I Introduction
Optimization to a problem is a process for seeking better or best alternative solution from a number of possible solutions [1]. As the analytical optimal solution is difficult to obtain even for relatively simple application problems, the need for numerical optimization algorithm arises from almost every field of engineering design, systems operation, decision making, and computer science [3, 2, 4]. In global optimization problems, the particular challenge is that an algorithm may be trapped in the local optima of the objective function when the dimension is high and there are numerous local optima [5].
Typical conventional search methods include steepest descent methods, conjugate gradient, quadratic programming, and linear approximation methods. These strategies rely on local information of the objective function to decide on their next move in the neighborhood of visited solutions. Their main advantage is the efficiency, however, they tend to be sensitive to starting point selection, and more likely to settle at nonglobal optima than modern stochastic algorithms [1].
Modern stochastic algorithms such as evolutionary algorithms (EAs) draw inspiration from biological evolution. They guide the evolution of a set of randomly selected individuals through a number of generations in approaching the global optimum solution, making use of competitive selection, recombination, crossover, mutation or other stochastic operators to generate new solutions [1, 4]. They only require information of the objective function itself, and other accessory properties such as differentiability or continuity are not necessary. And EAs essentially work with building blocks, which increase exponentially as the evolution through generations proceeds. This results an efficient exploitation of the given search space.
Modern stochastic optimizers include simulated annealing, Tabu search, genetic algorithms, evolutionary programming, evolution strategies, differential evolution, and others
[6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 15, 18, 19, 20, 16, 21]. Most of the successful applications of EAs are limited to problems with dimensions below 30 [14, 17, 15, 16]. Only in the last decade, did researchers begin to test their EAs on problems with more than 30 dimensions [22, 23, 24, 25, 26, 27, 28, 31, 29, 30, 5].To deal with these high dimensional and complex problems effectively and enhance EAs, many researchers have tried to combine techniques from other research fields into EAs. The combination of evolutionary algorithm with local search approach is known as Memetic or Hybrid algorithm [32]. Several new designed hybrid algorithms have been applied to practical problems [34, 35, 36, 37, 33]. The studies on hybrid algorithm have demonstrated that they converge to high quality solutions more efficiently than their conventional counterparts [1]. The purpose of this paper is to develop a more efficient hybrid EA for high dimensional optimization problems.
Several local search methods have been successfully combined into EAs. A robust stochastic genetic algorithm (StGA) for global numerical optimization is given in [4]
, where a stochastic coding scheme based on Gaussian distribution is proposed. A mutation operator based on Cauchy distribution was proposed as a “fast evolutionary programming”
[17], and a further generalization of the mutation operator with Lévy distribution was given in [31]. These algorithms are based on the assumptions about the sampling distributions. In order to avoid the influence of distributional assumption, an nonparameterized importance sampling method is proposed in this paper.Experimental design methods have been successfully combined into EAs [3]. Zhang and Leung were the first to combine the orthogonal design into EAs for a discrete optimization problem [38], and Li and Smith used Latin squares to improve EAs [39]. Tsai et al. combined the Taguchi method into a genetic algorithm [40]. Other researchers set up a marginal model to estimate the distribution of globally optimal solutions for any problem and obtained good results [42, 41]. On the other hand, the estimation of marginal distribution is not enough for high dimensional optimization problems, due to the number of possible combinations increases exponentially with larger scale of problems.
A relatively simple method is proposed to estimate the joint distribution of optimal solutions in this paper. It is supposed that the interval which makes an individual a smaller value of fitness than the value of a similar individual should be given a larger value of probability in the estimated joint distribution, therefore a set of genetics is selected from the visited solutions to give a score for each interval, and those intervals with scores beyond the quantiles are regarded as good intervals for each dimension. On the other hand, the solutions with smaller values of fitness are regarded as good genetics, and those good individuals with more elements falling into good intervals are more likely to be optimal solutions, which should be given a larger probability of selection. At the same time, those good intervals with more good genetics appearing should be given a larger probability of selection. It is a cross validation between good intervals and the pool of good genetics which determines the importance sampling probabilities for good intervals and good genetics in this paper.
