional functions (i.e., F11-F20).For F1, the PR result is only 0.900 when μ is 0, implying that by directly assigning range-exceeded solutions to x nbest , the population diversity may decrease and thus degrade the algorithmic performance.As μ increases, the PR results are all equal to 1.000 on F2-F6 and F10.However, for the high-dimensional functions, such as F7, F9, F12, F15, F17, and F20, the best values of PR are achieved when μ is 1E-04.It indicates that a suitable value of μ is helpful for the LBPADE to locate more global optima.Moreover, the effect of μ on LBPADE becomes more and more obvious as the function dimension increases.

G. Comparisons With Winners of CEC Competitions
In the above experiments, LBPADE shows general better performance than the compared state-of-the-art multimodal algorithms when dealing with MMOPs.To further prove the effective performance of LBPADE, we compare LBPADE with the winners of the CEC'2013 and CEC'2015 competitions on multimodal optimization, which are nearest-better clustering (NEA2) [38] and the niching migratory multiswarm optimizer (NMMSO) [39], respectively.Meanwhile, we also compare LBPADE with a dynamic archive-niching DE algorithm (dADE2) [40].For simplicity and convenience, we directly cite the results of these algorithms from the corresponding competitions, where the results of NEA2 and dADE2 are from https://github.com/mikeagn/CEC2013/tree/master/NichingCompetition2013FinalData and the results of NMMSO are from https://github.com/mikeagn/CEC2013/tree/master/NichingCompetition2015FinalData.
Tables S.VI-S.X in the supplementary material present the comparison results with respect to PR and SR between LBPADE and these algorithms (NEA2, NMMSO, and dADE2).The results of accuracy level from 1.0E-01 to 1.0E-05 are shown in Tables S.VI-S.X in the supplementary material, respectively.The best PR are highlighted in boldface, and the last row of the table called #Best counts the number of the best PR each algorithm obtains on the total 20 functions, namely, the number of bolded PR.
From Tables S.VI-S.X in the supplementary material, we can draw the following conclusions.
1) LBPADE performs the best in all compared algorithms at the accuracy levels ε = 1.0E-01 and 1.0E-02.More specifically, at ε = 1.0E-01,LBPADE can find all of the peaks in each run except for F9, F12, and F19.Particularly, at these accuracy levels, LBPADE is much better than NEA2, NMMSO, and dADE2 on F15-F20 that have many local optima.2) At accuracy level ε = 1.0E-03,LBPADE and NMMSO generally perform better than NEA2 and dADE2.They both perform best in 12 of all test problems.But the difference is that LBPADE has great advantages in dealing with high-dimensional problems, while NMMSO performs better in low-dimensional problems.For example, LBPADE performs better than NMMSO on F15-F20 which are in high dimension.3) At the last two accuracy levels, LBPADE performs better than dADE2, it also remains at its competitive performance with NEA2 and NMMSO.Even though the number of #Best for LBPADE, NEA2, and NMMSO is 8, 12, and 12 at ε = 1.0E-04 and 8, 10, and 11 at ε = 1.0E-05, respectively, LBPADE can achieve very similar performance to NEA2 and NMMSO on most of those functions.For example, with respect to PR at ε = 1.0E-04, on F16 and F18, LBPADE achieves 0.667 and 0.667, respectively, which is very similar to NEA2 with 0.673 and 0.667, respectively.Overall, we can see that LBPADE is competitive against the winners of the CEC'2013 and the CEC'2015 competitions.

V. CONCLUSION
In this paper, a novel LBP-based niching strategy is proposed, which forms a niche for each individual according to its local information by simulating the LBP operator in the image processing.Meanwhile, the NGI mutation strategy and the APS technique are incorporated, which results in the LBPADE algorithm, which not only can enhance the population diversity but also can accelerate convergence speed.The experimental results show that the proposed LBPADE can outperform a number of state-of-the-art multimodal optimization algorithms on benchmark problems.
In the future, we will extend the LBPADE to solve some real-world problems in the domains with potential multimodal optimization requirements, including resourceconstrained project scheduling [41]; electricity markets [42]; energy resource management [43]; optical networks [44]; cloud computing [45], [46]; and image processing [47], [48].For example, how to detect multiple equilibriums simultaneously is a key challenging economic game problem in electricity markets.Herein, we can use the following two main advantages of LBPADE to solve this problem.One is that the NGI mutation strategy of LBPADE can locate multiple optima simultaneously and avoid solutions trapped into local optima; and the other is that the APS of LBPADE does not need to input the extra sensitive parameters when dealing with this problem.So LBPADE has a promising ability to solve these real-world problems.Besides, we will try to use the parallel/distributed computing resources to reduce the runtime of LBPADE and further improve the performance of LBPADE.

Abstract-
The multimodal optimization problem (MMOP) requires the algorithm to find multiple global optima of the problem simultaneously.In order to solve MMOP efficiently, a novel differential evolution (DE) algorithm based on the local binary pattern (LBP) is proposed in this paper.The LBP makes use of the neighbors' information for extracting relevant pattern information, so as to identify the multiple regions of interests, which is similar to finding multiple peaks in MMOP.Inspired by the principle of LBP, this paper proposes an LBP-based adaptive DE (LBPADE) algorithm.It enables the LBP operator to form multiple niches, and further to locate multiple peak regions in MMOP.Moreover, based on the LBP niching information, we develop a niching and global interaction (NGI) mutation strategy and an adaptive parameter strategy (APS) to fully search the niching areas and maintain multiple peak regions.The proposed NGI mutation strategy incorporates information from both the niching and the global areas for effective exploration, while APS adjusts the parameters of each individual based on its own LBP information and guides the individual to the promising direction.The proposed LBPADEalgorithm is evaluated on the I. INTRODUCTION

Fig. 2 .
Fig. 2. Distribution of individuals in the current niche.(a) Concentrated and uniform.(b) Scattered and uneven.

TABLE II PR
AND SR (TEST FUNCTIONS F1-F20)

TABLE III AveFEs
OF DIFFERENT ALGORITHMS

TABLE IV EXPERIMENTAL
RESULT IN PR AND SR BETWEEN LBPADE AND THEIR VARIANTS WITH DIFFERENT MUTATION STRATEGIES ON F1-F20 WITH ACCURACY LEVEL ε = 1.0E-04 Table V shows the PR and SR results of different LBPADE variants.In Table V, boldface denotes the best PR values among LBPADE and its five variants.For F1 to F6, the PR values are 1.000 across all algorithms except for LBP-JADE, which might cause the adaptive parameter mechanisms introduced

TABLE V EXPERIMENTAL
RESULT IN PR AND SR RESULT IN LBPADE WITH DIFFERENT PARAMETER SETTINGS ON F1-F20 WITH ACCURACY LEVEL ε = 1.0E-04