Solving largescale global optimization problems using enhanced adaptive differential evolution algorithm
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Abstract
This paper presents enhanced adaptive differential evolution (EADE) algorithm for solving highdimensional optimization problems over continuous space. To utilize the information of good and bad vectors in the DE population, the proposed algorithm introduces a new mutation rule. It uses two random chosen vectors of the top and bottom 100p% individuals in the current population of size NP, while the third vector is selected randomly from the middle [NP2(100p%)] individuals. The mutation rule is combined with the basic mutation strategy DE/rand/1/bin, where the only one of the two mutation rules is applied with the probability of 0.5. This new mutation scheme helps to maintain effectively the balance between the global exploration and local exploitation abilities for searching process of the DE. Furthermore, we propose a novel selfadaptive scheme for gradual change of the values of the crossover rate that can excellently benefit from the past experience of the individuals in the search space during evolution process which, in turn, can considerably balance the common tradeoff between the population diversity and convergence speed. The proposed algorithm has been evaluated on the 7 and 20 standard highdimensional benchmark numerical optimization problems for both the IEEE CEC2008 and the IEEE CEC2010 Special Session and Competition on LargeScale Global Optimization. The comparison results between EADE and its version and the other stateofart algorithms that were all tested on these test suites indicate that the proposed algorithm and its version are highly competitive algorithms for solving largescale global optimization problems.
Keywords
Evolutionary computation Global optimization Differential evolution Novel mutation Selfadaptive crossoverIntroduction
Differential evolution (DE)
This section provides a brief summary of the basic Differential Evolution (DE) algorithm. In simple DE, generally known as DE/rand/1/bin [23, 24], an initial random population consists of NP vectors \({\vec {X}},\forall \quad i=1,2,\ldots ,NP\), is randomly generated according to a uniform distribution within the lower and upper boundaries (\(x_j^\mathrm{L} ,x_j^\mathrm{U})\). After initialization, these individuals are evolved by DE operators (mutation and crossover) to generate a trial vector. A comparison between the parent and its trial vector is then done to select the vector which should survive to the next generation [9]. DE steps are discussed below:
Initialization
Mutation
Crossover
Selection
Related work

Cooperative Coevolution (CC) framework algorithms or divideandconquer methods.

Non Cooperative Coevolution (CC) framework algorithms or no divideandconquer methods.
EADE algorithm
In this section, we outline a novel DE algorithm, EADE, and explain the steps of the algorithm in details.
Novel mutation scheme
Parameter adaptation schemes in EADE
The successful performance of DE algorithm is significantly dependent upon the choice of its three control parameters: The scaling factor F, crossover rate CR, and population size NP [23, 25]. In fact, they have a vital role, because they greatly influence the effectiveness, efficiency, and robustness of the algorithm. Furthermore, it is difficult to determine the optimal values of the control parameters for a variety of problems with different characteristics at different stages of evolution. In the proposed HDE algorithm, NP is kept as a userspecified parameter, since it highly depends on the problem complexity. Generally speaking, F is an important parameter that controls the evolving rate of the population, i.e., it is closely related to the convergence speed [15]. A small F value encourages the exploitation tendency of the algorithm that makes the search focus on neighborhood of the current solutions; hence, it can enhance the convergence speed. However, it may also lead to premature convergence [43]. On the other hand, a large F value improves the exploration capability of the algorithm that can makes the mutant vectors distribute widely in the search space and can increase the diversity of the population [43]. However, it may slow down the search [43] with respect to the scaling factors in the proposed algorithm, at each generation G, the scale factors F1 and F2 of each individual target vector are independently generated according to uniform distribution in (0,1) to enrich the search behavior. The constant crossover (CR) reflects the probability with which the trial individual inherits the actual individual’s genes, i.e., which and how many components are mutated in each element of the current population [17, 43]. The constant crossover CR practically controls the diversity of the population [44]. As a matter of fact, if \(\hbox {CR}\) is high, this will increase the population diversity. Nevertheless, the stability of the algorithm may reduce. On the other hand, small values of CR increase the possibility of stagnation that may weak the exploration ability of the algorithm to open up new search space. In addition, CR is usually more sensitive to problems with different characteristics such as unimodality and multimodality, and separable and nonseparable problems. For separable problems, CR from the range (0, 0.2) is the best, while for multimodal, parameter dependent problems, CR in the range (0.9,1) is suitable [45]. On the other hand, there are wide varieties of approaches for adapting or selfadapting control parameters values through optimization process. Most of these methods based on generating random values from uniform, normal, or Cauchy distributions or by generating different values from predefined parameter candidate pool besides use the previous experience (of generating better solutions) to guide the adaptation of these parameters [11, 15, 16, 17, 19, 45, 46, 47, 48, 49]. The presented work proposed a novel selfadaptation scheme for CR. The core idea of the proposed selfadaptation scheme for the crossover rate CR is based on the following fundamental principle. In the initial stage of the search process, the difference among individual vectors is large, because the vectors in the population are completely dispersed or the population diversity is large due to the random distribution of the individuals in the search space that requires a relatively smaller crossover value. Then, as the population evolves through generations, the diversity of the population decreases as the vectors in the population are clustered, because each individual gets closer to the best vector found so far. Consequently, to maintain the population diversity and improve the convergence speed, crossover should be gradually utilized with larger values along with the generations of evolution increased to preserve well genes in so far as possible and promote the convergence performance. Therefore, the population diversity can be greatly enhanced through generations. However, there is no an appropriate CR value that balances both the diversity and convergence speed when solve a given problem during overall optimization process. Consequently, to address this problem and following the SaDE algorithm [15], in this paper, a novel adaptation scheme for CR is developed that can benefit from the past experience through generations of evolutionary.
Experimental study
Benchmark functions
 1.
Separable functions \(F_{1}\)–\(F_{3}\);
 2.
Partially separable functions, in which a small number of variables are dependent, while all the remaining ones are independent (\(m=50\)) \(F_{4}\)–\(F_{8}\);
 3.
Partially separable functions that consist of multiple independent subcomponents, each of which is mnonseparable (\(m=50\)) \(F_{9}\)–\(F_{18}\);
 4.
Fully nonseparable functions \(F_{19}\)–\(F_{20};\)
 1.
Separable functions: \(F_{1}, F_{4}, F_{5}\, \mathrm{and}\, F_{6}\);
 2.
nonseparable functions \(F_{2}, F_{3} \,\mathrm{and}\, F_{7}.\)
Parameter settings and involved algorithms

Cooperative Coevolution with Delta Grouping for LargeScale Nonseparable Function Optimization (DECCDML) [28].

Largescale optimization by Differential Evolution with Landscape modality detection and a diversity archive (LMDEa) [34].

LargeScale Global Optimization using Selfadaptive Differential Evolution Algorithm (jDElsgo) [35].

DE Enhanced by Neighborhood Search for LargeScale Global Optimization (SDENS) [36].

A competitive swarm optimizer for largescale optimization (CEO) [39].

A social learning particle swarm optimization algorithm for scalable optimization (SLPSO) [40].

Cooperatively coevolving particle swarms for largescale optimization (CCPSO2) [50].

A simple modification in CMAES achieving linear time and space complexity (sepCMAES) [51].

Solving largescale global optimization using improved particle swarm optimizer (EPUSPSO) [52].

Multilevel cooperative coevolution for largescale optimization (MLCC) [31].

Dynamic multiswarm particle swarm optimizer with local search for largescale global Optimization (DMSLPSO) [53].
To compare the solution quality from a statistical angle of different algorithms and to check the behavior of the stochastic algorithms [54], the results are compared using multiproblem Wilcoxon signedrank test at a 0.05 significance level. Wilcoxon signedrank test is a nonparametric statistical test that allows us to judge the difference between paired scores when it cannot make the assumption required by the pairedsample t test, such as that the population should be normally distributed, where R\(^{+}\) denotes the sum of ranks for the test problems in which the first algorithm performs better than the second algorithm (in the first column), and R\(^{}\) represents the sum of ranks for the test problems in which the first algorithm performs worse than the second algorithm (in the first column). Larger ranks indicate larger performance discrepancy. The numbers in Better, Equal, and Worse columns denote the number of problems in which the first algorithm is better than, equal, or worse than the second algorithm. As a null hypothesis, it is assumed that there is no significance difference between the mean results of the two samples. Whereas the alternative hypothesis is that there is significance in the mean results of the two samples, the number of test problems \(N=20\) for 1.25e\(+\)05, 6.00e\(+\)05, and 3.00e\(+\)006 Function evaluations for CEC’2010, while the number of test problems \(N=7\) for 5.00e\(+\)005, 2.50E\(+\)06, and 5.00e\(+\)006 Function evaluations with \(D=100\), \(D=500\), \(D=1000\) for CEC’2008 and 5% significance level. Use the smaller of the values of the sums as the test value and compare it with the critical value or use the p value and compare it with the significance level. Reject the null hypothesis if the test value is less than or equal to the critical value or if the p value is less than or equal to the significance level (5%). Based on the result of the test, one of three signs (\(+\), −, and \(\approx \)) is assigned for the comparison of any two algorithms (shown in the last column), where (\(+\)) sign means the first algorithm is significantly better than the second, (−) sign means that the first algorithm is significantly worse than the second, and (\(\approx \)) sign means that there is no significant difference between the two algorithms. In addition, to obtain the final rankings of different algorithms for all functions, the Friedman test is used at a 0.05 significance level. All the p values in this paper were computed using SPSS (the version is 20.00).
To perform comprehensive evaluation and to assess the effectiveness of the proposed selfadaptive crossover rate scheme and new mutation scheme, another version of EADE, named EADE*, has been tested and compared against EADE and other DEbased algorithms. EADE* is the same as EADE except that the new mutation scheme is only used.
Experimental results and discussions

For many test functions, the worst results obtained by the proposed algorithms are better than the best results obtained by other algorithms with all FEs.

For many test functions, there are continuous improvement in the results obtained by our proposed algorithms, especially EADE and EADE*, with all FEs, while the results with FEs = 6.0E\(+\)05 are very close to the results with FEs = 3.0E\(+\)06 obtained by some of the compared algorithms which indicate that our proposed approaches are scalable enough and can balance greatly the exploration and exploitation abilities for solving highdimensional problems until the maximum FEs are reached.

For many functions, the remarkable performance of EADE and EADE* with FEs = 1.20E\(+\)05, and FEs = 6.0E\(+\)05 compared to the performance of other algorithms shows its fast convergence behavior. Thus, our proposed algorithms can perform well and achieve good results within limited number of function evaluations which is very important issue when dealing with realworld problems.

EADE and EADE* got very close to the optimum of singlegroup mnonseparable multimodal functions F\(_{6}\) in all statistical results with 1.20E\(+\)05 FEs.

EADE and LMDEa, among all other algorithms, got very close to the optimum in all runs of singlegroup mnonseparable multimodal functions F\(_{8}\) with 3.0E\(+\)06 FEs.