Many stochastic algorithms do not memorize places where they have visited, and the information about the evaluated solutions is not taken into consideration for further search. In order to improve the efficiency of EA, a genetic algorithm that adaptively mutates and never revisits was proposed by [43]. And an evolutionary algorithm based on the entire previous search history (HdEA) was proposed in [44]. However, there are more and more visited solutions needed to be memorized as algorithm proceeds, such that the requirement of memory may be extremely large. In order to use the information provided by the previous search process, and to avoid the extra requirement of memory ability, only part of the visited solutions are selected and used to give scores for the intervals in this paper. They are updated from one generation to the next, and the requirement of memory is a parameter which can be adjusted during the process of algorithm design.
Premature population convergence about a local optimum is a common problem of traditional genetic algorithms [45]. It is a result of individuals hastily congregating within a small region of the search space [46]. Maintaining a diverse population is very important for evolutionary algorithms, which means that the selection of individuals can not only dependent on their fitness scores, and other principle such as the diversity proposed in [46] should be taken into consideration. The distributions of importance sampling for individuals and intervals are determined through a cross validation mechanism between the pool of good genetics and the good intervals in this paper, which is not related to the values of fitness directly. And a purely random EA is combined into the proposed algorithm to maintain the diversity of individuals in this paper.
test functions and benchmark evolutionary algorithms are selected to evaluate the performance of the proposed algorithm. There are new optimal solutions found in our numerical investigations, solutions similar to the best results reported in the literature, and solutions closed to the best results. On the other hand, there are test functions where the proposed algorithm can not find the optimal solutions efficiently. However, the proposed algorithm has the smallest number of fitness values which are different from the optimal solutions with respect to the order of magnitude among the algorithms considered in this paper.
The remainder of this paper is structured as follows. Section II describes the problem of optimization for multidimensional functions. The details of hybrid EA are given in Section III. Section IV is devoted to the empirical investigations of the proposed algorithm through 30 test functions. And conclusions and discussions are given in Section V.
Ii Optimization problem
The problem we consider is an unconstrained global optimization problem
(1) 
where
is a vector with
elements, is a subset of , where and are the lower and upper boundaries of respectively. The value of objective function at point is called the fitness value of in this paper. The purpose of optimization is to find the solutions which make the objective function reach its minimum value.Iii Hybrid EA with importance sampling
Canonical EA is an optimization algorithm based on population, where individuals are used to generate the offspring generation with genetic operators, such as mutation, crossover, and selection. The individuals with smaller values of fitness are survival from the evolution of population. While the information provided by those individuals which are not survival is completely dropped in further searching process. Some researchers have suggested to use those information efficiently to improve the performance of EAs [43, 44]
. Following this line, the information obtained in the process of searching is used to design new crossover operator, mutation operator, interpolation operator with importance sampling method in this paper.
Iiia Initiation
The individuals of first generation are randomly generated within the search space, where the size of the first generation is a predetermined parameter. There are individuals chosen to be the pool of good genetics. The range of search in each dimension is partitioned into subintervals with equal length. And is the base number of new generated individuals, where the numbers of new generated solutions for crossover, mutation and interpolation operators are several times of respectively in the following sections. There are only four parameters needed to be determined before the application of the hybrid EA with importance sampling method (HisEA). Figure 1 is the flow chart of HisEA.
IiiB Fitness scores of individuals
Suppose the individuals in the current pool of good genetics are , whose values of fitness are with increasing order, and the maximum value of fitness in the current search history is denoted as . The score for the ith individual is defined as
(2) 
which indicates that the individual with smaller value of fitness will be given a relatively larger score among the current pool of good genetics. And will be updated in the following search process, such that the score for each individual is changeable to update the new information achieved in the process of search.
IiiC Scores of intervals
As each dimension of the search space has been partitioned into equal subintervals, the length of one interval for the ith dimension is
(3) 
where is the kth interval of the ith dimension, and are the partition points of this dimension, and
(4) 
IiiC1 Selection of scoring genetics for intervals
A pool of genetics is selected from all of the evaluated solutions to give a score for each interval of every dimension with the following algorithm.

