Chaotic dynamic weight grey wolf optimizer for numerical function optimization

Abstract

The grey wolf optimizer (GWO) algorithm is a recently proposed optimization technique based on the social leadership and hunting behavior of grey wolves in nature. Due to its small number of control parameters, ease of implementation and high level of exploration and exploitation, the GWO has attracted the interest of researchers from different fields. However, the GWO has problems with its position-updated equation, which is good for exploitation but not conducive to exploration because it can prematurely convergence to local optima. To overcome this drawback, a chaotic dynamic weight grey wolf optimizer (CDGWO) is proposed. In the CDGWO algorithm, a new position-updated equation is presented by applying a chaotic map and dynamic weight to guide the search process for potential candidate solutions. In addition, a nonlinear control parameter strategy is designed to balance the exploration and exploitation and accelerate the convergence speed of the GWO algorithm. The search accuracy and performance of the modified position-updated equation and the nonlinear control parameter strategy are verified using 19 well-known classical benchmark functions. The experimental results show that, for almost all benchmark functions, the CDGWO algorithm gives competitive results in terms of convergence, solution quality and local optimal avoidance compared with other nature-inspired optimizations and GWO variants.

Keywords

Grey wolf optimizer chaos theory nonlinear control parameters global optimization

1 Introduction

Large-scale and complex optimization problems are common in engineering design, information science, logistics scheduling, artificial intelligence and economic management. In the past, solving these real-world optimization problems were the work of mathematicians and engineers. They developed many traditional mathematical methods for solving optimization problems by enumerating all the possible solutions [1]. Recently, however, huge time complexity of the traditional methods is encountered frequently when coping with these optimization problems. These new realities necessitate a simple, powerful and accurate optimization technique. Fortunately, algorithms to rapidly approximate (heuristic) and solve these problems have been proposed and are known as swarm intelligence (SI). SI algorithms are inspired by the collective survival, hunting, navigation or foraging behavior of individuals in nature, and they include particle swarm optimization (PSO) [2], ant colony optimization (ACO) [3], artificial bee colony (ABC) [4], the whale optimization algorithm (WOA) [5], the grey wolf optimizer (GWO) [6], the cuckoo search algorithm (CSA) [7, 8], the flower pollination algorithm (FPA) [9], harmony search (HS) [10], and so on. These algorithms have shown considerable success in solving nonconvex, nondifferentiable optimization problems and have attracted increasing attention in recent years [11 –13]. The most exciting examples are function optimization [14, 15], feature selection [16, 17], economic load dispatch [18, 19], structural optimization [20], logic circuit design [21] and artificial neural networks [22, 23].

Randomness in SI algorithms plays a huge role because it affects the exploration and exploitation in the search process [1, 24]. In SI algorithms, exploration refers to moves for discovering entirely new regions of a search space, whereas exploitation refers to moves that focus on searching the vicinity of the known solutions to find better solutions during the search process [1]. Based on these, one encouraging research trend is to study the stochastics of SI algorithms in a global search process. In recent years, with the rapid development of nonlinear dynamics, the applications of chaotic theory have drawn attention of researchers from various areas [25]. Empirical studies have demonstrated that the randomness of the chaotic sequences is competitive compared to the traditional random methods. The advantages of these sequences including high-level mixing, higher diversity and mobility [15] and are well suited to enhancing the abilities of exploration and exploitation for SI algorithms. Therefore, the introduction of the chaos concept in swarm intelligence based algorithms is very promising, such as the butterfly optimization algorithm (BOA) [26], particle swarm optimization (PSO) [15, 27], biogeography-based optimization (BBO) [28], the gravitational search algorithm (GSA) [29], and harmony search (HS) [30], as well as the krill herd (KH) [31], firefly (FA) [32] and Cuckoo search algorithms (CSA) [33].