The performance of EADE and EADE* performs well in all types of problems which indicate that it is less affected than the most of other algorithms by the characteristics of the problems.
Experimental comparisons between EADE, EADE*, and stateoftheart algorithms, FES \(=\) 1.20E\(+\)05
\(F_{1}\)  \(F_{2}\)  \(F_{3}\)  \(F_{4}\)  \(F_{5}\)  \(F_{6}\)  \(F_{7}\)  

EADE  
Best  3.45E\(+\)07  7.51E\(+\)03  6.79E\(+\)00  8.85E\(+\)12  2.48E\(+\)08  1.86E\(+\)01  1.14E\(+\)09 
Median  3.96E\(+\)07  7.67E\(+\)03  7.00E\(+\)00  1.30E\(+\)13  2.95E\(+\)08  1.89E\(+\)01  1.94E\(+\)09 
Worst  4.54E\(+\)07  7.92E\(+\)03  7.32E\(+\)00  3.13E\(+\)13  3.11E\(+\)08  2.14E\(+\)01  3.38E\(+\)09 
Mean  3.94E\(+\)07  7.70E\(+\)03  7.00E \(+\) 00  1.48E\(+\)13  2.87E\(+\)08  1.93E \(+\) 01  2.11E\(+\)09 
Std  2.80E\(+\)06  1.08E\(+\)02  1.51E−01  6.09E\(+\)12  1.80E\(+\)07  9.98E−01  8.17E\(+\)08 
EADE*  
Best  3.21E\(+\)07  7.94E\(+\)03  6.83E\(+\)00  3.94E\(+\)12  4.58E\(+\)07  2.00E\(+\)01  1.28E\(+\)08 
Median  3.58E\(+\)07  8.19E\(+\)03  7.08E\(+\)00  5.27E\(+\)12  1.56E\(+\)08  2.03E\(+\)01  2.63E\(+\)08 
Worst  4.07E\(+\)07  8.32E\(+\)03  7.73E\(+\)00  1.01E\(+\)13  2.39E\(+\)08  2.04E\(+\)01  1.06E\(+\)09 
Mean  3.61E\(+\)07  8.16E\(+\)03  7.12E\(+\)00  6.29E\(+\)12  1.52E \(+\) 08  2.03E\(+\)01  4.22E\(+\)08 
Std  2.44E\(+\)06  1.25E\(+\)02  2.28E−01  2.01E\(+\)12  6.45E\(+\)07  1.06E−01  3.10E\(+\)08 
LMDEa  
Best  4.40E\(+\)08  9.68E\(+\)03  1.43E\(+\)01  2.60E\(+\)13  2.35E\(+\)08  4.09E\(+\)04  6.05E\(+\)09 
Median  4.92E\(+\)08  9.84E\(+\)03  1.51E\(+\)01  6.08E\(+\)13  2.98E\(+\)08  6.44E\(+\)04  1.46E\(+\)10 
Worst  6.07E\(+\)08  1.02E\(+\)04  1.55E\(+\)01  9.40E\(+\)13  3.41E\(+\)08  1.53E\(+\)05  2.34E\(+\)10 
Mean  5.08E\(+\)08  9.89E\(+\)03  1.51E\(+\)01  6.25E\(+\)13  2.94E\(+\)08  6.90E\(+\)04  1.52E\(+\)10 
Std  4.76E\(+\)07  1.37E\(+\)02  2.50E−01  1.72E\(+\)13  2.51E\(+\)07  2.30E\(+\)04  3.94E\(+\)09 
SDENS  
Best  3.93E\(+\)09  1.16E\(+\)04  1.99E\(+\)01  3.90E\(+\)13  3.14E\(+\)08  9.88E\(+\)05  3.07E\(+\)10 
Median  4.74E\(+\)09  1.19E\(+\)04  2.01E\(+\)01  4.60E\(+\)13  3.32E\(+\)08  2.03E\(+\)06  3.57E\(+\)10 
Worst  6.19E\(+\)09  1.20E\(+\)04  2.02E\(+\)01  7.90E\(+\)13  3.41E\(+\)08  2.39E\(+\)06  4.70E\(+\)10 
Mean  5.01E\(+\)09  1.19E\(+\)04  2.01E\(+\)01  5.10E\(+\)13  3.29E\(+\)08  1.84E\(+\)06  3.75E\(+\)10 
Std  9.18E\(+\)08  9.89E\(+\)01  1.17E−01  1.46E\(+\)13  1.04E\(+\)07  4.77E\(+\)05  5.46E\(+\)09 
jDElsgo  
Best  2.78E\(+\)09  1.06E\(+\)04  1.81E\(+\)01  8.06E\(+\)13  2.98E\(+\)08  3.36E\(+\)06  2.89E\(+\)10 
Median  3.72E\(+\)09  1.09E\(+\)09  1.88E\(+\)01  1.43E\(+\)14  3.38E\(+\)08  4.24E\(+\)06  5.40E\(+\)10 
Worst  4.89E\(+\)09  1.13E\(+\)04  1.97E\(+\)01  2.30E\(+\)14  3.75E\(+\)08  4.84E\(+\)06  7.23E\(+\)10 
Mean  3.70E\(+\)09  1.09E\(+\)04  1.87E\(+\)01  1.40E\(+\)14  3.39E\(+\)08  4.26E\(+\)06  5.39E\(+\)10 
Std  5.11E\(+\)08  1.75E\(+\)02  4.46E−01  3.69E\(+\)13  1.82E\(+\)07  3.91E\(+\)05  1.07E\(+\)10 
DECCDML  
Best  2.28E\(+\)08  5.51E\(+\)03  8.22E\(+\)00  3.80E\(+\)13  1.43E\(+\)08  1.25E\(+\)06  2.65E\(+\)09 
Median  2.85E\(+\)08  5.76E\(+\)03  9.71E\(+\)00  6.40E\(+\)13  2.85E\(+\)08  1.96E\(+\)06  5.50E\(+\)09 
Worst  7.02E\(+\)08  5.96E\(+\)03  1.01E\(+\)01  1.20E\(+\)14  5.21E\(+\)08  2.00E\(+\)07  1.17E\(+\)10 
Mean  4.09E\(+\)08  5.75E \(+\) 03  9.51E\(+\)00  6.76E\(+\)13  3.00E\(+\)08  2.70E\(+\)06  5.97E\(+\)09 
Std  1.75E\(+\)08  1.35E\(+\)02  5.55E−01  2.02E\(+\)13  9.31E\(+\)07  3.62E\(+\)06  2.49E\(+\)09 
MASWchains  
Best  2.15E\(+\)07  3.32E\(+\)03  1.13E\(+\)01  1.22E\(+\)12  9.35E\(+\)07  2.02E\(+\)01  4.54E\(+\)06 
Median  2.76E\(+\)07  3.75E\(+\)03  1.15E\(+\)01  2.04E\(+\)12  2.64E\(+\)08  2.08E\(+\)01  4.91E\(+\)06 
Worst  3.51E\(+\)07  1.00E\(+\)04  1.22E\(+\)01  3.35E\(+\)12  3.42E\(+\)08  1.16E\(+\)06  5.71E\(+\)06 
Mean  2.83E \(+\) 07  5.09E\(+\)03  1.16E\(+\)01  2.12E \(+\) 12  2.52E\(+\)08  8.14E\(+\)04  4.90E \(+\) 06 
Std  3.06E\(+\)06  2.38E\(+\)03  2.68E−01  6.21E\(+\)11  6.49E\(+\)07  2.84E\(+\)05  2.59E\(+\)05 
\(F_{8}\)  \(F_{9}\)  \(F_{10}\)  \(F_{11}\)  \(F_{12}\)  \(F_{13}\)  \(F_{14}\)  

EADE  
Best  4.32E\(+\)07  2.05E\(+\)09  1.25E\(+\)04  7.47E\(+\)01  2.66E\(+\)06  5.06E\(+\)06  4.09E\(+\)09 
Median  4.54E\(+\)07  2.67E\(+\)09  1.29E\(+\)04  9.71E\(+\)01  3.00E\(+\)06  9.22E\(+\)06  4.49E\(+\)09 
Worst  1.04E\(+\)08  3.38E\(+\)09  1.33E\(+\)04  1.34E\(+\)02  4.13E\(+\)06  1.47E\(+\)07  5.41E\(+\)09 
Mean  5.31E \(+\) 07  2.63E\(+\)09  1.29E\(+\)04  1.01E\(+\)02  3.10E\(+\)06  9.03E\(+\)06  4.59E\(+\)09 
Std  2.04E\(+\)07  3.30E\(+\)08  2.46E\(+\)02  1.94E\(+\)01  4.04E\(+\)05  2.49E\(+\)06  3.92E\(+\)08 
EADE*  
Best  1.28E\(+\)07  1.58E\(+\)09  6.36E\(+\)03  1.71E\(+\)02  1.41E\(+\)06  1.40E\(+\)07  3.05E\(+\)09 
Median  9.20E\(+\)07  1.98E\(+\)09  6.58E\(+\)03  1.87E\(+\)02  1.54E\(+\)06  2.04E\(+\)07  3.71E\(+\)09 
Worst  1.54E\(+\)08  2.55E\(+\)09  7.07E\(+\)03  2.06E\(+\)02  1.78E\(+\)06  1.41E\(+\)08  4.14E\(+\)09 
Mean  8.05E\(+\)07  2.02E\(+\)09  6.63E\(+\)03  1.88E\(+\)02  1.56E\(+\)06  3.71E\(+\)07  3.67E\(+\)09 
Std  4.12E\(+\)07  2.87E\(+\)08  1.85E\(+\)02  9.52E\(+\)00  1.08E\(+\)05  3.30E\(+\)07  2.99E\(+\)08 
LMDEa  
Best  6.79E\(+\)07  4.06E\(+\)09  1.22E\(+\)04  2.16E\(+\)02  2.67E\(+\)06  2.11E\(+\)07  7.54E\(+\)09 
Median  1.71E\(+\)08  5.25E\(+\)09  1.27E\(+\)04  2.26E\(+\)02  2.84E\(+\)06  2.87E\(+\)07  9.59E\(+\)09 
Worst  4.46E\(+\)08  6.33E\(+\)09  1.34E\(+\)04  2.29E\(+\)02  3.05E\(+\)06  4.58E\(+\)07  1.16E\(+\)10 
Mean  1.88E\(+\)08  5.21E\(+\)09  1.27E\(+\)04  2.25E\(+\)02  2.85E\(+\)06  2.90E\(+\)07  9.64E\(+\)09 
Std  9.84E\(+\)07  5.87E\(+\)08  2.67E\(+\)02  2.90E\(+\)00  1.07E\(+\)05  5.51E\(+\)06  8.85E\(+\)08 
SDENS  
Best  6.05E\(+\)08  1.13E\(+\)10  1.37E\(+\)04  2.27E\(+\)02  2.71E\(+\)06  1.70E\(+\)10  1.42E\(+\)10 
Median  6.23E\(+\)08  1.52E\(+\)10  1.38E\(+\)04  2.27E\(+\)02  2.83E\(+\)06  1.91E\(+\)10  1.73E\(+\)10 
Worst  1.20E\(+\)09  1.89E\(+\)10  1.42E\(+\)04  2.28E\(+\)02  3.29E\(+\)06  2.01E\(+\)10  2.31E\(+\)10 
Mean  7.71E\(+\)08  1.56E\(+\)10  1.39E\(+\)04  2.27E\(+\)02  2.95E\(+\)06  1.88E\(+\)10  1.84E\(+\)10 
Std  2.27E\(+\)08  2.77E\(+\)09  2.51E\(+\)02  3.49E−01  2.37E\(+\)05  1.07E\(+\)09  3.56E\(+\)09 
jDElsgo  
Best  1.04E\(+\)09  1.43E\(+\)10  1.31E\(+\)04  2.02E\(+\)02  2.76E\(+\)06  2.23E\(+\)09  1.95E\(+\)10 
Median  2.29E\(+\)09  1.59E\(+\)10  1.43E\(+\)04  2.20E\(+\)02  3.18E\(+\)06  3.70E\(+\)09  2.32E\(+\)10 
Worst  5.42E\(+\)09  2.07E\(+\)10  1.51E\(+\)04  2.26E\(+\)02  3.65E\(+\)06  5.44E\(+\)09  2.76E\(+\)10 
Mean  2.39E\(+\)09  1.64E\(+\)10  1.43E\(+\)04  2.19E\(+\)02  3.15E\(+\)06  3.76E\(+\)09  2.32E\(+\)10 
Std  9.13E\(+\)08  1.73E\(+\)09  4.38E\(+\)02  5.92E\(+\)00  2.19E\(+\)05  1.04E\(+\)09  2.03E\(+\)09 
DECCDML  
Best  2.23E\(+\)09  4.09E\(+\)09  1.32E\(+\)04  1.02E\(+\)02  4.07E\(+\)06  1.09E\(+\)08  1.26E\(+\)10 
Median  4.92E\(+\)09  4.91E\(+\)09  1.39E\(+\)04  1.22E\(+\)02  4.68E\(+\)06  1.82E\(+\)08  1.37E\(+\)10 
Worst  1.35E\(+\)10  5.54E\(+\)09  1.45E\(+\)04  1.70E\(+\)02  5.35E\(+\)06  3.72E\(+\)08  1.51E\(+\)10 
Mean  5.57E\(+\)09  4.89E\(+\)09  1.38E\(+\)04  1.24E\(+\)02  4.70E\(+\)06  2.11E\(+\)08  1.37E\(+\)10 
Std  2.56E\(+\)09  3.77E\(+\)08  3.24E\(+\)02  1.38E\(+\)01  2.99E\(+\)05  9.68E\(+\)07  6.86E\(+\)08 
MASWchains  
Best  3.30E\(+\)07  4.48E\(+\)08  3.62E\(+\)03  5.01E\(+\)01  2.20E\(+\)05  7.64E\(+\)05  8.16E\(+\)08 
Median  4.17E\(+\)07  5.60E\(+\)08  4.15E\(+\)03  6.41E\(+\)01  2.40E\(+\)05  9.04E\(+\)05  8.81E\(+\)08 
Worst  8.55E\(+\)08  6.45E\(+\)08  1.00E\(+\)04  7.13E\(+\)01  2.62E\(+\)05  1.11E\(+\)06  1.04E\(+\)09 
Mean  1.21E\(+\)08  5.54E \(+\) 08  5.12E \(+\) 03  6.31E \(+\) 01  2.40E \(+\) 05  9.13E \(+\) 05  8.95E \(+\) 08 
Std  2.11E\(+\)08  5.20E\(+\)07  2.20E\(+\)03  5.53E\(+\)00  1.26E\(+\)04  8.09E\(+\)04  6.60E\(+\)07 
\(F_{15}\)  \(F_{16}\)  \(F_{17}\)  \(F_{18}\)  \(F_{19}\)  \(F_{20}\)  

