Adaptive Differential Evolution with the Stagnation Termination Mechanism

Abstract

Differential evolution is capable of searching for the optimum for different types of optimization problems with a few inputs, which has gained worldwide popularity. In this paper, we propose a parameters adaptation scheme based on the stagnation ratio (PASR), which regards the stagnation ratio (STR) as the indicator for adjusting the control parameters and greediness parameter. To be specific, when the STR is larger than the predefined threshold, exploration is advocated. In this circumstance, larger control parameters and a greediness parameter are adopted. However, when the STR is smaller than the predefined threshold, exploitation is preferred. In this case, smaller control parameters and a greediness parameter are utilized. Further, when the stagnation lasts for a long period, a generation-based selection (GBS) scheme is developed to help it escape from the local optimum and stagnation. Comparative experiments have been implemented on the CEC2017 to testify the effectiveness of adaptive differential evolution with the stagnation termination mechanism (STMDE) and its components. The competitiveness of the STMDE is also verified via comparing it to top-performing DE variants in the practical optimization problem selected from the CEC2011.

1. Introduction

Differential evolution (DE) [1] has been demonstrated to be a simple and competitive optimization method in dealing with real-world problems. Four operators are included in the DE, namely, initialization, mutation, crossover and selection. The initial population is generated first, which goes through the other three processes iteratively until the stopping criterion is met. In the mutation, new genetic materials derive from the perturbation of the current population via the differential vector(s). In the crossover, the offspring inherits the genes from the parent or the mutant vector, which is affected by the crossover rate. In the selection, the one with better fitness between the parent and the offspring can enter into the next round of evolution. There is no doubt that DE can handle a large number of optimization problems effectively. However, premature convergence and stagnation cannot be neglected. To address this problem, researchers have become devoted to developing top-performing DE variants via adjusting control parameters adaptively [2,3,4,5,6,7,8,9,10,11,12,13] and modifying the classic selection [14,15,16,17,18,19,20].

On the one hand, the adaptive adjustment of control parameters is beneficial in producing promising offspring according to the evolutionary requirement [2,3,4,5,6,7,8,9,10,11,12,13]. Plenty of works pay more attention to fitness in the adjustment of control parameters. For example, fitness is regarded as one of the effective metrics in [2,3,4,5,6,7]. Moreover, information from the solution space is also included in the adaptation of control parameters [8,9,10,11]. Finally, some powerful techniques are combined with the production of control parameters to achieve better the performance of DE, such as genetic programming [12], chaotic local search [13] and so on.

On the other hand, appropriate selection ensures that potential individuals can be included in the next round of evolution. However, classic selection regards fitness as the only metric for filtering out superior individuals, which may cause premature convergence and stagnation. During the past few decades, researchers have put forward some reasonable selection operators with the utilization of selection thresholds [14,15], advanced techniques like neural networks [20] and so on.

To enhance the DE, we propose adaptive differential evolution with the stagnation termination mechanism (STMDE), which focuses on both the mutation and selection processes. First, an adaptation scheme for control parameters is developed. Moreover, the greediness parameter of the mutation strategy is adjusted according to the evolutionary state. Finally, a generation-based selection (GBS) is put forward to confront the stagnation. Substantial experiments on CEC2017 [21] verify the effectiveness of the proposed STMDE. The real-world optimization problem from the CEC2011 [22] is also included to testify the effectiveness of the STMDE.

The organization of this paper is as follows. Section 2 states a brief review of classic and recent works related to the parameter adaptation and selection of DE. Section 3 illustrates the proposed STMDE concretely. Section 4 shows the experimental results and performance analysis of the STMDE. Section 5 draws the conclusion of the paper.

2. Background

2.1. Differential Evolution

In this section, the initialization, mutation, crossover and selection of the DE are described in detail.

• Initialization: Before evolution, an initial population with the size of NP is produced. Each individual is generated according to the following equation.

$$x_{i,j,0}=x_{j,\min }+ran{d}_{i,j}[0,1]\cdot (x_{j,\max }-x_{j,\min })$$

(1)

where x_j,min and x_j,max limit the minimal and maximal values of the j-th element of the i-th individual.

2.

Mutation: New genetic materials are introduced into the current population via differential vectors. Researchers have put forward plenty of mutation strategy with different searching characteristics. Several of the mutation strategies that are frequently used are displayed as follows.

DE/rand/1:

$$v_{i,g}=x_{r_1^i,g}+F\cdot \left(x_{r_2^i,g}-x_{r_3^i,g}\right)$$

(2)

DE/best/1:

$$v_{i,g}=x_{best,g}+F\cdot \left(x_{r_1^i,g}-x_{r_2^i,g}\right)$$

(3)

DE/rand/2:

$$v_{i,g}=x_{r_1^i,g}+F\cdot \left(x_{r_2^i,g}-x_{r_3^i,g}\right)+F\cdot \left(x_{r_4^i,g}-x_{r_5^i,g}\right)$$

(4)

DE/best/2:

$$v_{i,g}=x_{best,g}+F\cdot \left(x_{r_1^i,g}-x_{r_2^i,g}\right)+F\cdot \left(x_{r_3^i,g}-x_{r_4^i,g}\right)$$

(5)

where $r_1^i$, $r_2^i$, $r_3^i$, $r_4^i$ and $r_5^i$ are randomly generated integers. They are different from each other and the index i. In addition, x_best,g represents the best individual of the g-th generation.

3.

Crossover: The offspring inherits genes from both the parent and the mutant vector via the crossover. The crossover rate determines how many genes are from the parent and how many genes are from the mutant vector.

$$u_{j,i,g}=\{\begin{array}{cc}{v}_{j,i,g}& \mathrm{if} \mathrm{rand}(0,1)\le C{R}_i \mathrm{or} j=j_{rand}\\ x_{j,i,g}& \mathrm{otherwise}\end{array}$$

(6)

where CR_i is the crossover rate of the i-th individual; rand(0,1) denotes a random number between 0 and 1; j_rand represents a random integer between one and the problem size.

4.

Selection: Each offspring competes with its parent to be enrolled into the next generation based on their fitness. For the minimization problems, the smaller the fitness is, the better the individual is. The mathematical expression is given by

$$x_{i,g+1}=\{\begin{array}{cc}{u}_{i,g}& \mathrm{if} f\left(u_{i,g}\right)\le f\left(x_{i,g}\right)\\ x_{i,g}& otherwise\end{array}$$

(7)

where f(x_i,g) and f(u_i,g) denote the fitness of the i-th individual x_i,g and its corresponding trial vector u_i,g.

For the completeness of the paper, some special features of the SHADE [3] are also included here due to the fact that the SHADE is adopted as the baseline of the proposed STMDE in the following section.

Firstly, the SHADE adjusts the control parameters with the successful update of fitness considered, which is disclosed as follows.

• The generation of scale factor F

$$F_i=\mathrm{randc}\left({\mu }_F,0.1\right)$$

(8)

where the mean value ${\mu }_F$ of the distribution is 0.5 initially and changes with evolution when the historical set is not empty.

$${\mu }_F=\left(1-c\right)\cdot {\mu }_F+c\cdot {\mathrm{mean}}_{\mathrm{L}}\left(S_F\right)$$

(9)

where c is a constant between 0 and 1. The mean_L(S_F) denotes the Lehmer mean of the historical set of the scale factor F.

$${\mathrm{mean}}_L\left(S_F\right)={\displaystyle \frac{{\displaystyle \sum _{k=1}^{\left|S_F\right|}{\omega }_k\cdot F^2}}{{\displaystyle \sum _{k=1}^{\left|S_F\right|}{\omega }_k\cdot F}}}$$

(10)

where ${\omega }_k$ is a weighted factor and the calculation method of it is given in the following part.

2.

The generation of the crossover rate CR

$$C{R}_i={\mathrm{randn}}_i\left({\mu }_{CR},0.1\right)$$

(11)

where the mean value ${\mu }_{CR}$ of the distribution is 0.5 initially and is updated in each generation on the condition that the historical set is not empty. The update rule is shown as follows.

$${\mu }_{CR}=\left(1-c\right)\cdot {\mu }_{CR}+c\cdot {\mathrm{mean}}_A\left(S_{CR}\right)$$

(12)

where c is a constant between 0 and 1. The mean_A(S_CR) denotes the arithmetic mean of the historical set of the crossover rate CR.

$${\mathrm{mean}}_A\left(S_{CR}\right)={\displaystyle \sum _{k=1}^{\left|S_{CR}\right|}{\omega }_k\cdot S_{CR,k}}$$

(13)

The weighted factor ${\omega }_k$ in the Equations (10) and (13) is determined by fitness improvements, which can be obtained by,

$${\omega }_k={\displaystyle \frac{\Delta f_k}{{\displaystyle \sum _{k=1}^{\left|S_q\right|}\Delta f_k}}},q=\{CR,F\}$$

(14)

where the fitness improvement is calculated by $\Delta f_k=\left|f\left(u_{k,G}\right)-f\left(x_{k,G}\right)\right|$.

In addition, SHADE adopts a mutation strategy called “current-to-pbest”, which make use of the p × NP best individuals from the perspective of fitness to guide evolution. The mathematical form of “current-to-pbest” is given by,

$$v_{i,g}=x_{i,g}+F_i\cdot \left(x_{best,g}^p-x_{i,g}\right)+F_i\cdot \left(x_{r_1,g}-{\tilde{x}}_{r_2,g}\right)$$

(15)

where $x_{best,g}^p$ denotes a randomly selected individual from the first p × NP individuals.

2.2. Related Works

The simpleness and competitiveness of DE attract worldwide attention. However, its performance is heavily affected by the control parameters and selection. Therefore, researchers have put forward plenty of methods to generate appropriate parameters [2,3,4,5,6,7,8,9,10,11,12,13]. Further, the selection operator is also improved in order to filter out promising individuals [14,15,16,17,18,19,20]. Some works concerning the parameters adaptation and enhanced selection are listed as follows.

On the one hand, a brief review of adaptive parameters schemes is given. In general, fitness serves as an important guideline for tuning control parameters. JADE [2] takes advantage of successful control parameters achieving smaller fitness to lead evolution. SHADE [3] makes use of fitness improvements to enhance the parameters adaptation. Hip-DE [4] achieves effective adjustment for the distribution of control parameters by utilizing those parameters, obtaining better fitness. Li et al. introduced crossover rate sorting into the selection of control parameters to figure out the appropriate crossover rate for each individual based on fitness [5]. In [6], IMPEDE performs parameter adaptation via adding the weighed Lehmer mean strategy with the fitness improvements considered. In [7], the control parameters were obtained by fitness. Apart from fitness, researchers also introduced other information into the adaptive adjustment for control parameters, such as distance to the optimum [8], fitness differences between individuals [9] and so on. In [10], the control parameters are modified along with Euclidean distance. In [11], PFI-SHADE adjusts the control parameters to both the deviation of fitness and the dimension of the population. In addition, some powerful techniques are combined with parameter adaptation to enhance the DE [12,13]. For example, Stanovov et al. utilized genetic programming to obtain a top-performing parameter adaptation scheme [12]. In [13], Gao et al. incorporated a chaotic local search with JADE to address the premature convergence. Though researchers have become devoted to adjusting the control parameters based on fitness, fitness improvements, Euclidean distance and so on, the stagnation ratio of the population has not been fully considered. The stagnation ratio of the population can reflect its evolutionary state to some extent. Stagnation frequently happens, especially in the later evolutionary stage or in handling complex optimization problems. Capturing the stagnation ratio of the population may be beneficial to tuning the control parameters more reasonably and therefore attaining better performance.

On the other hand, some enhanced selection operators have been developed to accelerate convergence and cease stagnation. To get rid of stagnation, threshold-based selection was proposed [14,15]. Different from greedy selection, which is only determined by fitness, threshold-based selection may pick some inferior offspring for the next generation. Tian et al. proposed enhanced selection by taking both fitness and the position of individuals into consideration [16]. In addition, different from one-by-one selection, the current population and offspring are separated into several subsets and top-ranking individuals from each subset are selected [17]. In [18], the selection metric includes fitness ranking and Euclidean distance. In [19], the poor individual may be allowed to participate in the next generation. Apart from the information from the objective space and the solution space, Chen et al. proposed enhanced selection with the aid of a trained neural network [20]. Though plenty of competitive selection operators have been put forward, they pay more attention to filter out promising individuals from existing ones. However, the existing ones may not satisfy the optimization requirement when stagnation happens.