The focus of this paper is to introduce a novel, modified version of the grey wolf optimizer (GWO), a metaheuristic and global optimization algorithm originally proposed by Mirjalili et al. that mimics the leadership hierarchy and hunting mechanism of grey wolves in nature. Comparative numerical studies has verified that the GWO provides very competitive optimization results compared to other well-known meta-heuristics such as PSO, GSA, DE, EP, and ES [6 , 35]. Owning to its simple and easy implementation, the GWO has been used to solve many real world optimization problems such as high-dimensional numerical optimization [13, 36], large scale unit commitment problems [37], feature subset selection [38], combined heat and power dispatch problems [39], distributed compressed sensing [40], machinery fault diagnosis [41], two-stage assembly flow shop scheduling problems [42], multilevel thresholding [43], train multilayer perceptrons [23], optimal power flow [44], economic load dispatch problems [45], optimal reactive power dispatch problems [46], power system stabilizer design [47], and evolving kernel extreme learning machines [48], among others. However, the insufficiency that exists in the GWO is its position-updated equation that is only related to a few particles (the previous global best individual (wolf), the second-best individual (wolf), and the third-best individual (wolf)), which is good at exploitation but limits its exploration capability and causes premature convergence to local optima. This implies that, unbalance between exploration and exploitation occurs in GWO, and though it performs well in global search, it is not well-suited for performing a local search. To overcome this shortcoming, we introduce a chaos sequence into the original position-updated equation to create a new position-updated equation and then hybrid the new and the original to form a modified position-updated equation, which brings more information and produces a promising candidate individual to enhance the GWO algorithm exploration. We replaced the old (original position-updated) equation in the GWO algorithm with the new (modified position-updated) equation and developed a new algorithm named the chaotic dynamic weight GWO (CDGWO). In addition, to balance exploration and exploitation, a nonlinear control parameter strategy is presented. Thus, exploration capability at the initial stages and exploitation capability at the later stages were enhanced using the nonlinear control parameter. The modified position-updated equation increased the diversity of potential solution individuals such that it ensures the global optimum is obtained. The performance of the CDGWO is compared with the standard GWO and other state-of-the-art algorithms on 19 benchmark functions.

The rest of this paper is organized as follows. Section 2 presents a brief introduction of the standard GWO algorithm. In Section 3, the CDGWO algorithm is proposed based on the modified position-updated equation and the nonlinear control parameter strategy. The proposed CDGWO algorithm was tested using 19 test functions, and the results are discussed in Section 4. Finally, the conclusion and future work are given in Section 5.

2 Grey wolf optimizer algorithm

The grey wolf optimizer (GWO) is a recently proposed population-based optimization technique that mimics the social leadership and hunting behavior of grey wolves in nature. In the GWO algorithm, four wolf subgroups, each with a different fitness, form a population. The fittest wolf in the population is named alpha (α) while the second and third are called beta (β) and delta (δ), respectively. The other wolves in the population are considered to be omega (ω). The duty of the first three fittest wolves (α, β and δ) is to guide the other wolves’ (ω) search for promising areas using an encircling mechanism to hunt for food under the search space. The mathematical model to represent this is as follows [6]: $\vec{X} (t + 1) = {\vec{X}}_{p} (t) - \vec{A} \cdot | \vec{C} \cdot {\vec{X}}_{p} (t) - \vec{X} (t) |$ (1) where t indicates the current iteration, $\vec{X}$ is the position vector of a grey wolf, ${\vec{X}}_{p}$ is the position vector of the prey, $\vec{A} = 2 \vec{a} \cdot {\vec{r}}_{1} - \vec{a}$ and $\vec{C} = 2 \cdot {\vec{r}}_{2}$ are coefficient vectors, ${\vec{r}}_{1}$ , ${\vec{r}}_{2}$ are random vectors in [0, 1], and $\vec{a}$ linearly decreases from 2 to 0 as the number of iterations increases, i.e., $\vec{a} = 2 - \frac{2 t}{MaxIter}$ (2) where t indicates the current iteration and MaxIter is the maximum number of iterations.

The other wolves (ω) update their positions under the guidance of the first three fittest wolves (α, β and δ) as follows [6]: $\vec{X_{1}} = \vec{X_{α}} - \vec{A_{1} \cdot} | \vec{C_{1}} \cdot \vec{X_{β}} - \vec{X} |$ (3) $\vec{X_{2}} = \vec{X_{β}} - \vec{A_{2} \cdot} | \vec{C_{2}} \cdot \vec{X_{β}} - \vec{X} |$ (4) $\vec{X_{2}} = \vec{X_{δ}} - \vec{A_{3} \cdot} | \vec{C_{3}} \cdot \vec{X_{δ}} - \vec{X} |$ (5) $\vec{X} (t + 1) = \frac{1}{3} (\vec{X_{1}} + \vec{X_{2}} + \vec{X_{3}})$ (6)

The pseudo code of the GWO is described asAlgorithm 1 [6, 13].