EADE  
Best  1.36E\(+\)04  2.68E\(+\)02  3.12E\(+\)06  4.89E\(+\)09  2.22E\(+\)07  7.66E\(+\)09  
Median  1.40E\(+\)04  3.27E\(+\)02  3.68E\(+\)06  7.15E\(+\)09  2.51E\(+\)07  1.01E\(+\)10  
Worst  1.46E\(+\)04  3.74E\(+\)02  4.84E\(+\)06  1.00E\(+\)10  3.04E\(+\)07  1.26E\(+\)10  
Mean  1.40E\(+\)04  3.21E \(+\) 02  3.81E\(+\)06  7.00E\(+\)09  2.54E\(+\)07  9.97E\(+\)09  
Std  2.93E\(+\)02  3.73E\(+\)01  4.48E\(+\)05  1.47E\(+\)09  2.23E\(+\)06  1.30E\(+\)09  
EADE*  
Best  6.86E\(+\)03  3.91E\(+\)02  1.84E\(+\)06  1.48E\(+\)10  6.28E\(+\)06  1.68E\(+\)10  
Median  7.59E\(+\)03  3.99E\(+\)02  2.07E\(+\)06  1.81E\(+\)10  7.26E\(+\)06  2.83E\(+\)10  
Worst  1.21E\(+\)04  4.05E\(+\)02  2.26E\(+\)06  3.07E\(+\)10  8.11E\(+\)06  4.08E\(+\)10  
Mean  7.92E\(+\)03  3.98E\(+\)02  2.04E\(+\)06  1.96E\(+\)10  7.18E\(+\)06  2.75E\(+\)10  
Std  1.21E\(+\)03  3.65E\(+\)00  1.10E\(+\)05  4.53E\(+\)09  5.79E\(+\)05  5.74E\(+\)09  
LMDEa  
Best  1.32E\(+\)04  4.14E\(+\)02  4.21E\(+\)06  2.38E\(+\)09  8.75E\(+\)06  2.40E\(+\)09  
Median  1.37E\(+\)04  4.17E\(+\)02  4.70E\(+\)06  2.95E\(+\)09  9.96E\(+\)06  3.52E\(+\)09  
Worst  1.42E\(+\)04  4.18E\(+\)02  5.06E\(+\)06  3.97E\(+\)09  1.10E\(+\)07  4.80E\(+\)09  
Mean  1.37E\(+\)04  4.16E\(+\)02  4.65E\(+\)06  3.07E\(+\)09  1.00E\(+\)07  3.57E\(+\)09  
Std  2.66E\(+\)02  7.76E−01  2.37E\(+\)05  4.01E\(+\)08  5.88E\(+\)05  5.43E\(+\)08  
SDENS  
Best  1.36E\(+\)04  4.15E\(+\)02  3.84E\(+\)06  2.00E\(+\)11  1.19E\(+\)07  2.39E\(+\)11  
Median  1.45E\(+\)04  4.15E\(+\)02  4.25E\(+\)06  2.09E\(+\)11  1.57E\(+\)07  2.62E\(+\)11  
Worst  1.45E\(+\)04  4.15E\(+\)02  4.98E\(+\)06  2.35E\(+\)11  2.31E\(+\)07  2.82E\(+\)11  
Mean  1.43E\(+\)04  4.15E\(+\)02  4.31E\(+\)06  2.11E\(+\)11  1.67E\(+\)07  2.61E\(+\)11  
Std  3.72E\(+\)02  1.08E−01  4.04E\(+\)05  1.27E\(+\)10  3.71E\(+\)06  1.49E\(+\)10  
jDElsgo  
Best  1.44E\(+\)04  4.09E\(+\)02  4.28E\(+\)06  5.25E\(+\)10  2.07E\(+\)07  5.80E\(+\)10  
Median  1.55E\(+\)04  4.17E\(+\)02  4.79E\(+\)06  6.37E\(+\)10  2.88E\(+\)07  8.14E\(+\)10  
Worst  1.59E\(+\)04  4.24E\(+\)02  5.71E\(+\)06  8.76E\(+\)10  3.56E\(+\)07  1.11E\(+\)11  
Mean  1.54E\(+\)04  4.17E\(+\)02  4.85E\(+\)06  6.60E\(+\)10  2.85E\(+\)07  7.99E\(+\)10  
Std  3.33E\(+\)02  3.28E\(+\)00  3.53E\(+\)05  9.47E\(+\)09  3.38E\(+\)06  1.25E\(+\)10  
DECCDML  
Best  1.58E\(+\)04  3.22E\(+\)02  7.48E\(+\)06  1.56E\(+\)09  1.77E\(+\)07  2.04E\(+\)09  
Median  1.65E\(+\)04  3.73E\(+\)02  8.77E\(+\)06  3.30E\(+\)09  2.23E\(+\)07  3.93E\(+\)09  
Worst  1.73E\(+\)04  4.28E\(+\)02  1.01E\(+\)07  4.06E\(+\)09  2.72E\(+\)07  5.09E\(+\)09  
Mean  1.65E\(+\)04  3.75E\(+\)02  8.81E\(+\)06  3.08E\(+\)09  2.20E\(+\)07  3.84E\(+\)09  
Std  3.61E\(+\)02  3.60E\(+\)01  6.86E\(+\)05  7.84E\(+\)08  2.36E\(+\)06  7.72E\(+\)08  
MASWchains  
Best  3.94E\(+\)03  2.01E\(+\)02  5.80E\(+\)05  2.22E\(+\)04  3.23E\(+\)06  2.01E\(+\)03  
Median  4.29E\(+\)03  2.12E\(+\)02  6.78E\(+\)05  5.18E\(+\)04  3.63E\(+\)06  2.22E\(+\)03  
Worst  9.63E\(+\)03  2.31E\(+\)02  7.50E\(+\)05  8.33E\(+\)04  4.05E\(+\)06  4.69E\(+\)03  
Mean  4.83E \(+\) 03  2.13E \(+\) 02  6.78E \(+\) 05  5.14E \(+\) 04  3.63E \(+\) 06  2.43E \(+\) 03  
Std  1.51E\(+\)03  9.19E\(+\)00  3.52E\(+\)04  1.64E\(+\)04  1.94E\(+\)05  5.43E\(+\)02 
Experimental comparisons between EADE, EADE*, and stateoftheart algorithms, FES \(=\) 6.00E\(+\)05
\(F_{1}\)  \(F_{2}\)  \(F_{3}\)  \(F_{4}\)  \(F_{5}\)  \(F_{6}\)  \(F_{7}\)  

EADE  
Best  1.11E−02  1.94E\(+\)03  1.06E−04  4.63E\(+\)11  4.08E\(+\)07  1.59E\(+\)01  2.39E\(+\)05 
Median  1.38E−02  2.06E\(+\)03  2.01E−04  9.89E\(+\)11  6.37E\(+\)07  1.98E\(+\)01  4.24E\(+\)05 
Worst  2.19E−02  2.15E\(+\)03  4.90E−04  1.56E\(+\)12  1.06E\(+\)08  2.01E\(+\)01  6.62E\(+\)05 
Mean  1.45E−02  2.06E \(+\) 03  2.18E−04  9.17E\(+\)11  6.58E\(+\)07  1.96E\(+\)01  4.18E\(+\)05 
Std  3.17E−03  5.46E\(+\)01  1.04E−04  3.40E\(+\)11  1.70E\(+\)07  1.02E\(+\)00  9.99E\(+\)04 
EADE*  
Best  3.46E−02  2.21E\(+\)03  1.28E\(+\)00  3.54E\(+\)11  4.88E\(+\)07  1.65E\(+\)01  2.83E\(+\)05 
Median  4.57E−02  2.69E\(+\)03  1.63E\(+\)00  1.04E\(+\)12  5.77E\(+\)07  2.04E\(+\)01  3.72E\(+\)05 
Worst  1.26E−01  2.90E\(+\)03  1.87E\(+\)00  1.63E\(+\)12  8.56E\(+\)07  1.03E\(+\)06  6.42E\(+\)05 
Mean  5.45E−02  2.68E\(+\)03  1.56E\(+\)00  9.84E\(+\)11  6.25E \(+\) 07  4.11E\(+\)04  3.99E \(+\) 05 
Std  2.32E−02  1.88E\(+\)02  1.92E−01  3.84E\(+\)11  1.04E\(+\)07  2.05E\(+\)05  8.65E\(+\)04 
LMDEa  
Best  3.00E\(+\)02  3.23E\(+\)03  6.17E−01  1.65E\(+\)12  3.68E\(+\)07  4.73E\(+\)00  2.37E\(+\)07 
Median  4.32E\(+\)02  3.35E\(+\)03  9.23E−01  4.23E\(+\)12  6.17E\(+\)07  5.50E\(+\)00  5.03E\(+\)07 
Worst  6.90E\(+\)02  3.53E\(+\)03  1.10E\(+\)00  9.52E\(+\)12  1.50E\(+\)08  6.46E\(+\)00  1.34E\(+\)08 
Mean  4.59E\(+\)02  3.37E\(+\)03  9.16E−01  4.49E\(+\)12  7.21E\(+\)07  5.60E \(+\) 00  5.85E\(+\)07 
Std  1.09E\(+\)02  6.88E\(+\)01  1.10E−01  1.81E\(+\)12  2.74E\(+\)07  3.91E−01  2.99E\(+\)07 
SDENS  
Best  3.82E\(+\)06  7.00E\(+\)03  5.13E\(+\)00  8.47E\(+\)12  1.51E\(+\)08  1.38E\(+\)01  5.73E\(+\)09 
Median  4.59E\(+\)06  7.12E\(+\)03  6.27E\(+\)00  1.53E\(+\)13  1.83E\(+\)08  1.53E\(+\)01  7.73E\(+\)09 
Worst  1.95E\(+\)07  7.17E\(+\)03  6.76E\(+\)00  2.85E\(+\)13  2.12E\(+\)08  1.74E\(+\)01  1.36E\(+\)10 
Mean  7.87E\(+\)06  7.09E\(+\)03  6.12E\(+\)00  1.72E\(+\)13  1.81E\(+\)08  1.53E\(+\)01  9.28E\(+\)09 
Std  5.94E\(+\)06  6.76E\(+\)01  6.30E−01  6.68E\(+\)12  2.29E\(+\)07  1.18E\(+\)00  3.44E\(+\)09 
jDElsgo  
Best  7.04E\(+\)04  3.67E\(+\)03  9.70E−01  7.89E\(+\)12  1.42E\(+\)08  2.20E\(+\)01  3.36E\(+\)09 
Median  8.71E\(+\)04  3.93E\(+\)03  1.18E\(+\)00  1.29E\(+\)13  1.87E\(+\)08  3.24E\(+\)01  6.41E\(+\)09 
Worst  1.23E\(+\)05  4.20E\(+\)03  1.58E\(+\)00  2.67E\(+\)13  2.30E\(+\)08  3.14E\(+\)02  1.10E\(+\)10 
Mean  8.99E\(+\)04  3.95E\(+\)03  1.22E\(+\)00  1.39E\(+\)13  1.88E\(+\)08  5.97E\(+\)01  6.43E\(+\)09 
Std  1.39E\(+\)04  1.32E\(+\)02  1.38E−01  4.60E\(+\)12  2.31E\(+\)07  5.81E\(+\)01  2.123\(+\)09 
DECCDML  
Best  6.95E\(+\)01  2.51E\(+\)03  1.06E−02  7.92E\(+\)12  1.42E\(+\)08  4.59E\(+\)01  3.14E\(+\)08 
Median  4.63E\(+\)02  2.64E\(+\)03  1.83E−02  1.51E\(+\)13  2.85E\(+\)08  1.09E\(+\)02  5.42E\(+\)08 
Worst  1.22E\(+\)03  2.78E\(+\)03  2.20E−02  3.29E\(+\)13  5.20E\(+\)08  1.98E\(+\)07  9.17E\(+\)08 
Mean  6.02E\(+\)02  2.64E\(+\)03  1.81E−02  1.61E\(+\)13  2.99E\(+\)08  7.94E\(+\)05  5.84E\(+\)08 
Std  4.11E\(+\)02  5.88E\(+\)01  3.08E−03  6.19E\(+\)12  9.31E\(+\)07  3.97E\(+\)06  1.68E\(+\)08 
MASWchains  
Best  8.52E\(+\)02  2.36E\(+\)03  3.44E\(+\)00  4.29E\(+\)11  3.68E\(+\)07  3.61E\(+\)00  6.33E\(+\)04 
Median  1.55E\(+\)03  2.68E\(+\)03  3.83E\(+\)00  5.75E \(+\) 11  2.59E\(+\)08  1.78E\(+\)01  7.78E\(+\)05 
Worst  7.28E\(+\)03  2.97E\(+\)03  4.60E\(+\)00  7.42E\(+\)11  3.24E\(+\)08  1.16E\(+\)06  4.61E\(+\)06 
Mean  2.24E\(+\)03  2.67E\(+\)03  3.84E\(+\)00  5.79E \(+\) 11  2.17E\(+\)08  8.14E\(+\)04  8.35E \(+\) 05 
Std  1.71E\(+\)03  1.63E\(+\)02  2.13E−01  6.46E\(+\)10  8.56E\(+\)07  2.84E\(+\)05  9.08E\(+\)05 
\(F_{8}\)  \(F_{9}\)  \(F_{10}\)  \(F_{11}\)  \(F_{12}\)  \(F_{13}\)  \(F_{14}\)  