Initiation according to the first dimension. Denote the genetics in the first generation as , . For the kth interval of the first dimension , if , , where , then and are put into . In other words, the first two solutions whose elements of the first dimension are in the same interval are selected according to their fitness. And the solution with maximum value of fitness is included in .

Repeat step (1) for .

Update according to new evaluated solutions. If a new evaluated solution is belong to the kth interval, that is to say , the first two solutions with smaller values of fitness among the previous selected genetics and the new genetic are selected for the kth interval. And the solution with maximum value of fitness is updated.
IiiC2 Selection of good intervals
is used to give a score for each subinterval of every dimension, and denote the score for the jth interval of the ith dimension, , . The score matrix is used to determine the good intervals for each dimension with the following algorithm.

Initiation. Set , , .

For the ith dimension and the jth interval, find the genetics in whose ith elements are in the jth subinterval , denote as , where is the number of genetics appearing in the jth subinterval.

Case 1. , set .

Case 2. , select the first two genetics and according to their order in , and denote their weights as and (). For , denote and , set
(5)


Repeat step (2) for and .

Suppose is the quantile of the kth row of the score matrix . If , the subinterval is said to be a good interval for the kth dimension.

If and are both good intervals, set .

Repeat Step (5) until there is no more subinterval to be combined. And is said to be the mth good interval for the kth dimension.
IiiD Sampling probabilities of individuals
The individuals in the pool of good genetics are chosen according to their values of fitness. In order to describe the distributional information among all of the dimensions, the sampling probabilities of individuals are chosen in the following way. Denote the indicator function for . Let
(6) 
The score of is
(7) 
which is the number of elements falling into the good intervals.
Denote the probability that the ith individual is chosen among the individuals in the pool of good genetics,
(8) 
which means that it is more possible to be chosen for those individuals with more elements falling into the good intervals.
The sampling probabilities for individuals are not directly based on the values of fitness in this paper, which can be regarded as an alternative choice to maintain the diversity of population.
IiiE Sampling probabilities of intervals
There is a cross validation mechanism between the chosen good intervals and the individuals in the pool of good genetics, which is used to determine the sampling probabilities of the individuals in the previous section, and to determine the sampling probabilities of intervals in this section with the following algorithm.

Initiation. Let be the number of individuals falling into the kth good interval of the mth dimension, , and , where is the number of good intervals of the mth dimension.

Denote an individual in the pool of good genetics, if , where is some integer between 1 and , let , otherwise .

Repeat step (2) for .

Repeat step (2) and (3) for .

The sampling probability for the kth good interval of the mth dimension is
(9)
The estimated sampling probabilities for individuals and intervals are used to design a crossover operator, two kinds of mutation operators, and an interpolation operator with importance sampling method in the following sections.
IiiF Crossover operator with importance sampling
Crossover operator is used to generate new individuals from their parents. As the elements in the good intervals are more likely to be the optimal solutions, they will be kept in the offsprings, and those elements not in the good intervals are replaced by the elements of the other parent which are in the good intervals as the following algorithm.

Sampling two different individuals from the pool of good genetics with the importance sampling probabilities , say and .

Find the elements in the good intervals for and , whose positions are indicated by two indicators, denoted as and respectively, where means that the jth element in the ith individual is falling into the good intervals, , . Otherwise .

Generate one individual from with elements chosen from by the following algorithm:

If and , ;

If and , ;

If and , ;

If and , .


Repeat step (3) for .