3 The CDGWO algorithm

3.1 Improved position-updated equation

The high level of exploration and exploitation of the GWO algorithm has been proven in the literature [6, 23]; the excellent optimization performance of the GWO make it widely applicable to many real-world optimization problems, and many different GWO variants have been introduced in the literature [34 –36]. In the standard GWO algorithm, however, the deficiencies are that since all of the other individuals are attracted towards wolves α, β and δ, they are prone to converge prematurely without enough exploration of search space and to easy trapping in the local optimum solution. Therefore, the position updated equation described by Equation (6) is good at exploitation but poor at exploration [13]; that is, an imbalance between exploration and exploitation still persists in the standard GWO algorithm.

Algorithm 1.
Grey wolf optimizer algorithm.

Input: Population size N, the maximum number of iteration MaxIter

Output: Optimal individual position X_α, best fitness values f (X_α).

Initialize N individuals’ position randomly to construct a population.

Calculate the fitness value of each individual and find the α, β and δ.

Whilet > MaxIter or stopping criteria not met do

For each individual do

Update current individuals’ position according to Equation (6)

End

Update α, $\vec{A}$ and $\vec{C}$ .

Evaluate the fitness value of each individual.

Update the α, β and δ.

End

According to [49], the character of nonrepetition and the ergodicity of chaos means it can carry out overall searches with more diverse potential solutions and higher search speed than stochastic searches that depend on probability. Therefore, the chaotic map is introduced into the standard GWO algorithm to increase the diversity of potential solutions by modifying its position-updated equation. The standard position-updated equation, Equation (6), is modified as follows: $\vec{X^{'}} (t + 1) = \frac{1}{3} (\vec{X_{1}} + \vec{X_{2}} + \vec{X_{3}})$ (7)

$\begin{matrix} \vec{X^{″}} (t + 1) = \frac{1}{3} (\vec{R_{n}} \cdot (\vec{R_{n}} \cdot \vec{X_{1}} - \vec{X_{1}}) + \vec{R_{n}} \\ \cdot (\vec{R_{n}} \cdot \vec{X_{2}} - \vec{X_{2}}) + \vec{R_{n}} \cdot (\vec{R_{n}} \cdot \vec{X_{3}} - \vec{X_{3}})) \end{matrix}$ (8) $\vec{X} (t + 1) = w \cdot r_{3} \cdot \vec{X^{'}} + (1 - w) \cdot \vec{X^{″}}$ (9) $w = 1 - {(\frac{t}{MaxIter})}^{1.5}$ (10) where r₃ is a uniform random number in [0,1], and w is the dynamic weight that controls the influence of $\vec{X^{'}}$ and $\vec{X^{″}}$ and adjusts the exploration and exploitation abilities of Equation (9), t indicates the current iteration, MaxIter indicates the maximum number of iterations. $\vec{R_{n}}$ is a chaotic mapping vector and its mathematical expression is as follows: $\vec{R_{n}} = \vec{2 α^{'}} \cdot r_{n} - \vec{α^{'}}$ (11) where r_n is the optimization operator based on Cubic chaotic mapping [50], its mathematical model is as follows: $r_{n} = 4 r_{n - 1}^{3} - 3 r_{n - 1}, - 1 ⩽ r_{n - 1} ⩽ 1, r_{o} = 0.004$ (12)

According to Equation (9), the uniform random number r₃ is used to randomly disturb the position-updated equation of the standard GWO algorithm to increase the diversity of potential solutions. The second term on the right-hand side of Equation (9) is used to search deeply and enhance the exploration of the GWO algorithm through taking advantages of the characteristics of nonrepetition and the ergodicity of chaos. Comparing Equation (6) with Equation (9), the new candidate individual is generated by $\vec{X^{″}}$ , which helps avoid trapping in the local optimum solution at the latter stage of the GWO algorithm. Therefore, the position-updated equation presented by Equation (9) not only is random enough for exploration but also has improved the balance between exploration and exploitation. In general, the exploration ability of Equation (9) to avoid trapping in the local optimum solution is better than that of Equation (6).

3.2 Nonlinear control parameter design

In the search process of population-based optimization algorithms, both exploration and exploitation are necessary. Exploration refers to the ability to investigate the unknown regions in the search space to find the global optima, whereas exploitation refers to the ability to apply the information of the existing individuals to discover better individuals [1, 51]. The level of exploration and exploitation are not only affected by the diversity of potential solutions but also decided by specific operators that control the search speed in various stages.