EADE  
Best  2.58E\(+\)07  1.79E\(+\)08  3.11E\(+\)03  1.58E\(+\)02  3.40E\(+\)05  3.24E\(+\)03  7.66E\(+\)08 
Median  3.61E\(+\)07  2.20E\(+\)08  3.65E\(+\)03  1.67E\(+\)02  3.82E\(+\)05  5.06E\(+\)03  7.76E\(+\)08 
Worst  1.28E\(+\)08  2.62E\(+\)08  4.02E\(+\)03  1.77E\(+\)02  4.35E\(+\)05  1.85E\(+\)04  8.80E\(+\)08 
Mean  5.95E\(+\)07  2.21E\(+\)08  3.66E\(+\)03  1.68E\(+\)02  3.89E \(+\) 05  6.21E\(+\)03  8.13E\(+\)08 
Std  3.40E\(+\)07  2.28E\(+\)07  2.10E\(+\)02  6.33E\(+\)00  2.89E\(+\)04  3.96E\(+\)03  5.65E\(+\)07 
EADE*  
Best  9.90E\(+\)06  1.81E\(+\)08  3.73E\(+\)03  1.56E\(+\)02  3.25E\(+\)05  5.17E\(+\)03  6.75E\(+\)08 
Median  3.44E\(+\)07  2.27E\(+\)08  4.09E\(+\)03  1.72E\(+\)02  3.99E\(+\)05  1.89E\(+\)04  7.34E\(+\)08 
Worst  9.69E\(+\)07  4.29E\(+\)08  4.46E\(+\)03  1.84E\(+\)02  4.78E\(+\)05  4.79E\(+\)04  8.01E\(+\)08 
Mean  4.64E\(+\)07  2.40E\(+\)08  4.05E\(+\)03  1.73E\(+\)02  4.00E\(+\)05  2.01E\(+\)04  7.33E\(+\)08 
Std  2.99E\(+\)07  5.32E\(+\)07  1.49E\(+\)02  7.50E\(+\)00  3.52E\(+\)04  1.20E\(+\)04  3.85E\(+\)07 
LMDEa  
Best  3.19E\(+\)07  2.26E\(+\)08  7.82E\(+\)03  4.18E\(+\)01  3.91E\(+\)05  1.48E\(+\)03  6.49E\(+\)08 
Median  3.30E\(+\)07  2.76E\(+\)08  9.68E\(+\)03  6.70E\(+\)01  4.51E\(+\)05  2.07E\(+\)03  7.55E\(+\)08 
Worst  3.49E\(+\)07  3.25E\(+\)08  1.02E\(+\)04  1.14E\(+\)02  5.17E\(+\)05  5.72E\(+\)03  1.01E\(+\)09 
Mean  3.32E \(+\) 07  2.71E\(+\)08  9.65E\(+\)03  6.91E\(+\)01  4.50E\(+\)05  2.34E \(+\) 03  7.70E\(+\)08 
Std  8.55E\(+\)05  2.40E\(+\)07  4.76E\(+\)02  1.71E\(+\)01  3.21E\(+\)04  9.90E\(+\)02  7.45E\(+\)07 
SDENS  
Best  4.64E\(+\)07  1.78E\(+\)09  1.02E\(+\)04  2.25E\(+\)02  1.25E\(+\)06  4.37E\(+\)05  3.91E\(+\)09 
Median  6.40E\(+\)07  2.13E\(+\)09  1.09E\(+\)04  2.26E\(+\)02  1.30E\(+\)06  6.67E\(+\)05  5.02E\(+\)09 
Worst  1.09E\(+\)08  2.88E\(+\)09  1.15E\(+\)04  2.26E\(+\)02  1.42E\(+\)06  7.64E\(+\)05  6.93E\(+\)09 
Mean  7.41E\(+\)07  2.23E\(+\)09  1.10E\(+\)04  2.26E\(+\)02  1.32E\(+\)06  6.43E\(+\)05  5.14E\(+\)09 
Std  2.73E\(+\)07  3.70E\(+\)08  4.59E\(+\)02  3.83E−01  5.98E\(+\)04  1.10E\(+\)05  9.89E\(+\)08 
jDElsgo  
Best  3.57E\(+\)07  1.45E\(+\)09  7.66E\(+\)03  7.73E\(+\)01  8.61E\(+\)05  2.90E\(+\)04  3.61E\(+\)09 
Median  4.65E\(+\)07  1.64E\(+\)09  8.69E\(+\)03  1.14E\(+\)02  9.35E\(+\)05  4.95E\(+\)04  4.11E\(+\)09 
Worst  1.39E\(+\)08  1.82E\(+\)09  9.49E\(+\)03  1.49E\(+\)02  9.88E\(+\)05  9.02E\(+\)04  4.72E\(+\)09 
Mean  6.82E\(+\)07  1.66E\(+\)09  8.67E\(+\)03  1.17E\(+\)02  9.39E\(+\)05  5.32E\(+\)04  4.10E\(+\)09 
Std  3.53E\(+\)07  8.29E\(+\)07  3.99E\(+\)02  1.87E\(+\)01  2.96E\(+\)04  1.70E\(+\)04  2.89E\(+\)08 
DECCDML  
Best  4.19E\(+\)07  2.82E\(+\)08  1.25E\(+\)04  4.00E−01  3.64E\(+\)06  8.29E\(+\)02  9.82E\(+\)08 
Median  1.15E\(+\)08  3.85E\(+\)08  1.30E\(+\)04  7.09E−01  4.22E\(+\)06  1.71E\(+\)03  1.18E\(+\)09 
Worst  2.38E\(+\)08  4.21E\(+\)08  1.36E\(+\)04  1.72E\(+\)00  4.65E\(+\)06  1.47E\(+\)04  1.29E\(+\)09 
Mean  1.24E\(+\)08  3.73E\(+\)08  1.30E\(+\)04  7.66E−01  4.19E\(+\)06  3.15E\(+\)03  1.17E\(+\)09 
Std  5.40E\(+\)07  3.13E\(+\)07  2.93E\(+\)02  2.81E−01  2.18E\(+\)05  3.09E\(+\)03  8.20E\(+\)07 
MASWchains  
Best  3.42E\(+\)06  6.93E\(+\)07  2.79E\(+\)03  2.77E\(+\)01  1.39E\(+\)03  1.08E\(+\)03  1.51E\(+\)08 
Median  1.90E\(+\)07  8.08E\(+\)07  3.25E\(+\)03  3.79E\(+\)01  1.64E\(+\)03  3.06E\(+\)03  1.70E\(+\)08 
Worst  6.11E\(+\)08  1.00E\(+\)08  3.54E\(+\)03  5.15E\(+\)01  1.91E\(+\)03  1.07E\(+\)04  1.95E\(+\)08 
Mean  6.13E\(+\)07  8.18E\(+\)07  3.22E \(+\) 03  3.83E\(+\)01  1.63E \(+\) 03  4.34E\(+\)03  1.69E \(+\) 08 
Std  1.27E\(+\)08  8.36E\(+\)06  1.85E\(+\)02  7.23E\(+\)00  1.53E\(+\)02  3.21E\(+\)03  1.17E\(+\)07 
\(F_{15}\)  \(F_{16}\)  \(F_{17}\)  \(F_{18}\)  \(F_{19}\)  \(F_{20}\)  