Repeat step (1) to (4) times to generate a set of new genetics.
The proposed algorithm is based on the pool of good genetics, which are chosen according to their values of fitness. On the other hand, the two parents to generate new individuals are sampled with the importance sampling probabilities, which are not directly related to the values of fitness. And the result of crossover is related to the estimation of good intervals, which can be regarded as the estimation of the joint distribution of the optimal solutions. This is the difference between the proposed hybrid algorithm and the traditional EAs.
IiiG Mutation operators with importance sampling
There are two kinds of mutation operators proposed in this section, which are all based on the importance sampling probabilities.
IiiG1 Locally adjusting algorithm
There may be some individuals in the pool of good genetics whose elements are not all falling into the good intervals. In order to make those individuals look more like good genetics, a locally adjusting algorithm is proposed as the following steps.

Select one of the individuals in the pool of good genetics according to the probabilities , denote as , and denote the individual to be generated, set .

Mutation for the kth dimension. If there dose not exist any such that , select one of the good intervals of the kth dimension according to , denoted as , and is adjusted as , where
is an uniformly distributed random variable on
. 
Repeat step 2 for . If there is no dimension to be adjusted, there is no new genetic to be generated in this run.

Repeat times to generate a set of new individuals.
IiiG2 Entirely adjusting algorithm
Another mutation algorithm is proposed to explore the visited space as the following steps.

Select one of individuals in the pool of good genetics according to the probabilities , denote as , and denote the individual to be generated.

Mutation for the kth dimension with the following algorithm:

If there dose not exist any such that , select one of the good intervals with , denoted as , and is adjusted as .

If there exists some , such that , is adjusted as , where is an uniformly distributed random variable on .


Repeat step (2) for .

Repeat step (1) to (3) times to generate a set of new individuals.
The difference between these two kinds of mutation operators is that the elements falling into the good intervals are not adjusted by the locally adjusting algorithm, while which are adjusted by the entirely adjusted algorithm.
IiiH Interpolation operator with importance sampling
In order to search the space between two suboptimal solutions, an interpolation operator is adopted in this paper, where the estimated good intervals are used to guide the direction of search as the following steps.

Randomly choose two individuals in the pool of good genetics according to the probabilities ,, , , , denoted as and .

Generate the element for the ith dimension with the following algorithm:

If there exists two good intervals and such that and . Set , where denotes the largest integer which is less than or equal to , and generate the ith dimension for the new individual as
(10) where is uniformly distributed on .

If there exists one good interval such that , and no good interval to contain . The ith dimension for the new individual is
(11) 
If there exists one good interval such that , and no good interval to contain . The ith dimension for the new individual is
(12) 
If there exists no good interval to contain any of the two samples and , a good interval for the ith dimension is randomly selected according to the probabilities , , where is the number of good intervals for the ith dimension, denoted as . The ith dimension for the new individual is
(13)


Repeat Step 2 for to generate a new individual.

Repeat Step 1 to Step 3 times to generate a set of new individuals.
IiiI Random sampling
In order to explore the search space, there are two kinds of random sampling methods adopted in this paper, one of which is based on the probabilities of importance sampling, and the other one is not related to the information obtained in the process of searching.
IiiI1 Importance sampling algorithm
Importance sampling algorithm is designed to explore the search space, where the estimated distribution of optimal solutions is involved in the following steps.

For the ith dimension, one of the estimated good subinterval is sampled according to the probabilities , denoted as .

Randomly sampling one sample from as
(14) where is uniformly distributed within .

Repeat Step 1 to Step 2 for .

Repeat Step 1 to Step 3 times to generate a set of new individuals.
As more and more individuals are generated from the estimated good intervals, the resolution of these intervals is improved.
IiiI2 Purely random sampling
In order to keep the diversity of the chosen good genetics, and to reduce the risk of premature, a purely random sampling method is adopted as the following steps.

For the ith dimension, of individual is
(15) where and are the lower and upper boundaries for the ith dimension respectively.

Repeat Step 1 for .

Repeat Step 1 to Step 2 times to generate a set of new individuals.
IiiJ Purely random EA
In order to keep the diversity of genetics, and to escape the trap of local optimal solutions, an evolutionary algorithm with purely random crossover and mutation operators is adopted in this paper, which is dependent on the pool of good genetics, but does not use the information from the previous search process.
IiiJ1 Purely random crossover
A purely random crossover operator is adopted in this paper as following steps.