In a standard GWO algorithm, the specific control operator is $\vec{a}$ , and it linearly decreases from 2 to 0 as the number of iterations increases. According to [13], a larger control parameter $\vec{α}$ facilitates global exploration, whereas a smaller control parameter $\vec{α}$ facilitates local exploitation. Therefore, the control parameter $\vec{α}$ should be designed properly to reflect the actual search situation of the GWO algorithm. As we know, the search process of the GWO algorithm is both nonlinear and dynamic. At the initial stage of searching, the diversity of particles is higher and at this time a small control parameter $\vec{α}$ is needed to accelerate the convergence speeds. In contrast, at the latter stage of searching, only a large control parameter $\vec{α}$ is needed to slow the convergence speed so the search space can be searched fully. However, the control parameter $\vec{α}$ of the GWO algorithm linearly decreases, which is a deviation from reality. Based on the above considerations, we designed a new nonlinear increasing control parameter $\vec{α^{'}}$ to replace the linearly decreasing control parameter $\vec{α}$ presented by Equation (2) as follows: $\vec{α^{'}} = 2 \cdot (1 - {(\frac{t}{MaxIter})}^{μ})$ (13) where t indicates the current iteration, MaxIter indicates the maximum number of iterations, and μ is the nonlinear modulation index.

3.3 Implementation of the CDGWO algorithm

We introduced the proposed position-updated equation and the nonlinear control parameter strategy into the standard GWO algorithm to propose the CDGWO algorithm, but it should be emphasized that this method can be implemented in other GWO versions easily without changing their core framework. The basic steps of the CDGWO algorithm are as follows:

Initialize the parameters of CDGWO such as population size, the nonlinear modulation index μ and the maximum number of iterations, then initialize a population of wolves randomly based on the upper and lower bounds in the search space.

Calculate the fitness function value for each wolf.

Choose the best global wolf, second and third best wolves, and save them as α, β and δ, respectively.

Update the position of other wolves (ω) using Equation (9).

The nonlinear increasing control parameter $\vec{α^{'}}$ is updated using Equation (13) for better searching at different search stages of the CDGWO algorithm, and the chaotic mapping vector $\vec{R_{n}}$ is updated using Equation (11)

Update parameters $\vec{A}$ and $\vec{C}$ .

Go to step b) if t > MaxIter or the end criterion is not satisfied.

Return the position and fitness function value of the global-best wolf α.

Based on the above description, the CDGWO algorithm flowchart is shown in Fig. 1.

Fig.1

The flowchart of the CDGWO algorithm.

4 Numerical experiments and discussion

4.1 Benchmark functions and simulation settings

In this section, the proposed CDGWO algorithm is used to perform simulation experiments on 19 classical benchmark test functions collected from several references [6 , 15]. The selected benchmark test functions are listed in Table 1, where f_min denotes the global optima value. Furthermore, three real-world engineering applications were also applied to simulation experiments to validate the effectiveness of our algorithm. For the sake of clarity, the 19 benchmark test functions are divided into two groups. The first group includes 9 unimodal functions (f₁–f₉), which are suitable for benchmarking the exploitation of algorithms since they have one global optimum and no local optima [13]. The second group contains 10 complex multimodal functions (f₁₀–f₁₉) that are helpful for revealing the capability of exploration and local optima avoidance for algorithms with a large number of local optima, making it difficult to find the global optimization values [52].

Table 1
The19 classical benchmark functions used in the experiments of this paper

Function type Function name Function formula Search range f _min

Unimodal functions Sphere $f_{1} (x) = \sum_{i = 1}^{n} x_{i}^{2}$ [–100, 100] 0

Sum-Square $f_{2} (x) = \sum_{i = 1}^{n} {ix}_{i}^{2}$ [–10, 10] 0

Schwefel’s 2.22 $f_{3} (x) = \sum_{i = 1}^{n} | x_{i} | + \prod_{i = 1}^{n} | x_{i} |$ [–10, 10] 0

Rotated hyper-ellipsoid $f_{4} (x) = \sum_{i = 1}^{n} (\sum_{j = 1}^{i} x_{j})^{2}$ [–100, 100] 0