EADE  
Best  3.97E\(+\)03  2.74E\(+\)02  9.31E\(+\)05  1.67E\(+\)04  7.60E\(+\)06  7.13E\(+\)03  
Median  4.34E\(+\)03  3.24E\(+\)02  1.04E\(+\)06  4.47E\(+\)04  1.15E\(+\)07  9.68E\(+\)03  
Worst  9.42E\(+\)03  3.58E\(+\)02  1.14E\(+\)06  6.22E\(+\)04  1.50E\(+\)07  1.24E\(+\)04  
Mean  4.96E\(+\)03  3.21E\(+\)02  1.04E\(+\)06  4.39E\(+\)04  1.12E\(+\)07  9.73E\(+\)03  
Std  1.75E\(+\)03  2.21E\(+\)01  5.41E\(+\)04  1.29E\(+\)04  2.47E\(+\)06  1.64E\(+\)03  
EADE*  
Best  4.40E\(+\)03  3.44E\(+\)02  7.51E\(+\)05  1.70E\(+\)05  2.73E\(+\)06  2.12E\(+\)05  
Median  4.63E\(+\)03  3.57E\(+\)02  8.07E\(+\)05  5.63E\(+\)05  2.93E\(+\)06  4.61E\(+\)05  
Worst  4.81E\(+\)03  3.95E\(+\)02  8.98E\(+\)05  1.47E\(+\)06  3.19E\(+\)06  3.73E\(+\)07  
Mean  4.63E\(+\)03  3.63E\(+\)02  8.10E\(+\)05  5.81E\(+\)05  2.95E\(+\)06  3.13E\(+\)06  
Std  1.39E\(+\)02  1.55E\(+\)01  3.95E\(+\)04  3.75E\(+\)05  1.32E\(+\)05  9.47E\(+\)06  
LMDEa  
Best  1.14E\(+\)04  4.09E\(+\)02  1.27E\(+\)06  6.11E\(+\)03  2.91E\(+\)06  3.33E\(+\)03  
Median  1.21E\(+\)04  4.13E\(+\)02  1.38E\(+\)06  1.80E\(+\)04  3.15E\(+\)06  3.99E\(+\)03  
Worst  1.24E\(+\)04  4.14E\(+\)02  1.48E\(+\)06  3.24E\(+\)04  3.40E\(+\)06  4.56E\(+\)03  
Mean  1.20E\(+\)04  4.13E\(+\)02  1.38E\(+\)06  1.90E\(+\)04  3.17E\(+\)06  4.02E\(+\)03  
Std  3.04E\(+\)02  1.21E\(+\)00  5.84E\(+\)04  6.80E\(+\)03  1.44E\(+\)05  2.94E\(+\)02  
SDENS  
Best  7.32E\(+\)03  4.13E\(+\)02  1.96E\(+\)06  1.65E\(+\)08  4.92E\(+\)06  1.36E\(+\)08  
Median  1.18E\(+\)04  4.13E\(+\)02  2.02E\(+\)06  1.86E\(+\)08  5.39E\(+\)06  2.78E\(+\)08  
Worst  1.26E\(+\)04  4.14E\(+\)02  2.29E\(+\)06  3.00E\(+\)08  6.18E\(+\)06  3.52E\(+\)08  
Mean  1.03E\(+\)04  4.13E\(+\)02  2.07E\(+\)06  2.02E\(+\)08  5.41E\(+\)06  2.69E\(+\)08  
Std  2.29E\(+\)03  3.49E−01  1.17E\(+\)05  5.02E\(+\)07  4.31E\(+\)05  7.57E\(+\)07  
jDElsgo  
Best  1.10E\(+\)04  2.67E\(+\)02  1.81E\(+\)06  6.95E\(+\)05  4.16E\(+\)06  5.98E\(+\)05  
Median  1.22E\(+\)04  2.97E\(+\)02  1.97E\(+\)06  1.04E\(+\)06  5.48E\(+\)06  1.05E\(+\)06  
Worst  1.26E\(+\)04  3.53E\(+\)02  2.10E\(+\)06  1.39E\(+\)06  1.22E\(+\)07  1.66E\(+\)06  
Mean  1.20E\(+\)04  2.99E\(+\)02  1.95E\(+\)06  1.03E\(+\)06  6.09E\(+\)06  1.01E\(+\)06  
Std  5.30E\(+\)02  1.91E\(+\)01  6.54E\(+\)04  2.08E\(+\)05  1.65E\(+\)06  2.48E\(+\)05  
DECCDML  
Best  1.53E\(+\)04  3.46E\(+\)00  6.50E\(+\)06  5.64E\(+\)03  1.54E\(+\)07  1.43E\(+\)03  
Median  1.59E\(+\)04  8.65E\(+\)00  7.29E\(+\)06  1.48E\(+\)04  1.84E\(+\)07  1.67E\(+\)03  
Worst  1.67E\(+\)04  4.28E\(+\)02  7.99E\(+\)06  3.96E\(+\)04  2.46E\(+\)07  2.02E\(+\)03  
Mean  1.59E\(+\)04  4.47E \(+\) 01  7.27E\(+\)06  1.74E\(+\)04  1.87E\(+\)07  1.69E\(+\)03  
Std  3.63E\(+\)02  1.16E\(+\)02  3.77E\(+\)05  8.26E\(+\)03  1.99E\(+\)06  1.58E\(+\)02  
MASWchains  
Best  2.95E\(+\)03  8.51E\(+\)01  3.59E\(+\)04  1.80E\(+\)03  1.29E\(+\)06  1.02E\(+\)03  
Median  3.19E\(+\)03  9.71E\(+\)01  4.29E\(+\)04  3.89E\(+\)03  1.42E\(+\)06  1.18E\(+\)03  
Worst  3.45E\(+\)03  1.26E\(+\)02  5.01E\(+\)04  1.61E\(+\)04  1.58E\(+\)06  1.65E\(+\)03  
Mean  3.19E \(+\) 03  1.02E\(+\)02  4.31E \(+\) 04  5.53E \(+\) 03  1.41E \(+\) 06  1.21E \(+\) 03  
Std  1.46E\(+\)02  1.42E\(+\)01  3.42E\(+\)03  3.94E\(+\)03  7.44E\(+\)04  1.42E\(+\)02 
Experimental comparisons between EADE, EADE*, and stateoftheart algorithms, FES \(=\) 3.0E\(+\)06
\(F_{1}\)  \(F_{2}\)  \(F_{3}\)  \(F_{4}\)  \(F_{5}\)  \(F_{6}\)  \(F_{7}\)  

EADE  
Best  1.53E−23  3.80E\(+\)02  5.68E−14  4.94E\(+\)10  4.48E\(+\)07  1.89E\(+\)01  3.31E−03 
Median  1.23E−22  4.17E\(+\)02  6.39E−14  8.91E\(+\)10  8.66E\(+\)07  1.90E\(+\)01  3.60E−02 
Worst  2.75E−21  4.53E\(+\)02  6.39E−14  2.69E\(+\)11  1.69E\(+\)08  1.91E\(+\)01  1.92E\(+\)00 
Mean  4.70E−22  4.16E\(+\)02  6.25E−14  1.08E\(+\)11  8.79E\(+\)07  1.90E\(+\)01  2.11−001 
Std  8.65E−22  2.07E\(+\)01  2.62E−15  6.56E\(+\)10  3.11E\(+\)07  7.06E−02  4.87E−01 
EADE*  
Best  1.09E−22  7.43E\(+\)02  1.35E\(+\)00  6.80E\(+\)10  5.57E\(+\)07  1.95E\(+\)01  1.54E−02 
Median  3.27E−22  8.39E\(+\)02  1.68E\(+\)00  1.33E\(+\)11  7.96E\(+\)07  1.96E\(+\)01  3.88E−01 
Worst  1.19E−21  8.97E\(+\)02  1.74E\(+\)00  2.25E\(+\)11  1.19E\(+\)08  1.96E\(+\)01  1.85E\(+\)00 
Mean  4.18E−22  8.27E\(+\)02  1.60E\(+\)00  1.37E\(+\)11  8.48E\(+\)07  1.96E\(+\)01  4.18E−01 
Std  3.12E−22  4.89E\(+\)01  1.47E−01  3.79E\(+\)10  1.80E\(+\)07  2.22E−02  4.81E−01 
LMDEa  
Best  2.42E−24  5.31E\(+\)02  7.86E−14  1.14E\(+\)11  3.68E\(+\)07  4.00E−09  3.79E−02 
Median  7.36E−24  6.87E\(+\)02  8.79E−01  2.06E\(+\)11  6.07E\(+\)07  4.02E−09  1.91E−01 
Worst  1.55E−22  8.62E\(+\)02  1.09E\(+\)00  3.15E\(+\)11  1.13E\(+\)08  1.01E\(+\)00  6.42E−01 
Mean  1.35E−23  6.97E\(+\)02  6.44E−01  2.08E\(+\)11  6.62E \(+\) 07  2.63E−01  2.45E−01 
Std  2.91E−23  8.22E\(+\)01  4.46E−01  5.89E\(+\)10  2.02E\(+\)07  4.22E−01  1.68E−01 
SDENS  
Best  1.75E−06  2.14E\(+\)03  1.23E−05  3.26E\(+\)12  7.66E\(+\)07  1.53E−04  6.36E\(+\)07 
Median  2.54E−06  2.17E\(+\)03  2.35E−05  3.72E\(+\)12  1.17E\(+\)08  1.76E−04  8.57E\(+\)07 
Worst  1.16E−05  2.39E\(+\)03  5.50E−05  8.99E\(+\)12  1.52E\(+\)08  2.57E−04  2.39E\(+\)08 
Mean  5.73E−06  2.21E\(+\)03  2.70E−05  5.11E\(+\)12  1.18E\(+\)08  2.02E−04  1.20E\(+\)08 
Std  4.46E−06  8.95E\(+\)01  1.54E−05  2.16E\(+\)12  2.88E\(+\)07  4.29E−05  6.56E\(+\)07 
jDElsgo  
Best  4.78E−20  1.09E−11  1.63E−12  3.09E\(+\)10  7.42E\(+\)07  7.14E−09  2.69E−05 
Median  6.63E−20  4.69E−11  2.35E−12  8.28E\(+\)10  9.82E\(+\)07  7.22E−09  1.04E−04 
Worst  2.24E−19  1.12E\(+\)00  2.24E−11  1.34E\(+\)11  1.24E\(+\)08  2.10E−07  3.19E−01 
Mean  8.86E−20  1.25E−01  3.81E−12  8.06E \(+\) 10  9.72E\(+\)07  1.70E−08  1.31E−02 
Std  4.51E−20  3.45E−01  5.02E−12  3.08E\(+\)10  1.44E\(+\)07  4.03E−08  6.38E−02 
DECCDML  
Best  9.05E−27  1.62E\(+\)02  1.10E−13  1.38E\(+\)12  1.42E\(+\)08  3.55E−09  7.09E\(+\)07 
Median  1.22E−25  2.12E\(+\)02  1.14E−13  3.32E\(+\)12  2.85E\(+\)08  7.11E−09  1.23E\(+\)08 
Worst  7.12E−25  2.94E\(+\)02  1.35E−13  6.89E\(+\)12  5.20E\(+\)08  1.98E\(+\)07  4.82E\(+\)08 
Mean  1.93E−25  2.17E\(+\)02  1.18E−13  3.58E\(+\)12  2.99E\(+\)08  7.93E\(+\)05  1.39E\(+\)08 
Std  1.86E−25  2.98E\(+\)01  8.22E−15  1.54E\(+\)12  9.31E\(+\)07  3.97E\(+\)06  7.72E\(+\)07 
MASWchains  
Best  3.18E−15  7.04E\(+\)02  3.34E−13  3.04E\(+\)11  2.89E\(+\)07  8.13E−07  3.35E−03 
Median  1.50E−14  7.90E\(+\)02  6.11E−13  3.54E\(+\)11  2.31E\(+\)08  1.60E\(+\)00  9.04E\(+\)01 
Worst  8.15E−14  9.37E\(+\)02  1.58E−12  3.97E\(+\)11  2.90E\(+\)08  1.16E\(+\)06  2.68E\(+\)02 
Mean  2.10E−14  8.10E\(+\)02  7.28E−13  3.53E\(+\)11  1.68E\(+\)08  8.14E\(+\)04  1.03E\(+\)02 
Std  1.99E−14  5.88E\(+\)01  3.40E−13  3.12E\(+\)10  1.04E\(+\)08  2.84E\(+\)05  8.70E\(+\)01 
\(F_{8}\)  \(F_{9}\)  \(F_{10}\)  \(F_{11}\)  \(F_{12}\)  \(F_{13}\)  \(F_{14}\)  