Select two individuals in the pool of good genetics with equally possibility, denoted as and .

Denote and the new individuals to be generated.

For the ith dimension, randomly sample a number , where
is a binomial distributed variable
. The elements of and are determined with the following algorithm.
If , , and .

If , , and .


Repeat Step 3 for .

Repeat Step 1 to Step 4 times to generate a set of new individuals.
As the result of random trail is equally distributed between and , the crossover between and is purely random, which is designed to maintain the diversity of population in this paper.
IiiJ2 Purely random mutation
A similar algorithm for mutation is adopted in this paper, where the element of the solution is randomly selected to be mutated with the following algorithm.

Randomly select one individual in the pool of good genetics with equally probabilities, denoted as .

Randomly sample a value of . Denote the new genetic as , whose element in the ith dimension is determined by the following algorithm.

If , , where .

If , .


Repeat Step 2 for .

Repeat Step 1 to Step 3 times to generate a set of individuals.
The total number of new generated individuals with all of the previous operators is for each run of the hybrid algorithm, where individuals generated without the information obtained in the process of search are , which is designed to maintain the diversity of population.
IiiK Mature condition
A pool of good genetics is used to generate new individuals, whose values of fitness are evaluated and compared to their parents, and a new pool of good genetics is selected from the parents and offsprings according to their values of fitness. Denote the former pool of good genetics, and the new pool of good genetics. The stopping condition is based on the result of comparison between and with the following algorithm.
Select a set of quantiles, denoted as , where is the number of quantiles to be taken into consideration. The quantiles of for each dimension are denoted as
(16) 
where is the kth quantile for the ith dimension, , and . The similar quantiles for is denoted as . The difference between those two kinds of quantiles is
(17) 
and the hybrid EA is stopped when or the number of loops is beyond times, where the quantiles are those points from to with step length in this paper.
Iv Empirical investigations
To evaluate the performance of the proposed algorithm, the optimal values of fitness founded by HisEA are compare to their counterparts of 9 benchmark evolutionary algorithms for 30 test functions in this paper.
Iva Algorithms for comparison
IvA1 HdEA
HdEA is an evolutionary algorithm that uses the entire search history to improve its mutation strategy [44]. It uses the fitness function approximated from the search history to perform mutation. Since the proposed mutation operator is adaptive and parameterless, HdEA has only three control parameters: neighborhood size, population size, and crossover rate. The source code of HdEA is available at http://www.ee.cityu.edu.hk/ syyuen/Public/Code.html.
IvA2 RcgaUndx
Real Coded GA With UniModal Normal Distribution Crossover (RCGAUNDX) is a real coded GA that deals with continuous search spaces
[47, 44]. It applies the unimodal normal distribution crossover (UNDX) to preserve the statistics of the population. UNDX is a multiparent genetic operator in which the distribution of the corresponding offspring follows the distribution of the parents.IvA3 CmaEs
Covariance Matrix Adaptation Evolution Strategy (CMAES) is an evolution strategy that adapts the full covariance matrix of a normal search (mutation) distribution [48, 44]
. An important property of CMAES is its invariance against linear transformations of the search space. The underlying idea is to gather information about successful search steps to modify the covariance matrix of the mutation distribution in a derandomized, goal directed fashion. Changes to the covariance matrix are such that variances in directions of the search space that have previously been successful are increased, while those in other directions decrease passively. The accumulation of information over a number of search steps makes it possible to reliably adapt the covariance matrix even when using small populations. CMAES is designed with the emphasis that the same parameters are used in all applications in order to be “parameterless.” The source code of CMAES is taken from
[48] (Aug. 2007 version).IvA4 De
Differential evolution (DE ) is a stochastic search algorithm [49, 44]
. The basic idea behind DE is a scheme that generates trial parameter vectors. DE adds the weighted difference between two population vectors to a mutant vector, and the trial vector is the crossover between the mutant vector and the parent vector. By doing so, no separate probability distribution is used, which makes the scheme completely selforganizing.
IvA5 Ode
Oppositionbased differential evolution (ODE) utilizes the concept of oppositionbased learning (OBL) [50] to accelerate the convergence rate of DE. The main idea behind OBL is the simultaneous considerations of a solution and its corresponding opposite solution. ODE considers the evaluations of the opposite solution in a generation depending on a jumping rate [51, 50, 44].
IvA6 DEahcSPX
Differential Evolution With Adaptive HillClimbing Simplex Crossover (DEahcSPX) attempts to accelerate the classic DE by a local search strategy, named adaptive hillclimbing crossoverbased local search. It adopts the simplex crossover operation (SPX) to generate offspring individual for hillclimbing [44, 42, 50].
IvA7 Dpso
Dissipative Particle Swarm Optimization (DPSO) is a modified PSO which introduces random mutation that helps particles to escape from local minima. Its formula is described as follows: If
then where and are uniformly distributed random variables in the range , is the mutation rate to control the velocity, is a constant to control the extent of mutation, and is the maximum velocity [52, 44] .IvA8 Sepso
IvA9 Eda
EDA is based on undirected graphical model and Bayesian network. The source code of the EDA is taken from
[54] (Feb. 2009 version). The implementation is conceived to allow the user different combinations of selection, learning, sampling, and local search procedures [54, 44].Each of the above algorithms was executed to some of the test functions, and the results were reported in [44] and the references therein. We use existing results for a direct comparison in this section.
IvB Simulations and results
Function  