Schwefel’s 2.21 $f_{4} (x) = max_{i} {| x_{i} |, 1 ⩽ x_{i} ⩽ n}$ [–100, 100] 0

Rosenbrock $f_{6} (x) = \sum_{i = 1}^{n - 1} [100 (x_{i + 1} - x_{i}^{2})^{2} + (x_{i} - 1)^{2}]$ [–30, 30] 0

Quartic $f_{7} (x) = \sum_{i = 1}^{n} {ix}_{i}^{4}$ [–1.28, 1.28] 0

Noise $f_{8} (x) = \sum_{i = 1}^{n} {ix}_{i}^{4} + random [0, 1)$ [–1.28, 1.28] 0

Sum-Power $f_{9} (x) = \sum_{i = 1}^{n} {| x_{i} |}^{(i + 1)}$ [–1, 1] 0

Multimodal functions Rastrigin $f_{10} (x) = \sum_{i = 1}^{n} [x_{i}^{2} 10 cos (2 π_{i}) + 10]$ [–5.12, 5.12] 0

Ackley $f_{11} (x) = - 20 exp (- 0.2 \sqrt{\frac{1}{n}} \sum_{i = 1}^{n} x_{i}^{2})$ [–32, 32] 0

$- exp (\frac{1}{n} \sum_{i = 1}^{n} cos (2 π x_{i})) + 20 + e$

Griewank $f_{12} (x) = \frac{1}{4000} \sum_{i = 1}^{n} x_{i}^{2} - \prod_{i = 1}^{n} cos (\frac{x_{i}}{\sqrt{i}}) + 1$ [–600, 600] 0

Alpine $f_{13} (x) = \sum_{i = 1}^{n} | x_{i} \cdot sin (x_{i}) + 0.1 \cdot x_{i} |$ [–10, 10] 0

Inverted Cosine Mixture $f_{14} (x) = 0.1 n - (0.1 \sum_{i = 1}^{n} cos (5 π x_{i}) - \sum_{i = 1}^{n} x_{i}^{2})$ [–1, 1] 0

Zakharov $f_{15} (x) = \sum_{i = 1}^{n} x_{i}^{2} + {(\sum_{i = 1}^{n} 0.5 {ix}_{i})}^{2} + {(\sum_{i = 1}^{n} 0.5 {ix}_{i})}^{4}$ [–5, 10] 0

Pathological $f_{16} (x) = \sum_{i = 2}^{n} 0.5 + \frac{{sin}^{2} (\sqrt{100 x_{i - 1}^{2} + x_{i}^{2}}) - 0.5}{1 + 0.001 (x_{i - 1}^{2} - 2 x_{i - 1} x_{i} + x_{i}^{2})^{2}}$ [–100, 100] 0

Elliptic $f_{17} (x) = \sum_{i = 1}^{n} (10^{6})^{(i - 1) / (n - 1)} x_{i}^{2}$ [–5, 5] 0

Salomon $f_{18} (x) = 1 - cos (2 π \sqrt{\sum_{i = 1}^{n} x_{i}^{2}}) + 0.1 \sqrt{\sum_{i = 1}^{n} x_{i}^{2}}$ [–100, 100] 0

Schaffer $f_{19} (x) = 0.5 + \frac{{sin}^{2} (\sqrt{\sum_{i = 1}^{n} x_{i}^{2}}) - 0.5}{(1 + 0.001 (\sqrt{\sum_{i = 1}^{n} x_{i}^{2}}))^{2}}$ [–100, 100] 0