3. Proposed Method

In this section, we introduce the adaptive parameters scheme first, followed by generation-based selection, which can cease stagnation effectively and therefore improve the performance of DE.

3.1. Parameters Adaptation Based on the Stagnation Ratio

The parameters adaptation based on the stagnation ration (PASR) consists of two schemes, namely the adaptive control parameters scheme (CPSR) and the adaptive greediness parameter scheme (GPSR).

3.1.1. Adaptive Control Parameters Scheme

Adjusting the scale factor F and the crossover rate CR to the evolutionary status is beneficial to enhancing the performance of DE. In this section, we propose an adaptive control parameters scheme based on the stagnation ratio (CPSR). The pseudocode of it is shown in Algorithm 1.

Algorithm 1 CPSR
1	Generate a group of CR and a group of F with the size of NP, respectively;
2	Sort the CR in descending order and denote it as CRsort and divide it into two parts of equal size, namely CRL (the larger part) and CRS (the smaller part);
3	Sort the F in descending order and denote it as Fsort and divide it into two parts of equal size, namely FL (the larger part) and FS (the smaller part);
4	If STR > 0.5
5	Generate two groups of CR with the size of 0.55 × NP and 0.45 × NP via selecting randomly from CRL and CRS, respectively;
6	Generate two groups of F with the size of 0.6 × NP and 0.4 × NP via selecting randomly from FL and FS, respectively;
7	Else
8	Generate two groups of CR with the size of 0.45 × NP and 0.55 × NP via selecting randomly from CRL and CRS, respectively;
9	Generate two groups of F with the size of 0.6 × NP and 0.4 × NP via selecting randomly from FL and FS, respectively;
10	End If

Firstly, a group of CR and F with the size of NP are generated the same way as the SHADE (line 1). Afterwards, the group of CR is sorted in descending order and it is denoted as CR^sort, which is separated into two groups of equal size in the sequence (lines 2). The group of F is sorted in descending order and it is denoted as F^sort, which is separated into two groups of equal size in the sequence (line 3). If the stagnation ratio (STR) of the current population is larger than 0.5, the current evolutionary status may lack exploration. In this case, a group of CR with the size of 0.55 × NP is generated by sampling from the CR^L, while the group of CR with the size of 0.45 × NP is generated by sampling from the CR^S (line 5). The 0.55/0.45 in line 05 is the decomposition coefficient for CR (DCcr). Similarly, a group of F with the size of 0.60 × NP is generated by sampling from the F^L, while the group of F with the size of 0.40 × NP is generated by sampling from the F^S (line 6). The 0.55/0.45 in line 06 is the decomposition coefficient for F (DCf).

However, if the STR is smaller than 0.5, the evolutionary status may lack exploitation. In this circumstance, a group of CR with the size of 0.45 × NP is generated by sampling from the CR^L, while the group of CR with the size of 0.55 × NP is generated by sampling from the CR^S (line 8). Similarly, a group of F with the size of 0.40 × NP is generated by sampling from the F^L, while the group of F with the size of 0.60 × NP is generated by sampling from the F^S (line 9).

The reason why the CPSR adjusts control parameters in this way can be summarized as follows. On the one hand, when the population tends to be stagnant, exploration should be advocated and larger CR and F may be preferred. To this end, more CR and F values are obtained by randomly sampling from the CR^L and F^L subsets, respectively. On the other hand, exploitation is preferred if not many individuals are trapped into stagnation and smaller control parameters are appropriate. Therefore, more CR and F values are derived from CR^S and F^S subsets, respectively.

3.1.2. Adaptive Greediness Parameter Scheme

In this subsection, we propose an adaptive greediness parameter scheme based on the stagnation ratio (GPSR). When the STR of the current population is larger than the 0.5, exploration is advocated for (lines 1~2). Therefore, a larger greediness parameter p is adopted in this case, while exploitation is preferred in the opposite case (lines 3~4). To this end, a smaller greediness parameter p is considered to enhance exploitation. The concrete adjustment rules are displayed as follows (Algorithm 2).

Algorithm 2 GPSR
1	If STR > 0.5
2	p = 0.7
3	Else
4	p = 0.1
5	End If

3.2. Generation-Based Selection

Premature convergence and stagnation are two important factors that deteriorate the searching capability of DE. To tackle the aforementioned problems, generation-based selection (GBS) is developed, which comes into effect only when the stagnation counter Us reaches the predefined threshold. The pseudocode of the GBS mechanism is displayed in Algorithm 3.

Algorithm 3 GBS
Input:	The population size NP; the threshold of stagnation t3; the stagnation counter for the population at the g-th generation Us = [Us(x1,g), Us(x2,g), …, Us(xNP,g)]; the fitness of the current population Xg is represented as f(Xg) = {f(x1,g), f(x2,g), …, f(xNP,g)} and the fitness of its offspring Ug is denoted as f(Ug) = {f(u1,g), f(u2,g), …, f(uNP,g)};
Output:	The population for the next generation Xg+1;
1	Sort the fitness f(Xg) of the current population Xg in ascending order and find the best solution xbest, g of the g-th generation;
2	For i = 1:NP
3	If Us(xi,g) > T and f(xi,g) < f(ui,g) Then
4	xi,g+1 = xi,g + gp × (xbest,g − xi,g);
5	Us(xi,g) = 0;
6	Else If
7	Select the individual for the next generation according to Equation (7)
8	End If
9	End For

Firstly, sort the fitness of the whole population and distinguish the best individual x_best,g of the current generation (line 1). When the number of stagnations of the i-th individual x_i,g is larger than the threshold T and the offspring is still inferior to its parent at the g-th generation, a new offspring is produced to replace the stagnant individual, but without evaluating its fitness (lines 3~4) the stagnation counter is reset to be 0 (line 5). Otherwise, the classic selection is carried out to figure out the promising individual for the next generation (line 7). It is worth mentioning that the predefined parameter gp in line 04 ranges from 0 to 1 to generate potential offspring.

3.3. STMDE

The integrated STMDE is shown in Algorithm 4. Firstly, the initial population is produced according to Equation (1) (line 1), which is followed by iterations of mutation, crossover and selection until the optimization requirement is met (line 2). The current evolution status is disclosed by the STR, which guides the adjustment for control parameters and the greediness parameter p. To be specific, the scale factor F and the crossover rate CR for each individual are tuned according to Algorithm 1 (line 3). Other than the two control parameters, the greediness parameter p in the mutation strategy “current-to-pbest” also plays a core role in enhancing the performance of offspring generation. To this end, the greediness parameter p is also adaptive to the current evolution status as shown in Algorithm 2 (line 4). With suitable control parameters and greediness parameters selected, potential mutant vectors are produced via Equation (15) (line 5). In this paper, the binomial crossover operator as shown in Equation (6) is adopted to generate trial vectors (line 6). Afterwards, promising individuals will pass onto the next generation via Algorithm 3 (line 7). Finally, we update the STR in each generation via STR = Num_STG/NP, where Num_STG denotes the number of stagnant individuals in the current generation and NP represents the population size (line 8).

Algorithm 4 STMDE
Input:	Population size NP; the problem dimension D; the initial STR = 0;
Output:	The final population X;
1	Generate the initial population via Equation (1);
2	While the stopping condition is not met Do
3	Obtain the CR and F for each individual via Algorithm 1;
4	Select the appropriate p to achieve a better performance of the mutation strategy “current-to-pbest” via Algorithm 2;
5	Obtain the mutant vectors via Equation (15);
6	Obtain the offspring via the binomial crossover operator as shown in Equation (6);
7	Select the promising individual for the next generation via Algorithm 3;
8	Update STR via STR = Num_STG/NP;
9	End While

3.4. Time Complexity

The time complexity of sorting fitness, the crossover rate and the scale factor is O(3 × NP × log₂NP). The time complexity of calculating the STR is O(NP). Therefore, the overall overhead of the proposed mechanisms is O(3 × NP × log₂NP + NP).

4. Experimental Results

To better disclose the effectiveness of the STMDE, comprehensive experiments with the baseline and five competitive DE variants are implemented. Further, each component of the STMDE is also investigated to distinguish their contribution. All the comparative tests are carried out on the testsuite CEC2017 [21] with 51 independent runs to obtain the mean value and the standard deviation of solution error (SR). The measurement of performance is defined as the fitness difference between the best solution x^b found with 10⁴ × D function evaluations and the global optimum x*, which can be described as SR = f(x^b) − f(x*). Apart from the experiments on CEC2017, the competitiveness of handling a practical optimization problem from the CEC2011 [22] is also investigated.

To demonstrate the efficacy of the STMDE straightforwardly, a Wilcoxon rank-sum test [19] with a 5% significance level is introduced, in which the symbols “+”, “=” and “−” denote that the STMDE is superior (i.e., win, W), similar (i.e., tie, T) and inferior (i.e., lose, L) to the compared algorithm, respectively. All the experiments are carried out via MATLAB R2022b (MathWorks Inc., Natick, MA, USA) on a 3.6 GHz PC with 32GB RAM and an AMD Ryzen 7 3700X 8-Core processor in the Windows 10 edition.

4.1. Performance Comparison with Baseline

The performance comparison with the baseline on 30-D, 50-D and 100-D CEC2017 is shown in

Table 1. Performance comparison with the baseline on 30-D, 50-D and 100-D CEC2017.