EADE  
Best  2.40E−10  3.08E\(+\)07  2.53E\(+\)03  1.02E\(+\)02  1.67E\(+\)04  7.18E\(+\)02  1.28E\(+\)08 
Median  2.97E−05  3.59E\(+\)07  2.64E\(+\)03  1.15E\(+\)02  2.77E\(+\)04  9.89E\(+\)02  1.47E\(+\)08 
Worst  1.63E−03  4.29E\(+\)07  2.78E\(+\)03  1.22E\(+\)02  4.12E\(+\)04  1.36E\(+\)03  1.61E\(+\)08 
Mean  2.26E−04  3.67E\(+\)07  2.62E \(+\) 03  1.14E\(+\)02  2.80E\(+\)04  1.01E\(+\)03  1.46E\(+\)08 
Std  4.41E−04  3.48E\(+\)06  8.12E\(+\)01  6.43E\(+\)00  5.72E\(+\)03  1.83E\(+\)02  9.60E\(+\)06 
EADE*  
Best  5.49E−08  2.82E\(+\)07  2.78E\(+\)03  1.07E\(+\)02  2.75E\(+\)04  5.73E\(+\)02  1.19E\(+\)08 
Median  1.50E−03  3.22E\(+\)07  3.01E\(+\)03  1.23E\(+\)02  4.92E\(+\)04  1.04E\(+\)03  1.35E\(+\)08 
Worst  3.99E\(+\)06  3.86E\(+\)07  3.24E\(+\)03  1.34E\(+\)02  7.08E\(+\)04  1.39E\(+\)03  1.54E\(+\)08 
Mean  2.66E\(+\)05  3.29E\(+\)07  3.00E\(+\)03  1.23E\(+\)02  4.98E\(+\)04  1.06E\(+\)03  1.35E\(+\)08 
Std  1.03E\(+\)06  2.94E\(+\)06  1.07E\(+\)02  8.88E\(+\)00  1.32E\(+\)04  2.10E\(+\)02  1.12E\(+\)07 
LMDEa  
Best  7.80E−05  2.25E\(+\)07  2.45E\(+\)03  6.49E−11  1.35E\(+\)04  4.56E\(+\)02  7.65E\(+\)07 
Median  3.29E−04  2.65E\(+\)07  2.76E\(+\)03  2.74E\(+\)00  1.81E\(+\)04  5.64E\(+\)02  8.67E\(+\)07 
Worst  9.20E−04  2.98E\(+\)07  3.97E\(+\)03  5.40E\(+\)01  2.49E\(+\)04  9.38E\(+\)02  1.03E\(+\)08 
Mean  3.61E−04  2.64E\(+\)07  2.80E\(+\)03  1.19E\(+\)01  1.83E\(+\)04  5.95E\(+\)02  8.63E \(+\) 07 
Std  2.33E−04  1.89E\(+\)06  2.84E\(+\)02  1.50E\(+\)01  2.62E\(+\)03  1.06E\(+\)02  6.30E\(+\)06 
SDENS  
Best  3.96E\(+\)07  4.77E\(+\)08  5.78E\(+\)03  2.20E\(+\)02  3.80E\(+\)05  1.16E\(+\)03  1.61E\(+\)09 
Median  4.09E\(+\)07  5.75E\(+\)08  7.03E\(+\)03  2.21E\(+\)02  3.95E\(+\)05  1.80E\(+\)03  1.86E\(+\)09 
Worst  9.35E\(+\)07  6.38E\(+\)08  7.37E\(+\)03  2.22E\(+\)02  4.97E\(+\)05  4.13E\(+\)03  2.30E\(+\)09 
Mean  5.12E\(+\)07  5.63E\(+\)08  6.87E\(+\)03  2.21E\(+\)02  4.13E\(+\)05  2.19E\(+\)03  1.88E\(+\)09 
Std  2.12E\(+\)07  5.78E\(+\)07  5.60E\(+\)02  5.09E−01  4.28E\(+\)04  1.03E\(+\)03  2.33E\(+\)08 
jDElsgo  
Best  3.40E−03  2.36E\(+\)07  2.10E\(+\)03  1.27E\(+\)00  8.32E\(+\)03  4.79E\(+\)02  1.28E\(+\)08 
Median  1.25E\(+\)06  3.04E\(+\)07  2.66E\(+\)03  1.95E\(+\)01  1.18E\(+\)04  6.91E\(+\)02  1.72E\(+\)08 
Worst  8.08E\(+\)06  4.22E\(+\)07  3.30E\(+\)03  5.81E\(+\)01  1.71E\(+\)04  1.02E\(+\)03  2.02E\(+\)08 
Mean  3.15E\(+\)06  3.11E\(+\)07  2.64E\(+\)03  2.20E\(+\)01  1.21E\(+\)04  7.11E \(+\) 02  1.69E\(+\)08 
Std  3.27E\(+\)06  5.00E\(+\)06  3.19E\(+\)02  1.53E\(+\)01  2.04E\(+\)03  1.37E\(+\)02  2.08E\(+\)07 
DECCDML  
Best  7.34E\(+\)05  4.51E\(+\)07  1.21E\(+\)04  1.63E−13  3.46E\(+\)06  6.19E\(+\)02  1.54E\(+\)08 
Median  1.57E\(+\)07  5.97E\(+\)07  1.24E\(+\)04  1.78E−13  3.81E\(+\)06  1.06E\(+\)03  1.89E\(+\)08 
Worst  1.21E\(+\)08  7.09E\(+\)07  1.30E\(+\)04  2.03E−13  4.11E\(+\)06  2.09E\(+\)03  2.22E\(+\)08 
Mean  3.46E\(+\)07  5.92E\(+\)07  1.25E\(+\)04  1.80E−13  3.80E\(+\)06  1.14E\(+\)03  1.89E\(+\)08 
Std  3.56E\(+\)07  4.71E\(+\)06  2.66E\(+\)02  9.88E−15  1.50E\(+\)05  4.31E\(+\)02  1.49E\(+\)07 
MASWchains  
Best  1.54E\(+\)06  1.19E\(+\)07  1.81E\(+\)03  2.74E\(+\)01  2.65E−06  3.86E\(+\)02  2.79E\(+\)07 
Median  3.43E\(+\)06  1.40E\(+\)07  2.07E\(+\)03  3.75E\(+\)01  3.50E−06  1.07E\(+\)03  3.09E\(+\)07 
Worst  1.80E\(+\)08  1.62E\(+\)07  2.28E\(+\)03  5.11E\(+\)01  4.98E−06  2.92E\(+\)03  3.67E\(+\)07 
Mean  1.41E\(+\)07  1.41E \(+\) 07  2.07E\(+\)03  3.80E\(+\)01  3.62E−06  1.25E\(+\)03  3.11E \(+\) 07 
Std  3.68E\(+\)07  1.15E\(+\)06  1.44E\(+\)02  7.35E\(+\)00  5.92E−07  5.72E\(+\)02  1.93E\(+\)06 
\(F_{15}\)  \(F_{16}\)  \(F_{17}\)  \(F_{18}\)  \(F_{19}\)  \(F_{20}\)  

EADE  
Best  2.93E\(+\)03  2.86E\(+\)02  1.33E\(+\)05  1.74E\(+\)03  1.20E\(+\)06  1.93E\(+\)03  
Median  3.20E\(+\)03  2.99E\(+\)02  1.55E\(+\)05  2.27E\(+\)03  1.30E\(+\)06  2.06E\(+\)03  
Worst  3.41E\(+\)03  3.12E\(+\)02  1.66E\(+\)05  3.16E\(+\)03  1.38E\(+\)06  2.39E\(+\)03  
Mean  3.18E \(+\) 03  3.00E\(+\)02  1.52E\(+\)05  2.26E\(+\)03  1.29E\(+\)06  2.10E\(+\)03  
Std  1.33E\(+\)02  5.81E\(+\)00  1.14E\(+\)04  3.63E\(+\)02  8.25E\(+\)04  1.33E\(+\)02  
EADE*  
Best  3.18E\(+\)03  2.89E\(+\)02  1.23E\(+\)05  2.07E\(+\)03  1.35E\(+\)06  1.72E\(+\)03  
Median  3.50E\(+\)03  3.00E\(+\)02  1.59E\(+\)05  2.21E\(+\)03  1.47E\(+\)06  2.17E\(+\)03  
Worst  3.78E\(+\)03  3.10E\(+\)02  1.86E\(+\)05  2.94E\(+\)03  1.59E\(+\)06  2.53E\(+\)03  
Mean  3.50E\(+\)03  3.00E\(+\)02  1.59E\(+\)05  2.28E\(+\)03  1.47E\(+\)06  2.20E\(+\)03  
Std  1.54E\(+\)02  6.07E\(+\)00  2.28E\(+\)04  3.04E\(+\)02  8.18E\(+\)04  2.15E\(+\)02  
LMDEa  
Best  5.15E\(+\)03  3.75E\(+\)02  1.92E\(+\)05  1.34E\(+\)03  4.07E\(+\)05  1.11E\(+\)03  
Median  5.64E\(+\)03  3.85E\(+\)02  2.13E\(+\)05  1.65E\(+\)03  4.43E\(+\)05  1.38E\(+\)03  
Worst  6.34E\(+\)03  4.00E\(+\)02  2.43E\(+\)05  2.24E\(+\)03  4.93E\(+\)05  1.60E\(+\)03  
Mean  5.63E\(+\)03  3.87E\(+\)02  2.14E\(+\)05  1.68E\(+\)03  4.42E\(+\)05  1.38E\(+\)03  
Std  2.81E\(+\)02  5.24E\(+\)00  1.47E\(+\)04  2.09E\(+\)02  1.85E\(+\)04  1.16E\(+\)02  
SDENS  
Best  7.14E\(+\)03  4.03E\(+\)02  8.78E\(+\)05  1.16E\(+\)04  7.57E\(+\)05  9.81E\(+\)02  
Median  7.32E\(+\)03  4.09E\(+\)02  1.14E\(+\)06  3.32E\(+\)04  8.02E\(+\)05  9.83E\(+\)02  
Worst  7.44E\(+\)03  4.10E\(+\)02  1.18E\(+\)06  4.51E\(+\)04  1.19E\(+\)06  1.02E\(+\)03  
Mean  7.32E\(+\)03  4.08E\(+\)02  1.08E\(+\)06  3.08E\(+\)04  8.80E\(+\)05  9.90E\(+\)02  
Std  9.63E\(+\)01  2.53E\(+\)00  1.11E\(+\)05  1.22E\(+\)04  1.59E\(+\)05  1.62E\(+\)01  
jDElsgo  
Best  5.20E\(+\)03  7.30E\(+\)01  7.75E\(+\)04  1.31E\(+\)03  2.39E\(+\)05  1.24E\(+\)03  
Median  5.78E\(+\)03  1.46E\(+\)02  1.00E\(+\)05  1.88E\(+\)03  2.77E\(+\)05  1.55E\(+\)03  
Worst  6.84E\(+\)03  2.00E\(+\)02  1.28E\(+\)05  2.57E\(+\)03  3.21E\(+\)05  1.83E\(+\)03  
Mean  5.84E\(+\)03  1.44E\(+\)02  1.02E\(+\)05  1.85E\(+\)03  2.74E \(+\) 05  1.53E\(+\)03  
Std  4.48E\(+\)02  3.43E\(+\)01  1.26E\(+\)04  3.18E\(+\)02  2.12E\(+\)04  1.32E\(+\)02  
DECCDML  
Best  1.48E\(+\)04  2.74E−13  5.65E\(+\)06  1.64E\(+\)03  1.30E\(+\)07  9.69E\(+\)02  
Median  1.53E\(+\)04  3.20E−13  6.55E\(+\)06  2.21E\(+\)03  1.59E\(+\)07  9.75E\(+\)02  
Worst  1.62E\(+\)04  1.27E\(+\)00  7.63E\(+\)06  7.52E\(+\)03  2.16E\(+\)07  1.10E\(+\)03  
Mean  1.54E\(+\)04  5.08E−02  6.54E\(+\)06  2.47E\(+\)03  1.59E\(+\)07  9.91E \(+\) 02  
Std  3.59E\(+\)02  2.54E−01  4.63E\(+\)05  1.18E\(+\)03  1.72E\(+\)06  3.51E\(+\)01  
MASWchains  
Best  2.56E\(+\)03  8.51E\(+\)01  1.04E\(+\)00  7.83E\(+\)02  2.49E\(+\)05  9.25E\(+\)02  
Median  2.72E \(+\) 03  9.44E\(+\)01  1.26E\(+\)00  1.19E\(+\)03  2.85E\(+\)05  1.06E\(+\)03  
Worst  2.96E\(+\)03  1.24E\(+\)02  1.63E\(+\)00  2.55E\(+\)03  3.32E\(+\)05  1.21E\(+\)03  
Mean  2.74E \(+\) 03  9.98E\(+\)01  1.24E \(+\) 00  1.30E \(+\) 03  2.85E\(+\)05  1.07E\(+\)03  
Std  1.22E\(+\)02  1.40E\(+\)01  1.25E−01  4.36E\(+\)02  1.78E\(+\)04  7.29E\(+\)01 
Results of multipleproblem Wilcoxon’s test for EADE and EADE* versus LMDEa, SDENS, jDElsgo, and DECCDML over all functions at a 0.05 significance level with (1.25E\(+\)05 FES)
Algorithm  \(R^{+}\)  \(R^{}\)  p value  Better  Equal  Worse  Dec. 

EADE vs EADE*  77  133  0.296  9  0  11  \(\approx \) 
EADE vs LMDEa  148  62  0.108  14  0  6  \(\approx \) 
EADE vs SDENS  193  17  0.001  18  0  2  \(+\) 
EADE vs jDElsgo  210  0  0.000  20  0  0  \(+\) 
EADE vs DECCDML  162  48  0.033  16  0  4  \(+\) 
EADE vs MASWchains  22  188  0.002  3  0  17  − 
EADE* vs LMDEa  162  48  0.033  17  0  3  \(+\) 
EADE* vs SDENS  210  0  0.000  20  0  0  \(+\) 
EADE* vs jDElsgo  210  0  0.000  20  0  0  \(+\) 
EADE* vs DECCDML  164  46  0.028  15  0  5  \(+\) 
EADE* vs MASWchains  35  175  0.009  4  0  16  − 
Results of multipleproblem Wilcoxon’s test for EADE and EADE* versus LMDEa, SDENS, jDElsgo, and DECCDML over all functions at a 0.05 significance level with (6.00E\(+\)05 FES)
Algorithm  \(R^{+}\)  \(R^{}\)  p value  Better  Equal  Worse  Dec. 