Dimension  30  30  30  30  30  30  30  30  
Optimum  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  
HdEA  average  0.0000  0.0000  16920.2300  10.8802  21.1276  10.4615  0.0000  0.0000 
std. dev.  0.0000  0.0000  2818.2200  1.3212  13.7111  0.6028  0.0000  0.0000  
RCGA  average  0.0000  0.0000  2811.5869  91.8556  71.1275  8.8836  219.8173  1.6946 
UNDX  std. dev.  0.0000  0.0000  1668.9500  20.8748  67.2881  0.4382  12.6701  0.1151 
CMAES  average  0.0000  0.0000  0.0000  0.0000  0.0000  0.2303  53.6481  0.0014 
std. dev.  0.0000  0.0000  0.0000  0.0000  0.0000  0.0893  14.1662  0.0036  
DE  average  0.0221  0.3594  26073.0400  49.5082  614.4588  10.9846  25.7105  0.9931 
std. dev.  0.0048  0.0371  3339.0600  4.0114  112.1748  0.5641  2.6500  0.0301  
ODE  average  0.0000  0.0953  78.1691  0.0111  27.0344  8.5199  102.3277  0.0258 
std. dev.  0.0000  0.0486  45.1045  0.0808  0.7121  0.4121  37.9763  0.0326  
DEahcSPX  average  0.1075  0.0322  65.9908  15.5882  3160.5891  8.7536  40.8070  0.1613 
std. dev.  0.3990  0.1423  65.2220  3.5293  9387.6000  0.5200  26.7653  0.2685  
DPSO  average  3.3462  8.8458  1955.2153  10.5390  12789.4900  9.9202  125.4958  4.0567 
std. dev.  0.9949  1.1698  24.5853  1.3743  78.0953  0.8192  3.9027  0.8773  
SEPSO  average  2.8527  8.8184  2984.5912  13.8670  10464.0700  10.6244  112.9555  3.6995 
std. dev.  0.9324  1.3980  29.6756  1.7487  80.0175  0.8755  4.5789  0.9511  
EDA  average  3439.5320  22.2520  3749.1330  21.1420  30214.7330  100.0690  188.3840  30.5040 
std. dev.  1221.2100  5.4338  1294.7400  5.9330  12162.0800  52.0020  20.5550  10.9870  
HisEA  average  0.0000  0.0000  319.4241  0.9769  28.0095  0.0002  0.0800  0.0031 
std.dev.  0.0000  0.0000  242.1950  0.2552  1.5109  0.0004  0.2425  0.0042 
Average, Standard Deviation of the Best Fitness values for
.function  