	30-D					50-D					100-D
	SHADE			STMDE		SHADE			STMDE		SHADE			STMDE
	mean	std	sig	mean	std	mean	std	sig	mean	std	mean	std	sig	mean	std
F1	1.00 × 10−14	6.54 × 10−15	=	8.36 × 10−15	7.06 × 10−15	3.82 × 10−14	1.38 × 10−14	=	3.76 × 10−14	1.33 × 10−14	8.13 × 10−11	1.82 × 10−10	+	4.94 × 10−11	1.59 × 10−10
F3	6.80 × 10−14	2.28 × 10−14	=	6.58 × 10−14	2.38 × 10−14	5.98 × 10−12	1.70 × 10−11	+	4.24 × 10−13	2.50 × 10−13	7.96 × 10−8	1.96 × 10−7	−	2.62 × 10−3	1.14 × 10−2
F4	4.88 × 101	2.27 × 101	=	5.66 × 101	1.16 × 101	5.25 × 101	4.58 × 101	=	4.63 × 101	4.26 × 101	1.24 × 102	6.22 × 101	=	1.27 × 102	6.95 × 101
F5	1.54 × 101	2.89 × 100	+	1.17 × 101	3.48 × 100	3.84 × 101	5.82 × 100	+	2.83 × 101	8.71 × 100	1.00 × 102	1.21 × 101	=	9.98 × 101	1.52 × 101
F6	2.44 × 10−5	8.52 × 10−5	+	4.95 × 10−6	1.22 × 10−5	8.37 × 10−4	1.20 × 10−3	=	6.67 × 10−4	6.68 × 10−4	8.07 × 10−2	5.36 × 10−2	+	2.77 × 10−2	2.29 × 10−2
F7	4.47 × 101	2.63 × 100	+	4.08 × 101	3.88 × 100	8.28 × 101	5.07 × 100	+	7.50 × 101	7.02 × 100	2.08 × 102	1.48 × 101	+	1.94 × 102	1.79 × 101
F8	1.68 × 101	3.06 × 100	+	1.13 × 101	3.33 × 100	3.90 × 101	5.91 × 100	+	2.92 × 101	1.12 × 101	9.99 × 101	1.39 × 101	=	9.63 × 101	1.76 × 101
F9	2.30 × 10−2	6.98 × 10−2	=	1.23 × 10−2	3.11 × 10−2	1.36 × 100	1.07 × 100	+	4.82 × 10−1	4.95 × 10−1	2.74 × 101	1.15 × 101	+	1.46 × 101	7.64 × 100
F10	1.70 × 103	2.11 × 102	+	1.38 × 103	3.65 × 102	3.44 × 103	2.92 × 102	+	2.73 × 103	4.26 × 102	9.18 × 103	5.39 × 102	+	8.27 × 103	9.30 × 102
F11	3.46 × 101	2.64 × 101	+	2.27 × 101	2.31 × 101	1.25 × 102	2.93 × 101	+	1.08 × 102	2.65 × 101	1.13 × 103	1.96 × 102	+	8.81 × 102	2.42 × 102
F12	1.26 × 103	3.11 × 102	=	1.10 × 103	4.10 × 102	7.36 × 103	7.49 × 103	=	6.13 × 103	4.55 × 103	1.92 × 104	1.24 × 104	−	2.37 × 104	9.06 × 103
F13	4.15 × 101	2.76 × 101	+	2.39 × 101	9.97 × 100	2.27 × 102	1.54 × 102	+	1.38 × 102	8.51 × 101	1.93 × 103	3.84 × 103	=	3.31 × 102	1.69 × 102
F14	2.77 × 101	4.70 × 100	+	2.37 × 101	4.49 × 100	1.88 × 102	3.81 × 101	+	8.87 × 101	3.36 × 101	5.84 × 102	1.83 × 102	=	5.50 × 102	1.17 × 102
F15	1.81 × 101	1.03 × 101	+	9.05 × 100	4.56 × 100	2.21 × 102	1.03 × 102	+	1.43 × 102	7.94 × 101	3.19 × 102	8.06 × 101	−	4.49 × 102	2.28 × 102
F16	3.28 × 102	1.20 × 102	+	2.52 × 102	1.38 × 102	7.67 × 102	1.97 × 102	+	6.31 × 102	2.22 × 102	2.41 × 103	3.32 × 102	=	2.34 × 103	4.31 × 102
F17	5.20 × 101	1.26 × 101	+	3.55 × 101	9.66 × 100	5.87 × 102	1.25 × 102	+	4.42 × 102	1.35 × 102	1.82 × 103	2.99 × 102	+	1.65 × 103	3.72 × 102
F18	4.11 × 101	3.52 × 101	+	2.80 × 101	8.17 × 100	1.78 × 102	1.22 × 102	=	1.74 × 102	1.10 × 102	1.39 × 103	9.44 × 102	=	2.08 × 103	2.04 × 103
F19	1.29 × 101	3.52 × 100	+	7.39 × 100	2.12 × 100	1.30 × 102	4.53 × 101	+	9.69 × 101	3.87 × 101	3.18 × 102	4.89 × 102	=	2.41 × 102	5.01 × 101
F20	9.09 × 101	5.54 × 101	+	5.64 × 101	5.42 × 101	3.95 × 102	9.93 × 101	+	3.16 × 102	1.57 × 102	1.70 × 103	2.29 × 102	+	1.57 × 103	3.65 × 102
F21	2.18 × 102	3.12 × 100	+	2.13 × 102	4.01 × 100	2.36 × 102	5.09 × 100	+	2.31 × 102	8.33 × 100	3.32 × 102	1.43 × 101	+	3.25 × 102	1.51 × 101
F22	1.00 × 102	3.44 × 10−1	=	1.00 × 102	6.39 × 10−14	1.85 × 103	2.01 × 103	=	2.44 × 103	1.52 × 103	1.05 × 104	5.00 × 102	+	9.78 × 103	1.27 × 103
F23	3.64 × 102	4.52 × 100	+	3.60 × 102	4.91 × 100	4.61 × 102	6.82 × 100	+	4.57 × 102	1.32 × 101	6.12 × 102	1.44 × 101	+	5.98 × 102	1.95 × 101
F24	4.35 × 102	3.76 × 100	=	4.35 × 102	6.18 × 100	5.31 × 102	5.88 × 100	=	5.33 × 102	9.59 × 100	9.92 × 102	2.13 × 101	=	1.00 × 103	2.20 × 101
F25	3.87 × 102	2.04 × 10−1	+	3.87 × 102	5.10 × 10−2	5.38 × 102	3.22 × 101	=	5.25 × 102	3.43 × 101	7.43 × 102	5.48 × 101	=	7.40 × 102	4.27 × 101
F26	1.11 × 103	5.44 × 101	+	1.08 × 103	8.15 × 101	1.45 × 103	9.00 × 101	=	1.44 × 103	1.10 × 102	4.23 × 103	2.13 × 102	=	4.28 × 103	2.47 × 102
F27	5.05 × 102	6.01 × 100	=	5.05 × 102	5.52 × 100	5.43 × 102	2.59 × 101	+	5.33 × 102	1.31 × 101	6.70 × 102	3.48 × 101	+	6.47 × 102	2.51 × 101
F28	3.43 × 102	5.83 × 101	=	3.41 × 102	5.41 × 101	4.97 × 102	1.68 × 101	+	4.80 × 102	2.40 × 101	5.34 × 102	3.77 × 101	=	5.32 × 102	3.00 × 101
F29	4.63 × 102	1.10 × 101	+	4.47 × 102	1.90 × 101	4.99 × 102	8.17 × 101	+	4.04 × 102	7.97 × 101	2.13 × 103	2.94 × 102	+	1.91 × 103	2.94 × 102
F30	2.07 × 103	9.51 × 101	=	2.04 × 103	7.55 × 101	6.53 × 105	6.51 × 104	=	6.68 × 105	8.37 × 104	2.75 × 103	7.83 × 102	+	2.53 × 103	1.86 × 102
W/T/L	19/10/0					19/10/0					14/12/3

. On the whole, the STMDE performs better than the baseline on the 30-D, 50-D and 100-D test functions with the metric “W/T/L” of “19/10/0”, “19/10/0” and “14/12/3”, respectively. It can be observed that the STMDE does not lose to any test functions when the problem size is 30 or 50. With the problem size increasing, the performance of the STMDE degrades slightly. To be specific, the STMDE is inferior to three benchmark functions, namely F3, F12 and F15. When the stagnation lasts for a long period, the GBS may encourage the population to approach the best individual in the current generation. Therefore, the STMDE may converge to the local optimum on some test functions with multiple local optima that are far from the global optimum.

From the perspective of function properties, the STMDE excels in simple multi-modal (F4~F10), hybrid (F11~F20) and composition (F21~F30) functions. To be specific, the STMDE is superior to SHADE on five, five and four simple multi-modal functions and performs similarly on others when the problem size is 30, 50 and 100, respectively. The STMDE is advantaged over all 30-D hybrid functions other than F12. In the 50-D hybrid functions, the STMDE shows a similar performance with SHADE on F12 and F18, while it is better than SHADE on the rest of hybrid functions. With the problem size increasing, the STMDE can search for better solutions on F11, F17 and F20 and lose to SHADE on F12 and F15. On all considered composition functions, the STMDE is more competitive than SHADE in 16 functions and is similar on the others.

The reason why the STMDE can achieve a better performance than SHADE on complicated optimization problems is that the STMDE can identify the stagnation in a timely manner and adjust the control parameters and the greediness parameter to produce promising offspring. Further, when the individual traps into the stagnation for a long period, the GBS scheme takes effect to cease the stagnation, which enables limited computational resources to be allocated to potential individuals. However, stagnation seldom happens on unimodal functions, especially when the problem size is relatively small. Therefore, the proposed STMDE shows no advantage over SHADE on 30-D unimodal functions while the superiority can be observed on 50-D F1 and 100-D F1.

4.2. Performance Comparison with State-of-the-Art DEs

To investigate the performance of the proposed STMDE, performance comparisons with five state-of-the-art (SOTA) DEs on 30-D, 50-D and 100-D CEC2017 benchmark functions are given in Table 2, Table 3 and Table 4. The five SOTA DEs are ACos-JADE, IMPEDE, Db-SHADE, PFI-SHADE and CUSDE, which are described briefly as follows.

• ACos-JADE [23]: This this DE variant incorporates the original and eigen coordinate systems to enhance the JADE.
• IMPEDE [6]: this DE variant is based on multiple swarms and takes advantage of inferior individuals to produce promising offspring.
• Db-SHADE [10]: this DE variant adjusts the weighted factor for control parameters based on the Euclidean distance to enhance the SHADE.
• PFI-SHADE [11]: This this DE variant takes the population feature information into account to achieve effective adjustment for control parameters.
• CUSDE [24]: This this DE variant utilize the number of consecutively fail updates to lead the following evolution.

4.2.1. Performance Comparison with SOTA DEs on 30-D Test Functions

We investigate the performance of the STMDE on 30-D CEC2017 functions. It can be observed that the STMDE prevails over compared DE variants with the metric “W/L” of “18/5”, “19/2”, “12/1”, “20/0” and “19/7”, respectively. It is worth mentioning that the STMDE does not lose to PFI-SHADE in any cases. Compared with ACos-JADE, the STMDE shows advantages in dealing with hybrid functions with eight cases winning and none of the cases losing. In comparison to IMPEDE, the STMDE can achieve better or at least similar performance, except for with the F6 and F9. Compared to Db-SHADE, the STMDE is more competitive in handling simple multi-modal functions with five of the cases winning and none of the cases losing out of the seven cases. Compared with PFI-SHADE, the STMDE shows superiority in each kind of test function, especially in unimodal function, simple multi-modal functions and hybrid functions. To be specific, the STMDE achieves a better performance than PFI-SHADE in all unimodal functions. Further, the STMDE performs more competitively than PFI-SHADE in five multi-modal functions, eight hybrid functions and five composition functions. Compared with CUSDE, the STMDE is more advantaged on hybrid functions. Concretely, the STMDE achieves a better performance than CUSDE on all hybrid functions other than F20.

Overall, the STMDE performs better when encountering optimization problems that are difficult to handle, such as hybrid functions and composition functions. In complex optimization problems, the probability of trapping into stagnation is relatively high, while the STMDE can avoid stagnation via enhanced mutation and selection operators.

Table 2. Performance comparison with the SOTA DEs on 30-D CEC2017.