EADE vs EADE*  116  94  0.681  13  0  7  \(\approx \) 
EADE vs LMDEa  130  80  0.351  12  0  8  \(\approx \) 
EADE vs SDENS  198  12  0.001  18  0  8  \(+\) 
EADE vs jDElsgo  190  20  0.002  17  0  3  \(+\) 
EADE vs DECCDML  182  28  0.004  15  0  5  \(+\) 
EADE vs MASWchains  71  139  0.204  7  0  13  \(\approx \) 
EADE* vs LMDEa  147  63  0.117  13  0  7  \(\approx \) 
EADE* vs SDENS  172  38  0.000  13  0  7  \(+\) 
EADE* vs jDElsgo  183  27  0.004  15  0  5  \(+\) 
EADE* vs DECCDML  203  7  0.012  19  0  1  \(+\) 
EADE* vs MASWchains  61  149  0.10  6  0  14  \(\approx \) 
Results of multipleproblem Wilcoxon’s test for EADE and EADE* versus LMDEa, SDENS, jDElsgo, and DECCDML over all functions at a 0.05 significance level with (3.00E\(+\)06 FES)
Algorithm  \(R^{+}\)  \(R^{}\)  p value  Better  Equal  Worse  Dec. 

EADE vs EADE*  138  52  0.084  15  1  4  \(\approx \) 
EADE vs LMDEa  83  127  0.411  9  0  11  \(\approx \) 
EADE vs SDENS  188  22  0.002  17  0  3  \(+\) 
EADE vs jDElsgo  81  129  0.370  7  0  13  \(\approx \) 
EADE vs DECCDML  186  24  0.002  15  0  5  \(+\) 
EADE vs MASWchains  98  112  0.794  10  0  10  \(\approx \) 
EADE* vs LMDEa  51  159  0.044  4  0  16  − 
EADE* vs SDENS  185  25  0.003  16  0  4  \(+\) 
EADE* vs jDElsgo  66  144  0.145  4  0  16  \(\approx \) 
EADE* vs DECCDML  182  28  0.004  14  0  6  \(+\) 
EADE* vs MASWchains  79  131  0.332  7  0  13  \(\approx \) 
Average ranks for all algorithms across all problems and 1.2e\(+\)05, 6.0e\(+\)05, and 3.0e\(+\)06 function evaluations (FEs)
Rank  Algorithm  1.2e\(+\)05  6.0e\(+\)05  3.0e\(+\)06  Mean ranking 

1  EADE  3.00  2.90  3.33  3.08 
2  EADE*  2.65  3.35  4.53  3.51 
3  LMDEa  4.05  3.45  3.20  3.57 
4  SDENS  5.75  6.08  5.85  5.89 
5  jDElsgo  6.40  5.08  2.78  4.75 
6  DECCDML  4.70  4.65  5.03  4.79 
7  MASWchains  1.45  2.50  3.30  2.42 
Experimental comparisons between EADE and stateoftheart algorithms, \(D = 100\)
EADE  CEO  SLPSO  CCPSO2  sepCMAES  EPUSPSO  MLCC  DMSLPSO  

\(F_{1}\)  
Mean  0.00E \(+\) 00  9.11E−29  1.09E−27  7.73E−14  9.02E−15  7.47E−01  6.82E−14  0.00E \(+\) 00 
Std  0.00E \(+\) 00  1.10E−28  3.50E−28  3.23E−14  5.53E−15  1.70E−01  2.32E−14  0.00E \(+\) 00 
\(F_{2}\)  
Mean  3.58E−03  3.35E\(+\)01  9.45E−06  6.08E\(+\)00  2.31E\(+\)01  1.86E\(+\)01  2.53E\(+\)01  3.65E\(+\)00 
Std  3.49E−03  5.38E\(+\)00  4.97E−06  7.83E\(+\)00  1.39E\(+\)01  2.26E\(+\)00  8.73E\(+\)00  7.30E−01 
\(F_{3}\)  
Mean  9.36E\(+\)01  3.90E\(+\)02  5.74E\(+\)02  4.23E\(+\)02  4.31E \(+\) 00  4.99E\(+\)03  1.50E\(+\)02  2.83E\(+\)02 
Std  5.10E\(+\)01  5.53E\(+\)02  1.67E\(+\)02  8.65E\(+\)02  1.26E \(+\) 01  5.35E\(+\)03  5.72E\(+\)01  9.40E\(+\)02 
\(F_{4}\)  
Mean  0.00E \(+\) 00  5.60E\(+\)01  7.46E\(+\)01  3.98E−02  2.78E\(+\)02  4.71E\(+\)02  4.39E−13  1.83E\(+\)02 
Std  0.00E \(+\) 00  7.48E\(+\)00  1.21E\(+\)01  1.99E−01  3.43E\(+\)01  5.94E\(+\)01  9.21E−14  2.16E\(+\)01 
\(F_{5}\)  
Mean  0.00E \(+\) 00  0.00E \(+\) 00  0.00E \(+\) 00  3.45E−03  2.96E−04  3.72E−01  3.41E−14  0.00E \(+\) 00 
Std  0.00E \(+\) 00  0.00E \(+\) 00  0.00E \(+\) 00  4.88E−03  1.48E−03  5.60E−02  1.16E−14  0.00E \(+\) 00 
\(F_{6}\)  
Mean  1.42E−14  1.20E−14  2.10E−14  1.44E−13  2.12E\(+\)01  2.06E\(+\)00  1.11E−13  0.00E \(+\) 00 
Std  0.00E \(+\) 00  1.52E−15  5.22E−15  3.06E−14  4.02E−01  4.40E−01  7.87E−15  0.00E \(+\) 00 
\(F_{7}\)  
Mean  −1.17E\(+\)03  −7.28E \(+\) 05  −1.48E\(+\)03  −1.50E\(+\)03  −1.39E\(+\)03  −8.55E\(+\)02  −1.54E\(+\)03  −1.14E\(+\)03 
Std  1.83E\(+\)01  1.88E\(+\)04  1.90E\(+\)01  1.04E\(+\)01  2.64E\(+\)01  1.35E\(+\)01  2.52E\(+\)00  8.48E\(+\)00 
Experimental comparisons between EADE and stateoftheart algorithms, \(D = 500\)
EADE  CEO  SLPSO  CCPSO2  sepCMAES  EPUSPSO  MLCC  DMSLPSO  

\(F_{1}\)  
Mean  0.00E \(+\) 00  6.57E−23  7.24E−24  7.73E−14  2.25E−14  8.45E\(+\)01  4.30E−13  0.00E \(+\) 00 
Std  0.00E \(+\) 00  3.90E−24  2.05E−25  3.23E−14  6.10E−15  6.40E\(+\)00  3.31E−14  0.00E \(+\) 00 
\(F_{2}\)  
Mean  7.28E\(+\)01  2.60E \(+\) 01  3.47E\(+\)01  5.79E\(+\)01  2.12E\(+\)02  4.35E\(+\)01  6.67E\(+\)01  6.89E\(+\)01 
Std  4.35E\(+\)00  2.40E \(+\) 00  1.03E\(+\)00  4.21E\(+\)01  1.74E\(+\)01  5.51E−01  5.70E\(+\)00  2.01E\(+\)00 
\(F_{3}\)  
Mean  1.02E\(+\)03  5.74E\(+\)02  6.10E\(+\)02  7.24E\(+\)02  2.93E \(+\) 02  5.77E\(+\)04  9.25E\(+\)02  4.67E\(+\)07 
Std  106E\(+\)02  1.67E\(+\)02  1.87E\(+\)02  1.54E\(+\)02  2.59E \(+\) 01  8.04E\(+\)03  1.73E\(+\)02  5.87E\(+\)06 
\(F_{4}\)  
Mean  2.54E\(+\)01  3.19E\(+\)02  2.72E\(+\)03  3.98E−02  2.18E\(+\)03  3.49E\(+\)03  1.79E−11  1.61E\(+\)03 
Std  6.65E\(+\)00  2.16E\(+\)01  3.25E\(+\)02  1.99E−01  1.51E\(+\)02  1.12E\(+\)02  6.31E−11  1.04E\(+\)02 
\(F_{5}\)  
Mean  3.10E−16  2.22E−16  3.33E−16  1.18E−03  7.88E−04  1.64E\(+\)00  2.13E−13  0.00E \(+\) 00 
Std  4.68E−17  0.00E\(+\)00  0.00E\(+\)00  4.61E−03  2.82E−03  4.69E−02  2.48E−14  0.00E \(+\) 00 
\(F_{6}\)  
Mean  4.26E−14  4.13E−13  1.46E−13  5.34E−13  2.15E\(+\)01  6.64E\(+\)00  5.34E−13  2.00E\(+\)02 
Std  0.00E \(+\) 00  1.10E−14  2.95E−15  8.61E−14  3.10E−01  4.49E−01  7.01E−14  9.66E−02 
\(F_{7}\)  
Mean  −4.41E\(+\)03  −1.97E \(+\) 06  −5.94E\(+\)03  −7.23E\(+\)03  −6.37E\(+\)03  −3.51\(+\)03  −7.43E\(+\)03  −4.20E\(+\)03 
Std  4.11E\(+\)01  4.08E\(+\)04  1.72E\(+\)02  4.16E\(+\)01  7.59E\(+\)01  2.10E\(+\)01  8.03E\(+\)00  1.29E\(+\)01 
Experimental comparisons between EADE and stateoftheart algorithms, \(D= 1000\)
EADE  CEO  SLPSO  CCPSO2  sepCMAES  EPUSPSO  MLCC  DMSLPSO  

\(F_{1}\)  
Mean  4.06E−028  1.09E−21  7.10E−23  5.18E−13  7.81E−15  5.53E\(+\)02  8.46E−13  0.00E \(+\) 00 
Std  5.73E−028  4.20E−23  1.40E−24  9.61E−14  1.52E−15  2.86E\(+\)01  5.01E−14  0.00E \(+\) 00 
\(F_{2}\)  
Mean  8.97E\(+\)01  4.15E \(+\) 01  8.87E\(+\)01  7.82E\(+\)01  3.65E\(+\)02  4.66E\(+\)01  1.09E\(+\)02  9.15E\(+\)01 
Std  2.77E\(+\)00  9.74E−01  5.25E\(+\)00  4.25E\(+\)01  9.02E\(+\)00  4.00E−01  4.75E\(+\)00  7.14E−01 
\(F_{3}\)  
Mean  2.15E\(+\)03  1.01E\(+\)03  1.04E\(+\)03  1.33E\(+\)03  9.10E \(+\) 02  8.37E\(+\)05  1.80E\(+\)03  8.98E\(+\)09 
Std  1.51E\(+\)02  3.02E\(+\)01  5.14E\(+\)01  2.63E\(+\)02  4.54E \(+\) 01  1.52E\(+\)05  1.58E\(+\)02  4.39E\(+\)08 
\(F_{4}\)  
Mean  1.54E\(+\)02  6.89E\(+\)02  5.89E\(+\)02  1.99E−01  5.31E\(+\)03  7.58E\(+\)03  1.37E−10  3.84E\(+\)03 
Std  7.54E\(+\)00  3.10E\(+\)01  9.26E\(+\)00  4.06E−01  2.48E\(+\)02  1.51E\(+\)02  3.37E−10  1.71E\(+\)02 
\(F_{5}\)  
Mean  4.88E−16  2.26E−16  4.44E−16  1.18E−03  3.94E−04  5.89E\(+\)00  4.18E−13  0.00E \(+\) 00 
Std  6.08E−17  2.18E−17  0.00\(+\)E00  3.27E−03  1.97E−03  3.91E−01  2.78E−14  0.00E \(+\) 00 
\(F_{6}\)  
Mean  5.75E−14  1.21E−12  3.44E−13  1.02E−12  2.15E\(+\)01  1.89E\(+\)01  1.06E−12  7.76E\(+\)00 
Std  2.24E−15  2.64E−14  5.32E−15  1.68E−13  3.19E−01  2.49E\(+\)00  7.68E−14  8.92E−02 
\(F_{7}\)  
Mean  −7.99E\(+\)03  −3.83E \(+\) 06  −1.30E\(+\)04  −1.43E\(+\)04  −1.25E\(+\)04  −6.62E\(+\)03  −1.47E\(+\)04  −7.50E\(+\)03 
Std  3.09E\(+\)01  4.82E \(+\) 04  1.04E\(+\)02  8.27E\(+\)01  9.36E\(+\)01  3.18E\(+\)01  1.51E\(+\)01  1.63E\(+\)01 
Results of multipleproblem Wilcoxon’s test for EADE versus stateoftheart algorithms over all functions at a 0.05 significance level with (\(D = 100\))
Algorithm  \(R^{+}\)  \(R^{}\)  p value  Better  Equal  Worse  Dec. 