Dimension  30  30  2  2  2  30  30  30  
Optimum  0.0000  1.0316  0.3980  3.0000  0.0000  0.0000  0.0000  
HdEA  average  1.37E+04  0.00E+00  1.0316  4.01E01  4.41E+00  0.00E+00  0.00E+00  2.61E+02 
std. dev.  2.54E+01  0.00E+00  1.00E04  2.64E02  4.08E+00  0.00E+00  0.00E+00  3.50E+01  
RCGA  average  5.94E+03  2.07E+01  0.6587  4.60E01  5.61E+01  6.92E+07  3.33E01  2.94E+02 
UNDX  std. dev.  4.87E+02  9.04E02  3.12E01  5.65E02  2.96E+01  1.20E+07  9.54E02  3.69E+01 
CMAES  average  5.40E+03  2.13E+01  1.0235  3.98E01  7.32E+00  4.14E+01  1.15E02  0.00E+00 
std. dev.  9.56E+01  4.32E01  8.16E02  0.00E+00  1.66E+01  1.01E+02  1.15E01  0.00E+00  
DE  average  1.28E+04  1.83E+00  0.6695  1.52E+00  1.48E+01  4.66E+03  3.20E03  2.74E+02 
std. dev.  1.56E+02  3.32E01  3.21E01  1.35E+00  1.04E+01  9.26E+02  8.00E04  3.00E+01  
ODE  average  5.47E+03  9.90E03  1.0214  4.25E01  3.52E+00  1.16E+00  1.50E05  2.43E+01 
std. dev.  5.06E+02  1.17E02  1.09E02  2.77E02  4.77E01  1.25E+00  0.00E+00  8.07E+00  
DEahc  average  1.02E+04  2.82E+00  0.4882  6.68E+00  2.18E+01  3.82E+04  5.98E02  2.21E+00 
SPX  std. dev.  6.92E+02  4.03E+00  3.29E+00  1.93E+01  8.50E+01  2.01E+05  1.70E01  3.85E+00 
DPSO  average  5.18E+03  6.07E+00  1.0229  1.44E+00  3.14E+00  8.26E+06  1.31E+01  1.35E+02 
std. dev.  2.57E+01  8.21E01  1.17E01  1.62E+00  6.47E01  1.93E+03  1.66E+00  7.91E+00  
SEPSO  average  7.70E+03  6.44E+00  1.0252  4.10E01  3.06E+00  6.28E+06  1.40E+01  7.18E+01 
std. dev.  2.73E+01  1.00E+00  1.04E01  1.37E01  2.93E01  1.55E+03  1.45E+00  4.87E+00  
EDA  average  4.67E+03  1.02E+01  1.031  3.98E01  3.00E+00  6.75E+06  7.42E+04  6.97E+01 
std. dev.  7.03E+02  1.29E+00  1.20E03  0.00E+00  2.00E06  4.13E+06  5.60E+04  2.91E+01  
HisEA  average  12558.8751  0.0000  1.0316  0.3979  3.0000  2.46E+05  0.0000  0.3054 
std.dev.  29.1137  0.0000  0.0000  0.0000  0.0000  1.91E+05  0.0000  0.1539 
Function  