		ACos-JADE			IMPEDE			Db-SHADE			PFI-SHADE			CUSDE			STMDE
		mean	std	sig	mean	std	sig	mean	std	sig	mean	std	sig	mean	std	sig	mean	std
30-D	F1	5.57 × 10−16	2.79 × 10−15	−	9.47 × 10−15	6.77 × 10−15	=	7.24 × 10−15	7.17 × 10−15	=	1.23 × 10−14	4.94 × 10−15	+	9.47 × 10−15	6.77 × 10−15	=	8.36 × 10−15	7.06 × 10−15
	F3	0.00 × 100	0.00 × 100	−	1.19 × 10−6	8.35 × 10−6	+	6.80 × 10−14	2.55 × 10−14	=	1.40 × 103	1.00 × 104	+	1.16 × 101	8.79 × 100	+	6.58 × 10−14	2.38 × 10−14
	F4	5.34 × 101	1.94 × 101	=	4.67 × 101	2.42 × 101	=	5.56 × 101	1.41 × 101	=	5.53 × 101	1.40 × 101	=	5.92 × 101	1.81 × 100	=	5.66 × 101	1.16 × 101
	F5	2.50 × 101	3.96 × 100	+	4.09 × 101	5.16 × 100	+	1.43 × 101	3.02 × 100	+	1.41 × 101	2.49 × 100	+	1.79 × 102	1.02 × 101	+	1.17 × 101	3.48 × 100
	F6	4.57 × 10−13	7.47 × 10−13	−	1.75 × 10−8	1.83 × 10−8	−	7.95 × 10−6	1.43 × 10−5	+	3.20 × 10−5	8.74 × 10−5	+	1.15 × 10−8	2.09 × 10−8	−	4.95 × 10−6	1.22 × 10−5
	F7	5.42 × 101	4.19 × 100	+	7.51 × 101	5.59 × 100	+	4.55 × 101	3.24 × 100	+	4.44 × 101	2.70 × 100	+	2.08 × 102	1.17 × 101	+	4.08 × 101	3.88 × 100
	F8	2.37 × 101	4.44 × 100	+	3.99 × 101	4.33 × 100	+	1.56 × 101	2.70 × 100	+	1.55 × 101	2.66 × 100	+	1.76 × 102	7.26 × 100	+	1.13 × 101	3.33 × 100
	F9	1.59 × 10−2	6.95 × 10−2	−	1.24 × 10−2	6.55 × 10−2	−	2.12 × 10−2	8.01 × 10−2	=	4.25 × 10−2	1.19 × 10−1	=	4.46 × 10−15	2.23 × 10−14	−	1.23 × 10−2	3.11 × 10−2
	F10	1.95 × 103	2.33 × 102	+	3.14 × 103	2.62 × 102	+	1.71 × 103	2.47 × 102	+	1.68 × 103	2.05 × 102	+	6.80 × 103	2.80 × 102	+	1.38 × 103	3.65 × 102
	F11	2.43 × 101	1.94 × 101	+	2.95 × 101	1.89 × 101	+	2.98 × 101	2.73 × 101	=	2.94 × 101	2.36 × 101	+	5.25 × 101	2.39 × 101	+	2.27 × 101	2.31 × 101
	F12	1.26 × 103	4.59 × 102	=	1.06 × 103	3.47 × 102	=	1.24 × 103	3.69 × 102	=	1.14 × 103	3.74 × 102	=	7.15 × 103	6.20 × 103	+	1.10 × 103	4.10 × 102
	F13	6.62 × 101	4.98 × 101	+	5.20 × 101	3.59 × 101	+	2.67 × 101	1.05 × 101	=	3.60 × 101	1.80 × 101	+	8.01 × 101	9.57 × 100	+	2.39 × 101	9.97 × 100
	F14	2.47 × 101	5.16 × 100	+	3.34 × 101	3.40 × 100	+	2.59 × 101	2.62 × 100	+	2.94 × 101	6.02 × 100	+	6.21 × 101	5.86 × 100	+	2.37 × 101	4.49 × 100
	F15	1.97 × 101	1.28 × 101	+	1.36 × 101	1.02 × 101	+	1.18 × 101	8.57 × 100	=	2.21 × 101	1.48 × 101	+	3.73 × 101	4.45 × 100	+	9.05 × 100	4.56 × 100
	F16	2.76 × 102	1.56 × 102	=	4.34 × 102	1.43 × 102	+	2.95 × 102	1.31 × 102	=	2.35 × 102	1.41 × 102	=	5.86 × 102	4.12 × 102	+	2.52 × 102	1.38 × 102
	F17	6.38 × 101	2.61 × 101	+	1.02 × 102	1.53 × 101	+	5.28 × 101	2.61 × 101	+	4.64 × 101	1.04 × 101	+	7.38 × 101	7.69 × 100	+	3.55 × 101	9.66 × 100
	F18	3.36 × 101	1.20 × 101	+	2.67 × 101	2.79 × 100	=	3.55 × 101	2.37 × 101	=	7.27 × 101	5.90 × 101	+	3.65 × 101	3.94 × 100	+	2.80 × 101	8.17 × 100
	F19	1.24 × 101	4.58 × 100	+	1.39 × 101	1.97 × 100	+	8.27 × 100	1.97 × 100	+	1.41 × 101	1.02 × 101	+	1.33 × 101	5.01 × 100	+	7.39 × 100	2.12 × 100
	F20	1.10 × 102	5.54 × 101	+	1.22 × 102	4.13 × 101	+	7.68 × 101	5.11 × 101	+	5.15 × 101	2.93 × 101	+	2.92 × 101	3.16 × 101	−	5.64 × 101	5.42 × 101
	F21	2.26 × 102	4.40 × 100	+	2.40 × 102	5.29 × 100	+	2.17 × 102	3.56 × 100	+	2.16 × 102	3.28 × 100	+	3.65 × 102	1.06 × 101	+	2.13 × 102	4.01 × 100
	F22	1.01 × 102	4.14 × 100	=	1.00 × 102	1.24 × 10−13	=	1.00 × 102	8.92 × 10−14	=	1.00 × 102	1.37 × 10−13	=	1.00 × 102	1.75 × 10−13	=	1.00 × 102	6.39 × 10−14
	F23	3.72 × 102	6.46 × 100	+	3.89 × 102	5.95 × 100	+	3.63 × 102	4.70 × 100	+	3.64 × 102	5.35 × 100	+	5.19 × 102	1.07 × 101	+	3.60 × 102	4.91 × 100
	F24	4.39 × 102	5.17 × 100	+	4.55 × 102	5.28 × 100	+	4.36 × 102	3.73 × 100	=	4.35 × 102	3.45 × 100	=	5.88 × 102	9.15 × 100	+	4.35 × 102	6.18 × 100
	F25	3.87 × 102	7.56 × 10−2	=	3.87 × 102	1.15 × 10−1	+	3.87 × 102	3.85 × 10−2	−	3.87 × 102	3.63 × 10−1	+	3.87 × 102	2.19 × 10−2	−	3.87 × 102	5.10 × 10−2
	F26	1.18 × 103	8.10 × 101	+	1.14 × 103	3.37 × 102	+	1.06 × 103	5.53 × 101	=	1.08 × 103	5.60 × 101	=	2.46 × 103	1.52 × 102	+	1.08 × 103	8.15 × 101
	F27	4.99 × 102	9.69 × 100	−	5.02 × 102	6.58 × 100	=	5.04 × 102	6.13 × 100	=	5.05 × 102	6.91 × 100	=	4.88 × 102	7.73 × 100	−	5.05 × 102	5.52 × 100
	F28	3.38 × 102	5.73 × 101	=	3.30 × 102	4.90 × 101	=	3.26 × 102	5.18 × 101	=	3.47 × 102	5.73 × 101	=	3.17 × 102	4.00 × 101	−	3.41 × 102	5.41 × 101
	F29	4.77 × 102	3.32 × 101	+	5.20 × 102	2.71 × 101	+	4.57 × 102	2.00 × 101	+	4.60 × 102	3.11 × 101	+	5.35 × 102	1.07 × 102	+	4.47 × 102	1.90 × 101
	F30	2.16 × 103	1.67 × 102	+	2.06 × 103	1.63 × 102	=	2.06 × 103	1.32 × 102	=	2.13 × 103	1.39 × 102	+	2.00 × 103	6.46 × 101	−	2.04 × 103	7.55 × 101
	W/T/L	18/6/5			19/8/2			12/16/1			20/9/0			19/3/7

Table 2. Performance comparison with the SOTA DEs on 30-D CEC2017.

		ACos-JADE			IMPEDE			Db-SHADE			PFI-SHADE			CUSDE			STMDE
		mean	std	sig	mean	std	sig	mean	std	sig	mean	std	sig	mean	std	sig	mean	std
30-D	F1	5.57 × 10−16	2.79 × 10−15	−	9.47 × 10−15	6.77 × 10−15	=	7.24 × 10−15	7.17 × 10−15	=	1.23 × 10−14	4.94 × 10−15	+	9.47 × 10−15	6.77 × 10−15	=	8.36 × 10−15	7.06 × 10−15
	F3	0.00 × 100	0.00 × 100	−	1.19 × 10−6	8.35 × 10−6	+	6.80 × 10−14	2.55 × 10−14	=	1.40 × 103	1.00 × 104	+	1.16 × 101	8.79 × 100	+	6.58 × 10−14	2.38 × 10−14
	F4	5.34 × 101	1.94 × 101	=	4.67 × 101	2.42 × 101	=	5.56 × 101	1.41 × 101	=	5.53 × 101	1.40 × 101	=	5.92 × 101	1.81 × 100	=	5.66 × 101	1.16 × 101
	F5	2.50 × 101	3.96 × 100	+	4.09 × 101	5.16 × 100	+	1.43 × 101	3.02 × 100	+	1.41 × 101	2.49 × 100	+	1.79 × 102	1.02 × 101	+	1.17 × 101	3.48 × 100
	F6	4.57 × 10−13	7.47 × 10−13	−	1.75 × 10−8	1.83 × 10−8	−	7.95 × 10−6	1.43 × 10−5	+	3.20 × 10−5	8.74 × 10−5	+	1.15 × 10−8	2.09 × 10−8	−	4.95 × 10−6	1.22 × 10−5
	F7	5.42 × 101	4.19 × 100	+	7.51 × 101	5.59 × 100	+	4.55 × 101	3.24 × 100	+	4.44 × 101	2.70 × 100	+	2.08 × 102	1.17 × 101	+	4.08 × 101	3.88 × 100
	F8	2.37 × 101	4.44 × 100	+	3.99 × 101	4.33 × 100	+	1.56 × 101	2.70 × 100	+	1.55 × 101	2.66 × 100	+	1.76 × 102	7.26 × 100	+	1.13 × 101	3.33 × 100
	F9	1.59 × 10−2	6.95 × 10−2	−	1.24 × 10−2	6.55 × 10−2	−	2.12 × 10−2	8.01 × 10−2	=	4.25 × 10−2	1.19 × 10−1	=	4.46 × 10−15	2.23 × 10−14	−	1.23 × 10−2	3.11 × 10−2
	F10	1.95 × 103	2.33 × 102	+	3.14 × 103	2.62 × 102	+	1.71 × 103	2.47 × 102	+	1.68 × 103	2.05 × 102	+	6.80 × 103	2.80 × 102	+	1.38 × 103	3.65 × 102
	F11	2.43 × 101	1.94 × 101	+	2.95 × 101	1.89 × 101	+	2.98 × 101	2.73 × 101	=	2.94 × 101	2.36 × 101	+	5.25 × 101	2.39 × 101	+	2.27 × 101	2.31 × 101
	F12	1.26 × 103	4.59 × 102	=	1.06 × 103	3.47 × 102	=	1.24 × 103	3.69 × 102	=	1.14 × 103	3.74 × 102	=	7.15 × 103	6.20 × 103	+	1.10 × 103	4.10 × 102
	F13	6.62 × 101	4.98 × 101	+	5.20 × 101	3.59 × 101	+	2.67 × 101	1.05 × 101	=	3.60 × 101	1.80 × 101	+	8.01 × 101	9.57 × 100	+	2.39 × 101	9.97 × 100
	F14	2.47 × 101	5.16 × 100	+	3.34 × 101	3.40 × 100	+	2.59 × 101	2.62 × 100	+	2.94 × 101	6.02 × 100	+	6.21 × 101	5.86 × 100	+	2.37 × 101	4.49 × 100
	F15	1.97 × 101	1.28 × 101	+	1.36 × 101	1.02 × 101	+	1.18 × 101	8.57 × 100	=	2.21 × 101	1.48 × 101	+	3.73 × 101	4.45 × 100	+	9.05 × 100	4.56 × 100
	F16	2.76 × 102	1.56 × 102	=	4.34 × 102	1.43 × 102	+	2.95 × 102	1.31 × 102	=	2.35 × 102	1.41 × 102	=	5.86 × 102	4.12 × 102	+	2.52 × 102	1.38 × 102
	F17	6.38 × 101	2.61 × 101	+	1.02 × 102	1.53 × 101	+	5.28 × 101	2.61 × 101	+	4.64 × 101	1.04 × 101	+	7.38 × 101	7.69 × 100	+	3.55 × 101	9.66 × 100
	F18	3.36 × 101	1.20 × 101	+	2.67 × 101	2.79 × 100	=	3.55 × 101	2.37 × 101	=	7.27 × 101	5.90 × 101	+	3.65 × 101	3.94 × 100	+	2.80 × 101	8.17 × 100
	F19	1.24 × 101	4.58 × 100	+	1.39 × 101	1.97 × 100	+	8.27 × 100	1.97 × 100	+	1.41 × 101	1.02 × 101	+	1.33 × 101	5.01 × 100	+	7.39 × 100	2.12 × 100
	F20	1.10 × 102	5.54 × 101	+	1.22 × 102	4.13 × 101	+	7.68 × 101	5.11 × 101	+	5.15 × 101	2.93 × 101	+	2.92 × 101	3.16 × 101	−	5.64 × 101	5.42 × 101
	F21	2.26 × 102	4.40 × 100	+	2.40 × 102	5.29 × 100	+	2.17 × 102	3.56 × 100	+	2.16 × 102	3.28 × 100	+	3.65 × 102	1.06 × 101	+	2.13 × 102	4.01 × 100
	F22	1.01 × 102	4.14 × 100	=	1.00 × 102	1.24 × 10−13	=	1.00 × 102	8.92 × 10−14	=	1.00 × 102	1.37 × 10−13	=	1.00 × 102	1.75 × 10−13	=	1.00 × 102	6.39 × 10−14
	F23	3.72 × 102	6.46 × 100	+	3.89 × 102	5.95 × 100	+	3.63 × 102	4.70 × 100	+	3.64 × 102	5.35 × 100	+	5.19 × 102	1.07 × 101	+	3.60 × 102	4.91 × 100
	F24	4.39 × 102	5.17 × 100	+	4.55 × 102	5.28 × 100	+	4.36 × 102	3.73 × 100	=	4.35 × 102	3.45 × 100	=	5.88 × 102	9.15 × 100	+	4.35 × 102	6.18 × 100
	F25	3.87 × 102	7.56 × 10−2	=	3.87 × 102	1.15 × 10−1	+	3.87 × 102	3.85 × 10−2	−	3.87 × 102	3.63 × 10−1	+	3.87 × 102	2.19 × 10−2	−	3.87 × 102	5.10 × 10−2
	F26	1.18 × 103	8.10 × 101	+	1.14 × 103	3.37 × 102	+	1.06 × 103	5.53 × 101	=	1.08 × 103	5.60 × 101	=	2.46 × 103	1.52 × 102	+	1.08 × 103	8.15 × 101
	F27	4.99 × 102	9.69 × 100	−	5.02 × 102	6.58 × 100	=	5.04 × 102	6.13 × 100	=	5.05 × 102	6.91 × 100	=	4.88 × 102	7.73 × 100	−	5.05 × 102	5.52 × 100
	F28	3.38 × 102	5.73 × 101	=	3.30 × 102	4.90 × 101	=	3.26 × 102	5.18 × 101	=	3.47 × 102	5.73 × 101	=	3.17 × 102	4.00 × 101	−	3.41 × 102	5.41 × 101
	F29	4.77 × 102	3.32 × 101	+	5.20 × 102	2.71 × 101	+	4.57 × 102	2.00 × 101	+	4.60 × 102	3.11 × 101	+	5.35 × 102	1.07 × 102	+	4.47 × 102	1.90 × 101
	F30	2.16 × 103	1.67 × 102	+	2.06 × 103	1.63 × 102	=	2.06 × 103	1.32 × 102	=	2.13 × 103	1.39 × 102	+	2.00 × 103	6.46 × 101	−	2.04 × 103	7.55 × 101
	W/T/L	18/6/5			19/8/2			12/16/1			20/9/0			19/3/7