Function type	Function name	Function formula	Search range	f _min
Unimodal functions	Sphere	$f_{1} (x) = \sum_{i = 1}^{n} x_{i}^{2}$	[–100, 100]	0
	Sum-Square	$f_{2} (x) = \sum_{i = 1}^{n} {ix}_{i}^{2}$	[–10, 10]	0
	Schwefel’s 2.22	$f_{3} (x) = \sum_{i = 1}^{n} \| x_{i} \| + \prod_{i = 1}^{n} \| x_{i} \|$	[–10, 10]	0
	Rotated hyper-ellipsoid	$f_{4} (x) = \sum_{i = 1}^{n} (\sum_{j = 1}^{i} x_{j})^{2}$	[–100, 100]	0
	Schwefel’s 2.21	$f_{4} (x) = max_{i} {\| x_{i} \|, 1 ⩽ x_{i} ⩽ n}$	[–100, 100]	0
	Rosenbrock	$f_{6} (x) = \sum_{i = 1}^{n - 1} [100 (x_{i + 1} - x_{i}^{2})^{2} + (x_{i} - 1)^{2}]$	[–30, 30]	0
	Quartic	$f_{7} (x) = \sum_{i = 1}^{n} {ix}_{i}^{4}$	[–1.28, 1.28]	0
	Noise	$f_{8} (x) = \sum_{i = 1}^{n} {ix}_{i}^{4} + random [0, 1)$	[–1.28, 1.28]	0
	Sum-Power	$f_{9} (x) = \sum_{i = 1}^{n} {\| x_{i} \|}^{(i + 1)}$	[–1, 1]	0
Multimodal functions	Rastrigin	$f_{10} (x) = \sum_{i = 1}^{n} [x_{i}^{2} 10 cos (2 π_{i}) + 10]$	[–5.12, 5.12]	0
	Ackley	$f_{11} (x) = - 20 exp (- 0.2 \sqrt{\frac{1}{n}} \sum_{i = 1}^{n} x_{i}^{2})$	[–32, 32]	0
		$- exp (\frac{1}{n} \sum_{i = 1}^{n} cos (2 π x_{i})) + 20 + e$
	Griewank	$f_{12} (x) = \frac{1}{4000} \sum_{i = 1}^{n} x_{i}^{2} - \prod_{i = 1}^{n} cos (\frac{x_{i}}{\sqrt{i}}) + 1$	[–600, 600]	0
	Alpine	$f_{13} (x) = \sum_{i = 1}^{n} \| x_{i} \cdot sin (x_{i}) + 0.1 \cdot x_{i} \|$	[–10, 10]	0
	Inverted Cosine Mixture	$f_{14} (x) = 0.1 n - (0.1 \sum_{i = 1}^{n} cos (5 π x_{i}) - \sum_{i = 1}^{n} x_{i}^{2})$	[–1, 1]	0
	Zakharov	$f_{15} (x) = \sum_{i = 1}^{n} x_{i}^{2} + {(\sum_{i = 1}^{n} 0.5 {ix}_{i})}^{2} + {(\sum_{i = 1}^{n} 0.5 {ix}_{i})}^{4}$	[–5, 10]	0
	Pathological	$f_{16} (x) = \sum_{i = 2}^{n} 0.5 + \frac{{sin}^{2} (\sqrt{100 x_{i - 1}^{2} + x_{i}^{2}}) - 0.5}{1 + 0.001 (x_{i - 1}^{2} - 2 x_{i - 1} x_{i} + x_{i}^{2})^{2}}$	[–100, 100]	0
	Elliptic	$f_{17} (x) = \sum_{i = 1}^{n} (10^{6})^{(i - 1) / (n - 1)} x_{i}^{2}$	[–5, 5]	0
	Salomon	$f_{18} (x) = 1 - cos (2 π \sqrt{\sum_{i = 1}^{n} x_{i}^{2}}) + 0.1 \sqrt{\sum_{i = 1}^{n} x_{i}^{2}}$	[–100, 100]	0
	Schaffer	$f_{19} (x) = 0.5 + \frac{{sin}^{2} (\sqrt{\sum_{i = 1}^{n} x_{i}^{2}}) - 0.5}{(1 + 0.001 (\sqrt{\sum_{i = 1}^{n} x_{i}^{2}}))^{2}}$	[–100, 100]	0

To obtain a fair comparison of the GWO and CDGWO algorithm performance, the population size (N) was set as 30 for all functions and the maximum number of iterations (MaxIter) was 500 (i.e., the maximum number of fitness function evaluations (FFEs) is 15,000) on all of the simulations [13]. In addition, the other parameters of the CDGWO algorithm were set as follows: r_o = 0.004 and the nonlinear modulation index μ = 0.15. All the experiments were executed on the same machine with a Windows 10 Professional environment using Intel(R), Coretrademark 64×2 Dual Core system with 2.5 GHz and 4.00 GB RAM, and the codes are implemented using MATLAB R2015a.

4.2 Comparison of CDGWO with the standard GWO algorithm

In this section, we compare the performance of the proposed CDGWO algorithm with the standard GWO algorithm. Each benchmark test problem shown in Table 1 was run independently for 30 runs. The dimensions of each test problem were set to 30 and 60. The four criteria (best, mean, worst, and standard deviation) were taken from the literature [13] and were used for comparing the performance of the algorithms. The experimental results are shown in Table 2 and Fig. 2. In Table 2, the best value of each criteria obtained by the CDGWO and the standard GWO algorithms are shown in bold, “Dim” is the dimensions of the function, “Best” denotes the best value, “Mean” represents the average best values, “Worst” refers to the worst value, and “St. dev” is the standard deviation value.