EADE vs CEO  13  8  0.6  4  1  2  \(\approx \) 
EADE vs SLPSO  13  8  0.6  4  1  2  \(\approx \) 
EADE vs CCPSO2  21  7  0.237  6  0  1  \(\approx \) 
EADE vs SepCMAES  17  11  0.612  5  0  2  \(\approx \) 
EADE vs EPUSPSO  28  0  0.018  7  0  0  \(+\) 
EADE vs MLCC  21  7  0.237  6  0  1  \(\approx \) 
EADE vs DMSLPSO  14  1  0.080  4  1  2  \(\approx \) 
Furthermore, compared to the complicated structures and number of methods and number of control parameters used in other algorithms, we can see that our proposed EADE and EADE* are very simple and easy to be implemented and programmed in many programming languages. They only use very simple selfadaptive crossover rate with two parameters and a novel mutation rule with one parameters and basic mutation. Thus, they neither increase the complexity of the original DE algorithm nor the number of control parameters. To investigate and compare the performance of the proposed algorithms EADE and EADE* against other algorithms in statistical sense, multiproblem Wilcoxon signedrank test at a significance level 0.05 is performed on mean errors of all problems with (1.25E\(+\)05 FES, 6.00E\(+\)05 FES, and 3.00E\(+\)06 FES), and the results are presented in Tables 4, 5, and 6, respectively. Where R\(^{+}\) is the sum of ranks for the functions in which first algorithm outperforms the second algorithm in the row, and R\(^{}\) is the sum of ranks for the opposite. From Table 4 and 5, it can be obviously seen that EADE and EADE* are significantly better than SDENS, jDElsgo, and DECCDML algorithms. Moreover, there is no significant difference between EADE*, LMDEa, and EADE algorithm. However, MASW chains are significantly better than EADE and EADE* algorithms. Finally, from Table 5, it can be obviously seen that EADE and EADE* are significantly better than SDENS and DECCDML algorithms, EADE* is significantly worse than LMDEa algorithm. Besides, there is no significant difference between EADE*, LMDEa, and jDElsgo and EADE. From Tables 4 and 5, it is noteworthy that EADE* is better than all DE s algorithms (LMDEa, SDENS, jDElsgo, and DECCDML). Moreover, from Table 6, EADE* outperforms SDENS and DECCDML algorithms and it is competitive with jDElsgo algorithm which indicate that the new mutation scheme helps to maintain effectively the balance between the global exploration and local exploitation abilities for searching process of the DE during the search process. EADE outperforms SDENS and DECCDML algorithms, and it is competitive with jDElsgo, LMDEa, and MASWchains algorithms. Furthermore, the performance of all algorithms is analyzed using all function evaluations (Fes) and different categories of functions. Therefore, the mean aggregated rank of all the 6 algorithms across all problems (20) and all 1.2e\(+\)05, 6.0e\(+\)05, and 3.0e\(+\)06 function evaluations (FEs) is presented in Table 7. The best ranks are marked in bold and the second ranks are underlined. From Table 7, it can be clearly concluded that MASWchains is the best followed by EADE as second best among all algorithms while EADE* is ranked third. Note that the main contribution of this study is to propose a DE framework, and not to propose a “Best” algorithm or competitor to defeat other stateoftheart algorithms. However, it is noteworthy to mentioning that the performance of EADE considerably increases as the number of functions evaluation increases from 1.25E\(+\)05 to 3.00E\(+\)06 which means that it benefits from extra FES. Therefore, it can be obviously observed from Tables 4, 5, and 6 that EADE is inferior to MASW chains for 17, 13, and 10 functions in 1.25E\(+\)05, 6.00E\(+\)05, and 3.00E\(+\)06 FES, respectively. Thus, it can be concluded that the inferiority of the EADE algorithm against MASW chains algorithm considerably decreases as the FEs increases.
On the other hand, regarding CEC’2008 benchmark functions, Tables 8, 9, and 10 contain the results obtained by all algorithms in \(D=100\), \(D=500\), and \(D=1000\), respectively. It includes the obtained best and the standard deviations of error from optimum solution of EADE and other seven stateoftheart algorithms over 25 runs for all 7 benchmark functions. The results provided by these approaches were directly taken from references [39, 40]. For remarking the best algorithm, best mean for each function is highlighted in boldface.
Results of multipleproblem Wilcoxon’s test for EADE* versus stateoftheart algorithms over all functions at a 0.05 significance level with (\(D = 500\))
Algorithm  \(R^{+}\)  \(R^{}\)  p value  Better  Equal  Worse  Dec. 

EADE vs CEO  9  19  0.398  3  0  4  \(\approx \) 
EADE vs SLPSO  13  15  0.866  4  0  3  \(\approx \) 
EADE vs CCPSO2  6  22  0.176  3  0  4  \(\approx \) 
EADE vs SepCMAES  17  11  0.612  5  0  2  \(\approx \) 
EADE vs EPUSPSO  22  6  0.063  6  0  1  \(\approx \) 
EADE vs MLCC  6  22  0.176  3  0  4  \(\approx \) 
EADE vs DMSLPSO  18  3  0116  4  1  2  \(\approx \) 
Results of multipleproblem Wilcoxon’s test for EADE* versus stateoftheart algorithms over all functions at a 0.05 significance level with (\(D = 1000\))
Algorithm  \(R^{+}\)  \(R^{}\)  p value  Better  Equal  Worse  Dec. 

EADE vs CEO  9  19  0.398  3  0  4  \(\approx \) 
EADE vs SLPSO  9  19  0.398  3  0  4  \(\approx \) 
EADE vs CCPSO2  6  22  0.176  3  0  4  \(\approx \) 
EADE vs SepCMAES  17  11  0.612  5  0  2  \(\approx \) 
EADE vs EPUSPSO  25  3  0.063  6  0  1  \(\approx \) 
EADE vs MLCC  10  18  0.499  3  0  4  \(\approx \) 
EADE vs DMSLPSO  25  3  0.063  5  0  2  \(\approx \) 
Average ranks for all algorithms across all problems with D = 100, D = 500 and D = 1000
Rank  Algorithm  \(D=100\)  \(D=500\)  \(D=1000\)  Mean ranking 

1  EADE  2.57  3.93  3.86  3.45 
2  CEO  3.64  2.43  2.86  2.98 
3  SLPSO  3.93  3.71  3.29  3.64 
4  CCPSO2  5.14  4.36  4.00  4.5 
5  SepCMAES  5.43  5.29  5.71  5.47 
6  EPUSPSO  7.43  6.86  6.86  7.05 
7  MLCC  4.29  4.21  4.43  4.31 
8  DMSLPSO  3.57  5.21  5.00  4.59 
From Table 11, we can see that EADE obtains higher R\(^{+}\) values than R\(^{}\) in all cases, while slightly lower R\(^{+ }\) value than R\(^{}\) value in comparison with SaDE. However, from Tables 12 and 13, in the cases of EFADE versus CEO, SLPSO, and CCPSO2, they get higher R\(^{ }\) than R\(^{+}\) values. The reason is that EADE gains the performance far away of what these three algorithms do on function F\(_{7}\), resulting in higher ranking values. According to the Wilcoxon’s test at \(\alpha \) = 0.05, the significance difference can only be observed in EFADE versus EPUSPSO case. Besides, Table 14 lists the average ranks EADE and other algorithms according to Friedman test for D = 100, 500, and 1000, respectively. The best ranks are marked in bold and the second ranks are underlined. The p value computed through Friedman test is 0.01, 0.48, and 0.47, respectively. Thus, it can be concluded that there is a significant difference between the performances of the algorithms. It can be clearly seen from Table 14 that EADE gets the first ranking among all algorithms in 100dimensional functions, followed by CEO and SLPSO. Regarding 500D and 1000D problems, CEO gets the first ranking, followed by SLPSO and EADE. Furthermore, the performance of all algorithms is analyzed using all dimensions and different categories of functions. Therefore, the mean aggregated rank of all the 8 algorithms across all problems (7) and all dimensions (100D, 500D, and 100D) is presented in Table 12. From Table 12, it can be clearly concluded that CEO is the best followed by EADE as second best among all algorithms, while SLPSO is ranked third. Finally, it is noteworthy to highlighting that EADE has shown comparable performance to MLCC, CCPSO2, and DMSPSO, the three algorithms originally designed for solving largescale optimization problems. Plus, it also significantly outperforms sepCMAES and EPUSPSO algorithms.
Overall, from the above results, comparisons, and discussion, the proposed EADE algorithm is of better searching quality, efficiency, and robustness for solving unconstrained largescale global optimization problems. It is clear that the proposed EADE and EADE* algorithms perform well and it has shown its outstanding superiority with separable, nonseparable, unimodal, and multimodal functions with shifts in dimensionality, rotation, multiplicative noise in fitness, and composition of functions. Consequently, its performance is not influenced by all these obstacles. Contrarily, it greatly keeps the balance the local optimization speed and the global optimization diversity in challenging optimization environment with invariant performance. Besides, it can be obviously concluded from direct and statistical results that EADE and EADE* are powerful algorithms, and its performance is superior and competitive with the performance of thestateoftheart wellknown DEbased algorithms.
Conclusion
To efficiently concentrate the exploitation tendency of some subregion of the search space and to significantly promote the exploration capability in whole search space during the evolutionary process of the conventional DE algorithm, an enhanced adaptive Differential Evolution (EADE) algorithm for solving largescale global numerical optimization problems over continuous space was presented in this paper. The proposed algorithm introduces a new mutation rule. It uses two random chosen vectors of the top and bottom 100p% individuals in the current population of size NP, while the third vector is selected randomly from the middle (NP2p) individuals. The mutation rule is combined with the basic mutation strategy DE/rand/1/bin, where only one of the two mutation rules is applied with the probability of 0.5. Furthermore, we propose a novel selfadaptive scheme for gradual change of the values of the crossover rate that can excellently benefit from the past experience of the individuals in the search space during evolution process which, in turn, can considerably balance the common tradeoff between the population diversity and convergence speed. The proposed mutation rule was shown to enhance the global and local search capabilities of the basic DE and to increase the convergence speed. The algorithm has been evaluated on the standard highdimensional benchmark problems. The comparison results between EADE and EADE* and the other four stateofart DEbased algorithms that were all tested on this test suite on the IEEE congress on Evolutionary competition in 2008 and 2010 indicate that the proposed algorithm and its version are highly competitive algorithms for solving largescale global optimization problem. The experimental results and comparisons showed that the EADE and EADE* algorithms performed better in largescale global optimization problems with different types and complexity; they performed better with regard to the search process efficiency, the final solution quality, the convergence rate, and robustness, when compared with other algorithms. In fact, the performance of the EADE and EADE* algorithm was statistically superior to and competitive with other recent and wellknown DEs algorithms. Finally, to the best of our knowledge, this is the first study that uses all these different types of approaches (12) to carry out evaluation and comparisons on CEC’2008 and CEC’2010 benchmark problems. Virtually, this study aims to prove that EADE is a competitive and an efficient approach as well as being superior to the most recent techniques in the field of largescale optimization. Several current and future works can be developed from this study. First, current research effort focuses on how to control the scaling factors by selfadaptive mechanism and develop another selfadaptive mechanism for crossover rate. In addition, the new version of EADE combined with Cooperative Coevolution (CC) framework is being developed and will be experimentally investigated soon. Moreover, future research will investigate the performance of the EADE algorithm in solving constrained and multiobjective optimization problems as well as realworld applications such as data mining and clustering problems. In addition, largescale combinatorial optimization problems will be taken into consideration. Another possible direction is integrating the proposed novel mutation scheme with all compared and other selfadaptive DE variants plus combining the proposed selfadaptive crossover with other DE mutation schemes. In addition, the promising research direction is joining the proposed mutation with evolutionary algorithms, such as genetic algorithms, harmony search, and particle swarm optimization, as well as foraging algorithms such as artificial bee colony, bees algorithm, and ant colony optimization. The MATLAB source code of EADE is available upon request.
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