4.2.2. Performance Comparison with SOTA DEs on 50-D Test Functions

The performance of the STMDE on 50-D CEC2017 functions is investigated in this subsection. On the whole, the STMDE is advantaged over the five SOTA DEs. To be specific, the STMDE shows superiority over ACos-JADE, IMPEDE, Db-SHADE, PFI-SHADE and CUSDE on 22, 19, 10, 16 and 21 functions while performing worse on 4, 6, 4, 0, 8 functions. With respect to the function properties, the advantage is more significant in simple multi-modal functions and composition functions. On the one hand, the STMDE can achieve a better performance on 5, 5, 4, 5 and 5 functions compared to ACos-JADE, IMPEDE, Db-SHADE, PFI-SHADE and CUSDE out of seven simple multi-modal functions. On the other hand, the STMDE does not lose in any composition functions in comparison to ACos-JADE, IMPEDE and PFI-SHADE.

In summary, the STMDE shows advantages over compared DE variants in optimization problems with high complexity. When a large number of local optima exist, the searching process is hindered and stagnation happens, while the STMDE can generate appropriate control parameters and greediness parameters for each individual, which is beneficial to avoid the stagnation. In addition, the STMDE can also cease stagnation effectively by abandoning the stagnant one and generating a new one to replace it. It is worth noting that the new offspring may approach the best individual of the current population, which ensures their convergence.

Table 3. Performance comparison with the SOTA DEs on 50-D CEC2017.

		ACos-JADE			IMPEDE			Db-SHADE			PFI-SHADE			CUSDE			STMDE
		mean	std	sig	mean	std	sig	mean	std	sig	mean	std	sig	mean	std	sig	mean	std
50-D	F1	1.25 × 10−14	4.62 × 10−15	−	3.91 × 10−8	1.09 × 10−7	+	3.85 × 10−14	1.80 × 10−14	=	4.54 × 10−14	1.73 × 10−14	+	1.90 × 102	5.43 × 102	+	3.76 × 10−14	1.33 × 10−14
	F3	4.68 × 10−14	2.19 × 10−14	−	2.09 × 10−1	1.03 × 100	+	3.07 × 10−13	7.80 × 10−14	−	2.56 × 103	1.83 × 104	=	5.60 × 104	1.13 × 104	+	4.24 × 10−13	2.50 × 10−13
	F4	4.79 × 101	4.59 × 101	=	6.48 × 101	4.70 × 101	+	4.79 × 101	4.82 × 101	=	5.85 × 101	5.12 × 101	=	7.53 × 101	4.82 × 101	+	4.63 × 101	4.26 × 101
	F5	5.77 × 101	8.60 × 100	+	1.16 × 102	8.73 × 100	+	3.15 × 101	4.16 × 100	+	3.10 × 101	4.92 × 100	=	3.51 × 102	1.43 × 101	+	2.83 × 101	8.71 × 100
	F6	7.00 × 10−5	4.54 × 10−4	−	2.79 × 10−6	2.76 × 10−6	−	2.04 × 10−4	1.78 × 10−4	−	5.47 × 10−3	4.11 × 10−3	+	2.55 × 10−8	9.14 × 10−8	−	6.67 × 10−4	6.68 × 10−4
	F7	9.96 × 101	8.73 × 100	+	1.85 × 102	1.14 × 101	+	8.03 × 101	4.76 × 100	+	8.10 × 101	5.14 × 100	+	4.01 × 102	1.50 × 101	+	7.50 × 101	7.02 × 100
	F8	5.38 × 101	9.13 × 100	+	1.19 × 102	7.36 × 100	+	3.27 × 101	5.13 × 100	+	3.06 × 101	5.15 × 100	+	3.53 × 102	1.40 × 101	+	2.92 × 101	1.12 × 101
	F9	7.99 × 10−1	7.29 × 10−1	+	2.83 × 10−1	4.49 × 10−1	−	4.28 × 10−1	4.23 × 10−1	=	1.93 × 100	1.39 × 100	+	5.87 × 10−2	2.55 × 10−1	−	4.82 × 10−1	4.95 × 10−1
	F10	3.82 × 103	3.62 × 102	+	6.88 × 103	2.99 × 102	+	3.34 × 103	2.91 × 102	+	3.38 × 103	2.64 × 102	+	1.30 × 104	3.27 × 102	+	2.73 × 103	4.26 × 102
	F11	1.54 × 102	3.79 × 101	+	1.05 × 102	2.92 × 101	=	9.54 × 101	2.82 × 101	−	1.41 × 102	3.88 × 101	+	1.34 × 102	3.36 × 101	+	1.08 × 102	2.65 × 101
	F12	3.89 × 103	3.31 × 103	−	1.66 × 104	1.43 × 104	+	7.13 × 103	4.70 × 103	=	5.02 × 103	3.21 × 103	=	5.61 × 104	3.32 × 104	+	6.13 × 103	4.55 × 103
	F13	7.62 × 102	6.12 × 102	+	2.04 × 102	4.24 × 101	+	1.33 × 102	5.37 × 101	=	2.57 × 102	2.20 × 102	+	2.63 × 102	6.90 × 101	+	1.38 × 102	8.51 × 101
	F14	2.34 × 102	6.21 × 101	+	7.15 × 101	1.19 × 101	−	1.00 × 102	4.30 × 101	=	2.07 × 102	6.68 × 101	+	1.27 × 102	6.95 × 100	+	8.87 × 101	3.36 × 101
	F15	4.03 × 102	1.32 × 102	+	6.30 × 101	3.18 × 101	−	1.66 × 102	8.70 × 101	=	3.09 × 102	1.56 × 102	+	1.07 × 102	8.50 × 100	−	1.43 × 102	7.94 × 101
	F16	7.11 × 102	2.08 × 102	=	1.13 × 103	1.61 × 102	+	7.70 × 102	1.46 × 102	+	6.71 × 102	1.67 × 102	=	2.32 × 103	7.56 × 102	+	6.31 × 102	2.22 × 102
	F17	5.73 × 102	1.20 × 102	+	8.31 × 102	1.16 × 102	+	5.02 × 102	1.20 × 102	+	5.33 × 102	1.44 × 102	+	1.22 × 103	5.33 × 102	+	4.42 × 102	1.35 × 102
	F18	3.18 × 102	1.06 × 102	+	9.22 × 101	8.88 × 101	−	1.75 × 102	1.08 × 102	=	2.08 × 102	1.17 × 102	=	4.90 × 102	3.99 × 102	+	1.74 × 102	1.10 × 102
	F19	1.52 × 102	5.57 × 101	+	4.06 × 101	1.11 × 101	−	1.16 × 102	4.33 × 101	+	1.47 × 102	4.00 × 101	+	6.15 × 101	8.63 × 100	−	9.69 × 101	3.87 × 101
	F20	4.58 × 102	1.35 × 102	+	6.22 × 102	1.04 × 102	+	3.19 × 102	1.11 × 102	=	3.17 × 102	1.19 × 102	=	7.94 × 102	4.66 × 102	+	3.16 × 102	1.57 × 102
	F21	2.55 × 102	9.74 × 100	+	3.16 × 102	8.85 × 100	+	2.33 × 102	7.25 × 100	+	2.32 × 102	3.83 × 100	+	5.54 × 102	1.59 × 101	+	2.31 × 102	8.33 × 100
	F22	3.86 × 103	1.56 × 103	+	4.32 × 103	3.27 × 103	+	2.79 × 103	1.76 × 103	+	3.30 × 103	1.60 × 103	+	1.10 × 104	4.78 × 103	+	2.44 × 103	1.52 × 103
	F23	4.77 × 102	1.23 × 101	+	5.44 × 102	1.14 × 101	+	4.58 × 102	8.58 × 100	=	4.56 × 102	7.65 × 100	=	7.70 × 102	1.77 × 101	+	4.57 × 102	1.32 × 101
	F24	5.45 × 102	8.57 × 100	+	5.98 × 102	7.56 × 100	+	5.30 × 102	7.22 × 100	=	5.30 × 102	5.90 × 100	=	8.37 × 102	1.40 × 101	+	5.33 × 102	9.59 × 100
	F25	5.26 × 102	3.75 × 101	=	5.19 × 102	3.29 × 101	=	5.02 × 102	3.38 × 101	−	5.13 × 102	3.53 × 101	=	4.92 × 102	2.82 × 101	−	5.25 × 102	3.43 × 101
	F26	1.63 × 103	1.17 × 102	+	2.10 × 103	1.30 × 102	+	1.40 × 103	9.28 × 101	=	1.41 × 103	9.73 × 101	=	4.16 × 103	5.53 × 102	+	1.44 × 103	1.10 × 102
	F27	5.67 × 102	3.96 × 101	+	5.78 × 102	6.19 × 101	+	5.41 × 102	3.40 × 101	=	5.58 × 102	3.20 × 101	+	5.07 × 102	1.11 × 101	−	5.33 × 102	1.31 × 101
	F28	4.90 × 102	2.19 × 101	+	4.89 × 102	2.29 × 101	=	4.82 × 102	2.41 × 101	=	4.87 × 102	2.36 × 101	=	4.70 × 102	2.07 × 101	−	4.80 × 102	2.40 × 101
	F29	4.86 × 102	7.34 × 101	+	6.27 × 102	7.21 × 101	+	4.23 × 102	7.72 × 101	+	4.45 × 102	8.37 × 101	+	7.39 × 102	4.12 × 102	+	4.04 × 102	7.97 × 101
	F30	7.73 × 105	1.53 × 105	+	6.91 × 105	9.01 × 104	=	6.93 × 105	8.73 × 104	=	6.93 × 105	8.62 × 104	=	5.86 × 105	1.45 × 104	−	6.68 × 105	8.37 × 104
	W/T/L	22/3/4			19/4/6			10/15/4			16/13/0			21/0/8

Table 3. Performance comparison with the SOTA DEs on 50-D CEC2017.