Fig.2

Convergence plot of the GWO and CDGWO with D = 30 and D = 60 on fourteen typical functions.

Table 2

The statistical results obtained by the GWO and CDGWO algorithms for 19 test functions with 15,000 FFEs

Function	Dim	GWO				CDGWO
		Best	Mean	Worst	St.dev	Best	Mean	Worst	St.dev
f ₁	30	1.49E–28	1.69E–27	7.86E–28	2.25E–28	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	1.44E–28	1.90E–27	7.93E–28	2.85E–28	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₂	30	2.66E–29	1.69E–28	1.02E–27	3.07E–28	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	3.22E–18	2.45E–18	4.47E–18	1.38E–18	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₃	30	1.54E–17	1.01E–16	3.52E–16	1.02E–16	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	1.66E–11	4.05E–11	8.50E–11	2.02E–11	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₄	30	3.25E–28	1.62E–26	7.47E–26	2.66E–26	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	6.77E–17	2.91E–16	5.06E–16	1.51E–16	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₅	30	7.89E–08	4.74E–07	1.54E–06	4.31E–07	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	8.62E–04	4.10E–03	1.60E–02	5.10E–03	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₆	30	5.61E+01	5.72E+01	5.84E+01	8.94E–01	2.88E+01	2.89E+01	2.98E+01	5.40E–02
	60	2.61E+01	2.70E+01	2.80E+01	7.88E–01	5.88E+01	5.89E+01	5.90E+01	4.52E–02
f ₇	30	2.49E–54	6.93E–50	6.73E–49	2.12E–49	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	6.44E–36	9.08E–34	2.86E–33	1.14E–33	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₈	30	6.52E–04	1.90E–03	3.90E–03	9.89E–04	2.61E–07	6.26E–05	3.45E–04	7.59E–05
	60	1.20E–03	4.70E–03	9.30E–03	2.60E–03	7.58E–05	8.22E–05	3.72E–04	7.68E–05
f ₉	30	6.92E–112	3.37E–98	2.09E–97	6.82E–98	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	3.05E–96	8.14E–86	8.01E–85	2.53E–85	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₁₀	30	5.68E–14	3.27E+00	7.07E+00	3.18E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	2.84E–12	4.42E+00	1.46E+01	5.73E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₁₁	30	6.84E–14	1.01E–13	1.15E–13	1.30E–14	8.88E–16	8.88E–16	8.88E–16	0.00E+00
	60	2.33E–10	4.66E–10	8.02E–10	2.07E–10	8.88E–16	8.88E–16	8.88E–16	0.00E+00
f ₁₂	30	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	0.00E+00	3.37E–16	6.66E–16	2.96E–16	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₁₃	30	4.20E–17	4.91E–04	1.50E–03	5.37E–04	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	2.22E–10	1.60E–03	2.60E–03	8.73E–04	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₁₄	30	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	0.00E+00	1.69E–15	3.55E–15	1.35E–15	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₁₅	30	1.04E–09	1.13E–07	5.23E–07	1.87E–07	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	1.98E–01	3.07E–00	1.57E+01	4.98E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₁₆	30	9.39E+00	1.09E+01	1.17E+01	6.32E–01	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	2.48E+01	2.57E+01	2.65E+01	5.25E–01	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₁₇	30	1.30E–25	1.34E–24	4.00E–24	1.27E–24	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	2.31E–15	1.77E–14	4.51E–14	1.47E–14	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₁₈	30	9.99E–02	1.99E–01	2.99E–01	4.71E–02	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	2.00E–01	2.50E–01	3.00E–01	5.27E–02	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f ₁₉	30	9.70E–03	3.17E–02	3.72E–02	1.16E–02	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	60	3.72E–02	7.41E–02	7.82E–02	1.30E–02	0.00E+00	0.00E+00	0.00E+00	0.00E+00