Dimension  30  30  30  30  30  30  30  30  
Optimum  0.0000  0.0000  29.0000  0.0000  0.0000  4930.0000  
HdEA  average  0.0004  4.8663  24.9443  0.0000  25.3678  0.1626  8025.425  997867 
std. dev.  0.0002  0.3451  0.9411  0.0000  0.572  0.4616  4773.2  0.0271  
RCGA  average  10.2837  7.61  6.633  0.3566  8.8451  152.9672  62837.03  330 
UNDX  std. dev.  1.3909  0.5734  0.452  0.073  0.5691  17.6976  3012.7  0.0000 
CMAES  average  0.0025  13.7823  0.9678  0.4493  19.1834  0.023  2428.19  951 
std. dev.  0.0028  0.2792  0.732  0.258  1.8797  0.0472  0.0000  187570.7  
DE  average  0.1641  5.3987  18.8816  0.0027  18.3183  60.0966  122598.2  958473 
std. dev.  0.0486  0.5198  0.556  0.0006  0.6445  10.9953  25422.81  6695.09  
ODE  average  0.0299  0.0237  27.8856  0.000027  12.5543  26.0994  4930  610112 
std. dev.  0.0099  0.1431  1.8404  0  1.2739  12.9456  1162.05  37311.6  
DEahc  average  0.0013  4.6963  14.6745  0.1752  12.9365  37.1675  1911.297  996116 
SPX  std. dev.  0.0078  1.3226  4.0927  0.1499  2.0401  17.6322  4085.18  5578.9 
DPSO  average  5.758  11.8132  15.4114  0.6795  10.3292  135.0221  26342.66  342933 
std. dev.  1.3801  0.521  1.2455  0.4754  0.8904  6.6864  86.3998  217.3297  
SEPSO  average  9.5052  12.0147  16.9436  0.7684  10.9954  152.5561  30572.09  965431 
std. dev.  1.7957  0.532  1.2194  0.5383  0.9763  7.1358  100.8033  24.4007  
EDA  average  12.235  12.309  18.728  1.885  9.361  10.527  141156.77  286765.1 
std. dev.  3.110696  0.17794  4.185106  0.444738  0.75983  4.54527  65517.19  36881.35  
HisEA  average  0.0027  2.6454  28.9299  0.0000  25.9147  0.0000  2834.5739  984105.1432 
std.dev.  0.0018  0.5052  0.1942  0.0000  0.9994  0.0000  1627.9243  3769.7065 
function  

Dimension  30  30  30  30  30  30  
Optimum  0.9000  0.0000  3.5000  
HdEA  average  1.0004  1.2051  2E+34  1.521  29.559  20.6012 
std. dev.  0.0002  0.1365  3.60E+33  0.6748  0.0289  28.8726  
RCGA  average  6.9638  2.3127  1.3E+20  3.3678  10.2002  7440.466 
UNDX  std. dev.  0.3607  0.1817  2.10E+20  0.0231  0.5695  2193.41 
CMAES  average  8.142  1.1979  1.1E+29  2.6016  19.1408  319.3721 
std. dev.  5.7645  0.247  4.40E+29  1.5276  2.0299  102.4076  
DE  average  1.4523  3.6819  9E+29  2.1663  24.8678  830.2062 
std. dev.  0.076  0.2572  1.60E+30  0.1757  0.418  15.6812  
ODE  average  0.9107  0.4718  1.2E+24  3.5000  14.7206  766.9481 
std. dev.  0.0418  0.1056  7.10E+24  0.0000  1.0599  22.6609  
DEahcSPX  average  2.4177  0.4953  1.9E+24  3.3078  16.9775  9536.837 
std. dev.  0.8642  0.1756  1.20E+25  0.3708  2.3975  38054.29  
DPSO  average  4.8472  2.9157  2.7E+24  1.8368  13.6114  11716.56 
std. dev.  0.8796  0.6078  4.10E+12  0.7575  0.965  77.5304  
SEPSO  average  3.3681  3.3948  2.1E+25  2.3102  14.0837  11007.86 
std. dev.  0.7988  0.7648  8.60E+12  0.8684  1.1034  81.4433  
EDA  average  7.629  5.186  1.24E+22  1.222  10.781  985056.31 
std. dev.  0.5443  1.0724  8.90E+22  0.2879  0.7085  769131.6  
HisEA  average  1.0000  0.1859  6.25E+34  3.5000  28.4301  13.5723 
std.dev.  0.0000  0.0351  9.93E+30  0.0000  0.0179  11.8222 
IvB1 Test functions
30 wellknown real valued functions are used to evaluate the performance of HisEA in this paper. The test functions, the numbers of dimensions, and the ranges of search are as follows.
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