		ACos-JADE			IMPEDE			Db-SHADE			PFI-SHADE			CUSDE			STMDE
		mean	std	sig	mean	std	sig	mean	std	sig	mean	std	sig	mean	std	sig	mean	std
50-D	F1	1.25 × 10−14	4.62 × 10−15	−	3.91 × 10−8	1.09 × 10−7	+	3.85 × 10−14	1.80 × 10−14	=	4.54 × 10−14	1.73 × 10−14	+	1.90 × 102	5.43 × 102	+	3.76 × 10−14	1.33 × 10−14
	F3	4.68 × 10−14	2.19 × 10−14	−	2.09 × 10−1	1.03 × 100	+	3.07 × 10−13	7.80 × 10−14	−	2.56 × 103	1.83 × 104	=	5.60 × 104	1.13 × 104	+	4.24 × 10−13	2.50 × 10−13
	F4	4.79 × 101	4.59 × 101	=	6.48 × 101	4.70 × 101	+	4.79 × 101	4.82 × 101	=	5.85 × 101	5.12 × 101	=	7.53 × 101	4.82 × 101	+	4.63 × 101	4.26 × 101
	F5	5.77 × 101	8.60 × 100	+	1.16 × 102	8.73 × 100	+	3.15 × 101	4.16 × 100	+	3.10 × 101	4.92 × 100	=	3.51 × 102	1.43 × 101	+	2.83 × 101	8.71 × 100
	F6	7.00 × 10−5	4.54 × 10−4	−	2.79 × 10−6	2.76 × 10−6	−	2.04 × 10−4	1.78 × 10−4	−	5.47 × 10−3	4.11 × 10−3	+	2.55 × 10−8	9.14 × 10−8	−	6.67 × 10−4	6.68 × 10−4
	F7	9.96 × 101	8.73 × 100	+	1.85 × 102	1.14 × 101	+	8.03 × 101	4.76 × 100	+	8.10 × 101	5.14 × 100	+	4.01 × 102	1.50 × 101	+	7.50 × 101	7.02 × 100
	F8	5.38 × 101	9.13 × 100	+	1.19 × 102	7.36 × 100	+	3.27 × 101	5.13 × 100	+	3.06 × 101	5.15 × 100	+	3.53 × 102	1.40 × 101	+	2.92 × 101	1.12 × 101
	F9	7.99 × 10−1	7.29 × 10−1	+	2.83 × 10−1	4.49 × 10−1	−	4.28 × 10−1	4.23 × 10−1	=	1.93 × 100	1.39 × 100	+	5.87 × 10−2	2.55 × 10−1	−	4.82 × 10−1	4.95 × 10−1
	F10	3.82 × 103	3.62 × 102	+	6.88 × 103	2.99 × 102	+	3.34 × 103	2.91 × 102	+	3.38 × 103	2.64 × 102	+	1.30 × 104	3.27 × 102	+	2.73 × 103	4.26 × 102
	F11	1.54 × 102	3.79 × 101	+	1.05 × 102	2.92 × 101	=	9.54 × 101	2.82 × 101	−	1.41 × 102	3.88 × 101	+	1.34 × 102	3.36 × 101	+	1.08 × 102	2.65 × 101
	F12	3.89 × 103	3.31 × 103	−	1.66 × 104	1.43 × 104	+	7.13 × 103	4.70 × 103	=	5.02 × 103	3.21 × 103	=	5.61 × 104	3.32 × 104	+	6.13 × 103	4.55 × 103
	F13	7.62 × 102	6.12 × 102	+	2.04 × 102	4.24 × 101	+	1.33 × 102	5.37 × 101	=	2.57 × 102	2.20 × 102	+	2.63 × 102	6.90 × 101	+	1.38 × 102	8.51 × 101
	F14	2.34 × 102	6.21 × 101	+	7.15 × 101	1.19 × 101	−	1.00 × 102	4.30 × 101	=	2.07 × 102	6.68 × 101	+	1.27 × 102	6.95 × 100	+	8.87 × 101	3.36 × 101
	F15	4.03 × 102	1.32 × 102	+	6.30 × 101	3.18 × 101	−	1.66 × 102	8.70 × 101	=	3.09 × 102	1.56 × 102	+	1.07 × 102	8.50 × 100	−	1.43 × 102	7.94 × 101
	F16	7.11 × 102	2.08 × 102	=	1.13 × 103	1.61 × 102	+	7.70 × 102	1.46 × 102	+	6.71 × 102	1.67 × 102	=	2.32 × 103	7.56 × 102	+	6.31 × 102	2.22 × 102
	F17	5.73 × 102	1.20 × 102	+	8.31 × 102	1.16 × 102	+	5.02 × 102	1.20 × 102	+	5.33 × 102	1.44 × 102	+	1.22 × 103	5.33 × 102	+	4.42 × 102	1.35 × 102
	F18	3.18 × 102	1.06 × 102	+	9.22 × 101	8.88 × 101	−	1.75 × 102	1.08 × 102	=	2.08 × 102	1.17 × 102	=	4.90 × 102	3.99 × 102	+	1.74 × 102	1.10 × 102
	F19	1.52 × 102	5.57 × 101	+	4.06 × 101	1.11 × 101	−	1.16 × 102	4.33 × 101	+	1.47 × 102	4.00 × 101	+	6.15 × 101	8.63 × 100	−	9.69 × 101	3.87 × 101
	F20	4.58 × 102	1.35 × 102	+	6.22 × 102	1.04 × 102	+	3.19 × 102	1.11 × 102	=	3.17 × 102	1.19 × 102	=	7.94 × 102	4.66 × 102	+	3.16 × 102	1.57 × 102
	F21	2.55 × 102	9.74 × 100	+	3.16 × 102	8.85 × 100	+	2.33 × 102	7.25 × 100	+	2.32 × 102	3.83 × 100	+	5.54 × 102	1.59 × 101	+	2.31 × 102	8.33 × 100
	F22	3.86 × 103	1.56 × 103	+	4.32 × 103	3.27 × 103	+	2.79 × 103	1.76 × 103	+	3.30 × 103	1.60 × 103	+	1.10 × 104	4.78 × 103	+	2.44 × 103	1.52 × 103
	F23	4.77 × 102	1.23 × 101	+	5.44 × 102	1.14 × 101	+	4.58 × 102	8.58 × 100	=	4.56 × 102	7.65 × 100	=	7.70 × 102	1.77 × 101	+	4.57 × 102	1.32 × 101
	F24	5.45 × 102	8.57 × 100	+	5.98 × 102	7.56 × 100	+	5.30 × 102	7.22 × 100	=	5.30 × 102	5.90 × 100	=	8.37 × 102	1.40 × 101	+	5.33 × 102	9.59 × 100
	F25	5.26 × 102	3.75 × 101	=	5.19 × 102	3.29 × 101	=	5.02 × 102	3.38 × 101	−	5.13 × 102	3.53 × 101	=	4.92 × 102	2.82 × 101	−	5.25 × 102	3.43 × 101
	F26	1.63 × 103	1.17 × 102	+	2.10 × 103	1.30 × 102	+	1.40 × 103	9.28 × 101	=	1.41 × 103	9.73 × 101	=	4.16 × 103	5.53 × 102	+	1.44 × 103	1.10 × 102
	F27	5.67 × 102	3.96 × 101	+	5.78 × 102	6.19 × 101	+	5.41 × 102	3.40 × 101	=	5.58 × 102	3.20 × 101	+	5.07 × 102	1.11 × 101	−	5.33 × 102	1.31 × 101
	F28	4.90 × 102	2.19 × 101	+	4.89 × 102	2.29 × 101	=	4.82 × 102	2.41 × 101	=	4.87 × 102	2.36 × 101	=	4.70 × 102	2.07 × 101	−	4.80 × 102	2.40 × 101
	F29	4.86 × 102	7.34 × 101	+	6.27 × 102	7.21 × 101	+	4.23 × 102	7.72 × 101	+	4.45 × 102	8.37 × 101	+	7.39 × 102	4.12 × 102	+	4.04 × 102	7.97 × 101
	F30	7.73 × 105	1.53 × 105	+	6.91 × 105	9.01 × 104	=	6.93 × 105	8.73 × 104	=	6.93 × 105	8.62 × 104	=	5.86 × 105	1.45 × 104	−	6.68 × 105	8.37 × 104
	W/T/L	22/3/4			19/4/6			10/15/4			16/13/0			21/0/8

4.2.3. Performance Comparison with SOTA DEs on 100-D Test Functions

We investigate the performance of the STMDE on 100-D CEC2017 functions and the experimental results are given in Table 4. It can be observed that the STMDE is still superior in handling high-dimension optimization problems on the whole. When compared to ACos-JADE, IMPEDE, Db-SHADE, PFI-SHADE and CUSDE, the performance metric “W/L” is “19/6”, “21/7”, “8/7”, “16/2” and “23/4”, respectively. From the perspective of the function property, the STMDE shows advantages in handling simple multi-modal functions and composition function. On the one hand, with respect to simple multi-modal functions, the STMDE outperforms ACos-JADE, IMPEDE and CUSDE in five simple multi-modal functions. When compared with PFI-SHADE, the STMDE shows superiority in four test functions and loses in none of them. In comparison to Db-SHADE, the searching capacity is similar. On the other hand, with respect to composition functions, the STMDE shows advantages over compared DE variants on eight, nine, five, five, and seven test functions successively. Especially, the performance of the STMDE is not inferior to IMPEDE and PFI-SHADE on any benchmark functions.

Table 4. Performance comparison with the SOTA DEs on 100-D CEC2017.

		ACos-JADE			IMPEDE			Db-SHADE			PFI-SHADE			CUSDE			STMDE
		mean	std	sig	mean	std	sig	mean	std	sig	mean	std	sig	mean	std	sig	mean	std
100-D	F1	2.70 × 10−14	3.82 × 10−14	−	1.84 × 10−5	3.20 × 10−5	+	2.44 × 10−11	8.79 × 10−11	−	2.50 × 10−10	4.58 × 10−10	+	7.05 × 103	6.93 × 103	+	4.94 × 10−11	1.59 × 10−10
	F3	5.68 × 10−14	0.00 × 100	−	5.11 × 102	9.94 × 102	+	1.05 × 104	7.53 × 104	−	8.03 × 103	5.73 × 104	−	3.13 × 105	3.55 × 104	+	2.62 × 10−3	1.14 × 10−2
	F4	2.42 × 101	5.72 × 101	−	1.60 × 102	4.77 × 101	+	1.32 × 102	7.56 × 101	=	1.28 × 102	6.61 × 101	=	2.07 × 102	2.70 × 101	+	1.27 × 102	6.95 × 101
	F5	1.89 × 102	2.34 × 101	+	3.38 × 102	2.26 × 101	+	9.43 × 101	1.14 × 101	=	9.92 × 101	1.43 × 101	=	8.19 × 102	1.80 × 101	+	9.98 × 101	1.52 × 101
	F6	6.09 × 10−2	9.80 × 10−2	=	4.66 × 10−3	9.42 × 10−3	−	1.31 × 10−2	9.87 × 10−3	−	7.24 × 10−1	2.80 × 10−1	+	6.11 × 10−4	1.30 × 10−3	−	2.77 × 10−2	2.29 × 10−2
	F7	2.97 × 102	2.24 × 101	+	4.83 × 102	2.61 × 101	+	1.99 × 102	1.35 × 101	=	2.33 × 102	2.27 × 101	+	9.31 × 102	2.40 × 101	+	1.94 × 102	1.79 × 101
	F8	1.85 × 102	2.55 × 101	+	3.22 × 102	1.50 × 101	+	9.53 × 101	1.40 × 101	=	9.89 × 101	1.09 × 101	=	8.20 × 102	1.81 × 101	+	9.63 × 101	1.76 × 101
	F9	1.50 × 102	1.63 × 102	+	7.75 × 100	3.76 × 100	−	1.35 × 101	4.84 × 100	=	1.40 × 102	6.21 × 101	+	6.35 × 100	9.73 × 100	−	1.46 × 101	7.64 × 100
	F10	1.10 × 104	6.54 × 102	+	1.79 × 104	4.20 × 102	+	9.24 × 103	5.10 × 102	+	9.31 × 103	5.91 × 102	+	2.95 × 104	5.43 × 102	+	8.27 × 103	9.30 × 102
	F11	1.08 × 103	2.10 × 102	+	7.78 × 102	2.33 × 102	−	1.03 × 103	2.24 × 102	+	1.08 × 103	2.77 × 102	+	6.07 × 102	6.17 × 101	−	8.81 × 102	2.42 × 102
	F12	1.60 × 104	8.46 × 103	−	7.39 × 104	3.14 × 104	+	1.89 × 104	1.01 × 104	−	1.97 × 104	1.25 × 104	−	2.39 × 105	8.78 × 104	+	2.37 × 104	9.06 × 103
	F13	3.65 × 103	3.76 × 103	+	2.52 × 103	2.04 × 103	+	1.01 × 103	1.65 × 103	=	7.38 × 102	4.29 × 102	+	4.40 × 103	5.59 × 103	+	3.31 × 102	1.69 × 102
	F14	5.65 × 102	1.22 × 102	=	4.37 × 102	8.94 × 101	−	6.01 × 102	2.02 × 102	=	6.35 × 102	1.82 × 102	+	7.62 × 103	9.63 × 103	+	5.50 × 102	1.17 × 102
	F15	3.66 × 102	6.95 × 101	=	3.29 × 102	8.72 × 101	−	3.76 × 102	1.45 × 102	=	3.62 × 102	8.15 × 101	=	2.01 × 103	2.52 × 103	+	4.49 × 102	2.28 × 102
	F16	2.49 × 103	3.31 × 102	+	3.70 × 103	2.64 × 102	+	2.33 × 103	3.47 × 102	=	2.30 × 103	3.97 × 102	=	7.39 × 103	2.98 × 102	+	2.34 × 103	4.31 × 102
	F17	1.94 × 103	2.69 × 102	+	2.67 × 103	2.83 × 102	+	1.77 × 103	2.42 × 102	=	1.83 × 103	2.53 × 102	+	4.44 × 103	7.76 × 102	+	1.65 × 103	3.72 × 102
	F18	3.01 × 102	9.51 × 101	−	4.92 × 102	3.13 × 102	−	2.13 × 103	2.70 × 103	=	1.51 × 103	9.10 × 102	=	1.08 × 105	6.64 × 104	+	2.08 × 103	2.04 × 103
	F19	1.39 × 103	2.41 × 103	+	2.22 × 102	4.99 × 101	−	2.47 × 102	5.05 × 101	=	3.29 × 102	3.45 × 102	+	3.03 × 103	3.78 × 103	+	2.41 × 102	5.01 × 101
	F20	2.05 × 103	2.74 × 102	+	2.73 × 103	2.12 × 102	+	1.73 × 103	2.25 × 102	+	1.70 × 103	2.35 × 102	+	4.41 × 103	7.05 × 102	+	1.57 × 103	3.65 × 102
	F21	4.01 × 102	1.93 × 101	+	5.44 × 102	1.94 × 101	+	3.28 × 102	1.36 × 101	=	3.24 × 102	1.02 × 101	=	1.05 × 103	2.74 × 101	+	3.25 × 102	1.51 × 101
	F22	1.24 × 104	5.77 × 102	+	1.86 × 104	2.14 × 103	+	1.03 × 104	1.53 × 103	+	1.07 × 104	5.61 × 102	+	3.02 × 104	4.16 × 102	+	9.78 × 103	1.27 × 103
	F23	6.53 × 102	1.73 × 101	+	8.08 × 102	1.56 × 101	+	6.08 × 102	1.52 × 101	+	6.04 × 102	1.48 × 101	+	7.31 × 102	2.40 × 102	+	5.98 × 102	1.95 × 101
	F24	1.04 × 103	3.06 × 101	+	1.18 × 103	1.98 × 101	+	9.83 × 102	2.30 × 101	−	9.92 × 102	2.38 × 101	=	1.44 × 103	2.99 × 102	+	1.00 × 103	2.20 × 101
	F25	7.51 × 102	6.06 × 101	=	7.49 × 102	4.90 × 101	=	7.22 × 102	4.90 × 101	=	7.39 × 102	5.76 × 101	=	7.45 × 102	4.27 × 101	=	7.40 × 102	4.27 × 101
	F26	4.82 × 103	3.48 × 102	+	6.02 × 103	2.98 × 102	+	4.15 × 103	2.00 × 102	−	4.26 × 103	2.42 × 102	=	6.34 × 103	3.30 × 103	=	4.28 × 103	2.47 × 102
	F27	7.58 × 102	4.31 × 101	+	6.72 × 102	2.76 × 101	+	6.58 × 102	2.38 × 101	+	6.88 × 102	3.15 × 101	+	5.94 × 102	1.97 × 101	−	6.47 × 102	2.51 × 101
	F28	4.02 × 102	1.22 × 102	−	5.52 × 102	2.28 × 101	+	5.13 × 102	4.99 × 101	−	5.20 × 102	2.42 × 101	=	5.50 × 102	3.51 × 101	+	5.32 × 102	3.00 × 101
	F29	2.36 × 103	2.71 × 102	+	2.78 × 103	2.42 × 102	+	2.09 × 103	2.96 × 102	+	2.08 × 103	2.71 × 102	+	3.96 × 103	1.47 × 103	+	1.91 × 103	2.94 × 102
	F30	4.06 × 103	2.29 × 103	+	3.03 × 103	8.21 × 102	+	2.63 × 103	2.05 × 102	+	3.12 × 103	1.04 × 103	+	3.66 × 103	1.54 × 103	+	2.53 × 103	1.86 × 102
	W/T/L	19/4/6			21/1/7			8/14/7			16/11/2			23/2/4

Table 4. Performance comparison with the SOTA DEs on 100-D CEC2017.

		ACos-JADE			IMPEDE			Db-SHADE			PFI-SHADE			CUSDE			STMDE
		mean	std	sig	mean	std	sig	mean	std	sig	mean	std	sig	mean	std	sig	mean	std
100-D	F1	2.70 × 10−14	3.82 × 10−14	−	1.84 × 10−5	3.20 × 10−5	+	2.44 × 10−11	8.79 × 10−11	−	2.50 × 10−10	4.58 × 10−10	+	7.05 × 103	6.93 × 103	+	4.94 × 10−11	1.59 × 10−10
	F3	5.68 × 10−14	0.00 × 100	−	5.11 × 102	9.94 × 102	+	1.05 × 104	7.53 × 104	−	8.03 × 103	5.73 × 104	−	3.13 × 105	3.55 × 104	+	2.62 × 10−3	1.14 × 10−2
	F4	2.42 × 101	5.72 × 101	−	1.60 × 102	4.77 × 101	+	1.32 × 102	7.56 × 101	=	1.28 × 102	6.61 × 101	=	2.07 × 102	2.70 × 101	+	1.27 × 102	6.95 × 101
	F5	1.89 × 102	2.34 × 101	+	3.38 × 102	2.26 × 101	+	9.43 × 101	1.14 × 101	=	9.92 × 101	1.43 × 101	=	8.19 × 102	1.80 × 101	+	9.98 × 101	1.52 × 101
	F6	6.09 × 10−2	9.80 × 10−2	=	4.66 × 10−3	9.42 × 10−3	−	1.31 × 10−2	9.87 × 10−3	−	7.24 × 10−1	2.80 × 10−1	+	6.11 × 10−4	1.30 × 10−3	−	2.77 × 10−2	2.29 × 10−2
	F7	2.97 × 102	2.24 × 101	+	4.83 × 102	2.61 × 101	+	1.99 × 102	1.35 × 101	=	2.33 × 102	2.27 × 101	+	9.31 × 102	2.40 × 101	+	1.94 × 102	1.79 × 101
	F8	1.85 × 102	2.55 × 101	+	3.22 × 102	1.50 × 101	+	9.53 × 101	1.40 × 101	=	9.89 × 101	1.09 × 101	=	8.20 × 102	1.81 × 101	+	9.63 × 101	1.76 × 101
	F9	1.50 × 102	1.63 × 102	+	7.75 × 100	3.76 × 100	−	1.35 × 101	4.84 × 100	=	1.40 × 102	6.21 × 101	+	6.35 × 100	9.73 × 100	−	1.46 × 101	7.64 × 100
	F10	1.10 × 104	6.54 × 102	+	1.79 × 104	4.20 × 102	+	9.24 × 103	5.10 × 102	+	9.31 × 103	5.91 × 102	+	2.95 × 104	5.43 × 102	+	8.27 × 103	9.30 × 102
	F11	1.08 × 103	2.10 × 102	+	7.78 × 102	2.33 × 102	−	1.03 × 103	2.24 × 102	+	1.08 × 103	2.77 × 102	+	6.07 × 102	6.17 × 101	−	8.81 × 102	2.42 × 102
	F12	1.60 × 104	8.46 × 103	−	7.39 × 104	3.14 × 104	+	1.89 × 104	1.01 × 104	−	1.97 × 104	1.25 × 104	−	2.39 × 105	8.78 × 104	+	2.37 × 104	9.06 × 103
	F13	3.65 × 103	3.76 × 103	+	2.52 × 103	2.04 × 103	+	1.01 × 103	1.65 × 103	=	7.38 × 102	4.29 × 102	+	4.40 × 103	5.59 × 103	+	3.31 × 102	1.69 × 102
	F14	5.65 × 102	1.22 × 102	=	4.37 × 102	8.94 × 101	−	6.01 × 102	2.02 × 102	=	6.35 × 102	1.82 × 102	+	7.62 × 103	9.63 × 103	+	5.50 × 102	1.17 × 102
	F15	3.66 × 102	6.95 × 101	=	3.29 × 102	8.72 × 101	−	3.76 × 102	1.45 × 102	=	3.62 × 102	8.15 × 101	=	2.01 × 103	2.52 × 103	+	4.49 × 102	2.28 × 102
	F16	2.49 × 103	3.31 × 102	+	3.70 × 103	2.64 × 102	+	2.33 × 103	3.47 × 102	=	2.30 × 103	3.97 × 102	=	7.39 × 103	2.98 × 102	+	2.34 × 103	4.31 × 102
	F17	1.94 × 103	2.69 × 102	+	2.67 × 103	2.83 × 102	+	1.77 × 103	2.42 × 102	=	1.83 × 103	2.53 × 102	+	4.44 × 103	7.76 × 102	+	1.65 × 103	3.72 × 102
	F18	3.01 × 102	9.51 × 101	−	4.92 × 102	3.13 × 102	−	2.13 × 103	2.70 × 103	=	1.51 × 103	9.10 × 102	=	1.08 × 105	6.64 × 104	+	2.08 × 103	2.04 × 103
	F19	1.39 × 103	2.41 × 103	+	2.22 × 102	4.99 × 101	−	2.47 × 102	5.05 × 101	=	3.29 × 102	3.45 × 102	+	3.03 × 103	3.78 × 103	+	2.41 × 102	5.01 × 101
	F20	2.05 × 103	2.74 × 102	+	2.73 × 103	2.12 × 102	+	1.73 × 103	2.25 × 102	+	1.70 × 103	2.35 × 102	+	4.41 × 103	7.05 × 102	+	1.57 × 103	3.65 × 102
	F21	4.01 × 102	1.93 × 101	+	5.44 × 102	1.94 × 101	+	3.28 × 102	1.36 × 101	=	3.24 × 102	1.02 × 101	=	1.05 × 103	2.74 × 101	+	3.25 × 102	1.51 × 101
	F22	1.24 × 104	5.77 × 102	+	1.86 × 104	2.14 × 103	+	1.03 × 104	1.53 × 103	+	1.07 × 104	5.61 × 102	+	3.02 × 104	4.16 × 102	+	9.78 × 103	1.27 × 103
	F23	6.53 × 102	1.73 × 101	+	8.08 × 102	1.56 × 101	+	6.08 × 102	1.52 × 101	+	6.04 × 102	1.48 × 101	+	7.31 × 102	2.40 × 102	+	5.98 × 102	1.95 × 101
	F24	1.04 × 103	3.06 × 101	+	1.18 × 103	1.98 × 101	+	9.83 × 102	2.30 × 101	−	9.92 × 102	2.38 × 101	=	1.44 × 103	2.99 × 102	+	1.00 × 103	2.20 × 101
	F25	7.51 × 102	6.06 × 101	=	7.49 × 102	4.90 × 101	=	7.22 × 102	4.90 × 101	=	7.39 × 102	5.76 × 101	=	7.45 × 102	4.27 × 101	=	7.40 × 102	4.27 × 101
	F26	4.82 × 103	3.48 × 102	+	6.02 × 103	2.98 × 102	+	4.15 × 103	2.00 × 102	−	4.26 × 103	2.42 × 102	=	6.34 × 103	3.30 × 103	=	4.28 × 103	2.47 × 102
	F27	7.58 × 102	4.31 × 101	+	6.72 × 102	2.76 × 101	+	6.58 × 102	2.38 × 101	+	6.88 × 102	3.15 × 101	+	5.94 × 102	1.97 × 101	−	6.47 × 102	2.51 × 101
	F28	4.02 × 102	1.22 × 102	−	5.52 × 102	2.28 × 101	+	5.13 × 102	4.99 × 101	−	5.20 × 102	2.42 × 101	=	5.50 × 102	3.51 × 101	+	5.32 × 102	3.00 × 101
	F29	2.36 × 103	2.71 × 102	+	2.78 × 103	2.42 × 102	+	2.09 × 103	2.96 × 102	+	2.08 × 103	2.71 × 102	+	3.96 × 103	1.47 × 103	+	1.91 × 103	2.94 × 102
	F30	4.06 × 103	2.29 × 103	+	3.03 × 103	8.21 × 102	+	2.63 × 103	2.05 × 102	+	3.12 × 103	1.04 × 103	+	3.66 × 103	1.54 × 103	+	2.53 × 103	1.86 × 102
	W/T/L	19/4/6			21/1/7			8/14/7			16/11/2			23/2/4

From Table 2, Table 3 and Table 4, we can observe that the performance advantages degrade to some extend when the problem size increases. However, the STMDE is still better than the compared DE variants on the whole when the problem size is relatively high. Obviously, the STMDE benefits from the reasonable adjustment for parameters and the enhanced selection operation, which does not treat fitness as the only metric when the stagnation lasts for a long period.

4.2.4. Performance Rankings of STMDE and Compared DE Variants

We deploy the Friedman test [25] to verify the performance of the STMDE in this subsection, which is shown in Figure 1. It can be observed that the proposed STMDE performs best among all compared algorithms with the smallest performance ranking, namely 2.21. To show the statistical significance more clearly, we also present the critical difference diagram [26]. The critical difference diagram as shown in Figure 2 discloses that the STMDE is the most powerful optimization algorithm among all compared DE variants.

4.2.5. Experimental Time Complexity

The experimental time complexity is disclosed in this subsection. Firstly, the running time for the standard program T₀ is recorded. Further, the evaluation time of F18 for 200,000 times is denoted as T₁. Thirdly, the optimization time for F18 via the STMDE and its compared DE variants with the FES of 20,000 is denoted as T₂. To make the time complexity convinced, we implement five independent runs to obtain the average optimization time ${\overline{T}}_2$. Finally, the experimental time complexity can be calculated by $\left({\overline{T}}_2-T_1\right)/T_0$. Herein, the 30-D optimization problems are adopted to investigate the experimental time complexity.

For completeness, the standard program for obtaining T₀ is given as follows.

For i = 1:1,000,000

 x = 0.55;

 x = x + x; x = x/2; x = x ∗ x; x = sqrt(x);

 x = log(x); x = exp(x); x = x/(x + 2);

End For

T₀ and T₁ are 0.01 s and 0.87 s, respectively. From Figure 3, we can observe that ACos-JADE is the most time-consuming while CUSDE requires the least running time. The time complexity of the proposed STMDE is 31, which is second to IMPEDE with 85. The optimization time of Db-SHADE and PFI-SHADE are 20 and 27, respectively.

4.2.6. Effectiveness of Mechanisms

We develop three variants to investigate the effectiveness of the three novel mechanisms, which are illustrated as follows.

(1)

Variant 1: deactivate the CPSR scheme and other processes are the same as the STMDE;

(2)

Variant 2: deactivate the GPSR scheme and other processes are the same as the STMDE;

(3)

Variant 3: deactivate the GBS scheme and other processes are the same as the STMDE.

The overall experimental results of comparing the STMDE with three variants in the 30-D CEC2017 test suite are given in

Table 5. Performance comparison of STMDE with three variants on 30-D CEC2017.

Variant	Unimodal Functions	Simple Multi-Modal Functions	Hybrid Functions	Composition Functions	Overall
Variant 1	1/1	3/0	7/0	4/0	15/1
Variant 2	1/1	6/0	4/0	2/0	13/1
Variant 3	1/1	5/0	3/0	3/0	12/1

. In comparison with Variant1, Variant2 and Variant3, the performance metric “W/L” of the STMDE are “15/1”, “13/1” and “12/1”, respectively, which illustrates the effectiveness of the CPSR, GPSR and GBS schemes. From the perspective of function property, the CPSR, GPSR and GBS schemes are beneficial for accelerating the optimization of simple multi-modal functions, hybrid functions and composition functions. To be specific, the CPSR scheme contributes more greatly to hybrid functions with the metric “W/L” of “7/0”. Further, the GPSR and GBS schemes are more powerful in handling simple multi-modal functions with six and five cases winning respectively out of seven test functions and none of the cases losing.

The performance of three proposed schemes in unimodal functions is relatively insignificant due to the fact that such kinds of optimization problems are tractable and stagnation hardly happens.

4.2.7. Convergence Graphics

The convergence graphics of the STMDE and five SOTA DEs in 30-D test functions are given in Figure 4, which includes simple multi-modal function (F5), hybrid functions (F15 and F17) and composition function (F29). On the whole, the STMDE can obtain smaller error values than compared DE variants. On the 30-D F5, the STMDE shows advantage over others when the FES is larger than 1 × 10⁵ while ACos-JADE, Db-SHADE and PFI-SHADE can achieve better performance in the early evolutionary stage. On the 30-D F15, the STMDE achieves similar performance to Db-SHADE while PFI-SHADE, CUSDE and PaDE outperform the STMDE in the preliminary stage. Along with evolution, the STMDE converges more effectively. On the 30-D F17, the STMDE ranks fourthly when the FES is less than 1.5 × 10⁵, while it achieves a better converged population than any DE variant when the FES increases to 2.5 × 10⁵. On the 30-D F29, the STMDE shows competitiveness when the FES is more abundant. To be specific, the STMDE achieves the smallest error value when the FES is larger than 1.5 × 10⁵.

In general, the STMDE is more competitive in the later evolutionary stage. The reason can be summarized as follows. On the one hand, the adaptation of control parameters can be more appropriate with the successful memory considered. On the other hand, stagnation may occur frequently in the later evolutionary stage. The STMDE can adjust both control parameters and the greediness parameter to meet the optimization requirement disclosed by the stagnation ratio. In addition, the GBS scheme of the STMDE can cease the stagnation effectively. To this end, the STMDE can converge fast during the later evolution period.

4.2.8. Box Plots of Error

To show the performance comparisons of the proposed STMDE and its compared DE variants more straightforwardly, the box plots of error on four simple multi-modal functions (30-D F5, 30-D F7, 30-D F8 and 30-D F10), three hybrid functions (30-D F14, 30-D F17 and 30-D F19) and three composition functions (30-D F21, 30-D F23 and 30-D F29) are shown in Figure 5. It can be observed that the proposed STMDE converges better on F5, F8, F14, F21 and F29. In addition, the mean error and median error achieved by the STMDE is smaller than that of other DE variants. The reason why the STMDE achieves better performance is that it adopts effective parameter adaptation schemes based on the detection of stagnation. When most of the individuals tend to be stagnant, exploration is preferred. To this end, larger control parameters and greediness parameter are advocated. In addition, when the individual is trapped into stagnation for a long time, new offspring is generated to replace the stagnant one, which can terminate the stagnation and force evolution, but without the extra consumption of fitness evaluation.

4.2.9. The Analysis of Parameter Sensitivity

In this subsection, the parameter sensitivity is analyzed, including DCcr (as shown in Figure 6), DCf (as shown in Figure 7), T (as shown in Figure 8) and gp (as shown in Figure 9). 30-D benchmark functions from the CEC2017 test suite are utilized to investigate the parameter sensitivity. In addition, the Friedman test is implemented to figure out the performance rankings for different settings of parameters.

(1)

To investigate the parameter sensitivity of DCcr, the GBS scheme is deactivated. From Figure 6, it can be observed that a similar performance is achieved when the DCcr is 0.55/0.45, 0.60/0.40 and 0.65/0.35 (Note: the DCcr is 0.55/0.45 when the STR is larger than 0.5. And the setting is 0.45/0.55 when the STR is smaller than 0.5). Therefore, it is feasible to select one of the settings from 0.55/0.45, 0.60/0.40 and 0.65/0.35. It is worth mentioning that 0.55/0.45 is adopted in the STMDE.

(2)

Similarly, the parameter sensitivity of DCf is investigated via deactivating the GBS scheme. From Figure 7, similar performances can be observed when the DCf is 0.60/0.40, 0.65/0.35 and 0.70/0.30 (Note: the DCf is 0.60/0.40 when the STR is larger than 0.5. And the setting is 0.40/0.60 when the STR is smaller than 0.5). Therefore, it is reasonable to select one of the settings from 0.60/0.40, 0.65/0.35 and 0.70/0.30. It is worth mentioning that 0.60/0.40 is adopted in the STMDE.

(3)

To investigate the parameter sensitivity of threshold T, the PASR is deactivated. From Figure 8, it can be observed that the best performance is obtained when the threshold T is 128.

(4)

Apart from the threshold T, the parameter gp of the GBS scheme is also discussed in this subsection. It can be observed from Figure 9 that the performance rankings are similar when the values of gp are 0.5 and 0.7. In this paper, we set the gp as 0.7 when implementing the STMDE.

4.3. Performance Comparison on the Real-World Optimization Problem

The performance of the STMDE and its compared DE variants on handling the real-world optimization problem from the CEC2011 test suite [22] is investigated. The selected practical problem is spread spectrum radar polly phase code design (Problem No. 7 of the CEC2011 test suite), which includes numerous local optima and is piecewise smooth. The simulation results are presented as follows.

From

Table 6. Performance comparison in the practical optimization problem from the CEC2011.

	FES	5.00 × 104				1.00 × 105				1.50 × 105
DEs		mean	std	sig		mean	std	sig		mean	std	sig
SHADE		1.36 × 100	1.02 × 10−1	=		1.24 × 100	1.05 × 10−1	+		1.24 × 100	8.13 × 10−2	+
ACos-JADE		1.37 × 100	1.07 × 10−1	=		1.19 × 100	9.03 × 10−2	+		1.14 × 100	9.27 × 10−2	+
IMPEDE		1.37 × 100	5.39 × 10−2	=		1.19 × 100	1.00 × 10−1	+		1.13 × 100	9.57 × 10−2	+
Db-SHADE		1.36 × 100	1.03 × 10−1	=		1.25 × 100	1.12 × 10−1	+		1.22 × 100	1.11 × 10−1	+
PFI-SHADE		1.40 × 100	9.90 × 10−2	=		1.25 × 100	1.03 × 10−1	+		1.22 × 100	9.48 × 10−2	+
CUSDE		1.84 × 100	9.68 × 10−2	+		1.77 × 100	1.15 × 10−1	+		1.71 × 100	1.08 × 10−1	+
STMDE		1.37 × 100	1.12 × 10−1			1.00 × 100	2.93 × 10−1			8.95 × 101	2.47 × 10−1

, it can be observed that the STMDE can achieve a better performance than all the other compared DE variants when the maximal number of the fitness evaluation is 1 × 10⁵ or 1.5 × 10⁵. In the preliminary evolution period, stagnation hardly happens, while both the mutation and selection of the STMDE focuses on eliminating the stagnation and forcing evolution. Therefore, the advantage of the STMDE may not be significant in the early stage. However, the population may tend to be stagnant or be trapped into premature convergence along with the increase of generation. Therefore, the STMDE outperforms other competitive DE variants.

5. Conclusions

Aiming to cope with the stagnation frequently happening in the DE, we propose the CPSR scheme first, which includes the adjustment for the scale factor F and the crossover rate CR. Further, we propose the GPSR scheme to tune the greediness parameter p. The reasonable parameters adaptation benefits from the acquisition of the evolutionary state, which is revealed by the stagnation ratio STR. Apart from the effective mutation process, sensible selection can also help it to avoid stagnation. However, some parents may not be capable of producing promising offspring anymore. To this end, reproducing offspring for such kinds of parent can allow the skipping of greedy selection, in that the newly generated offspring can enter into the next generation without fitness evaluation. With the CPSR, GPSR and GBS schemes, the STMDE can achieve a more competitive performance, which has been verified via the substantial experiments implemented in the CEC2017. The effectiveness of the STMDE in handling a real-world optimization problem has also been testified by comparing it to competitive DE variants on the spread spectrum radar polly phase code design.