As seen from Table 2, for 9 unimodal functions, the CDGWO achieved better results than the GWO on all unimodal functions except for function f₆ and obtained theoretical optima (0) on 7 test functions (i.e., f₁, f₂, f₃, f₄, f₅, f₇ and f₉) with 15,000 FFEs, whereas the GWO provided better “Best”, “Mean” and “Worst” than CDGWO on function f₆ under Dim = 60. For 10 multimodal functions, the CDGWO algorithm obtained theoretical optima (0) on 9 multimodal test functions (i.e., f₁₀, f₁₂, f₁₃, f₁₄, f₁₅, f₁₆, f₁₇, f₁₈ and f₁₉), both the CDGWO and GWO achieved the theoretical optimal value with 2 functions (f₁₂ and f₁₄) under Dim = 30. Using the f₈ and f₁₁ functions, the CDGWO provided marginally better results than the GWO. Moreover, of 19 test functions, the CDGWO algorithm far outperformed the standard GWO algorithm for 16 test functions (i.e., f₁, f₂, f₃, f₄, f₅, f₇, f₉, f₁₂, f₁₃, f₁₄, f₁₅, f₁₆, f₁₇, f₁₈ and f₁₉).

The convergence curves of the GWO and CDGWO with D = 30 and D = 60 on fourteen typical problems (i.e., f₁, f₂, f₃, f₄, f₅, f₇, f₈, f₉, f₁₀, f₁₃, f₁₅, f₁₇, f₁₈ and f₁₉) are shown in Fig. 2. It should be noted that D appeared in Figs. 2 and 3 represents the dimensions of the problem. The CDGWO algorithm provided satisfactory performance on convergence speed and accuracy on all benchmark functions.

To demonstrate the flexible performance of the proposed method when dealing with high-dimensional problems (i.e., D = 200, 500 and 1000), the CDGWO and GWO were simulated on 19 high dimensional test functions from Table 1. A metric success rate (SR) (Equation (14)) is used for evaluating the performance of the algorithm [53]. $SR = \frac{z^{'}}{z} \times 100 {%$ (14) where z is the total number of experimental trials, and z′ is the number of experimental trials when an algorithm successfully converges to the global optimal value before reaching the maximum number of function calls. In this simulation, the global optimal value is considered to be the theoretical optima (0). The success rate (SR), mean (Mean) and standard deviation (St. dev) obtained by the CDGWO and GWO are recorded in Table 3 over 30 independent trials. In Table 3, “Dim” represents the dimensions of functions, “Fun” represents the function, “SR” denotes success rate, “St. dev” is the standard deviation value and “Mean” is the average best value.

As seen in Table 3, for 15 high-dimensional (Dim = 200, 500 and 1000) test functions (i.e., f₁–f₅, f₇, f₉, f₁₀, f₁₂–f₁₅ and f₁₇–f₁₉), the CDGWO algorithm directly finds the theoretical optimal value (0) and their “SR” are 100%. On the f₆ function, the CDGWO and GWO obtained similar “Mean” results under D = 200 and 500, and the three “St. dev” results obtained by CDGWO under Dim = 200, 500 and 1000 are marginally better than the GWO. The CDGWO method could not obtain the theoretical optimal value for the f₆, f₈ and f₁₁ functions, but it surpasses the GWO for these three functions. On the multimodal functions f₁₆, the CDGWO method obtained “SR” results are 100, 70 and 50 corresponding to Dim = 200, 500 and 1000. For 19 high-dimensional (Dim = 200, 500 and 1000) test functions, the GWO does not obtain any theoretical optimal value. Compared with the GWO algorithm, the CDGWO shows very good scalability to the search dimension, i.e., the optimization performance did not weaken seriously as the dimension increased. In addition, the convergence curves of two classical unimodal functions (f₁ and f₉) and two classical multimodal functions (f₁₀ and f₁₉) obtained by CDGWO are plotted in Fig. 3.

Fig.3

Convergence plot of the CDGWO with D = 200, 500, and 1000 on four typical functions.

Table 3

Results obtained by the GWO and CDGWO over 30 independent runs on dimensions 200, 500, and 1000 for 19 test functions with 15,000 FFEs

Fun	Algorithm	Dim=200			Dim=500			Dim=1000
		Mean	St. dev	SR(%)	Mean	St. dev	SR(%)	Mean	St. dev	SR(%)
f ₁	GWO	1.06E–07	5.71E–08	0	1.06E–03	6.75E–04	0	2.68E–01	9.32E–02	0
	CDGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₂	GWO	7.98E–08	3.90E–08	0	3.10E–03	8.74E–04	0	1.09E+00	2.25E–01	0
	CDGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₃	GWO	3.21E–05	7.93E–06	0	1.11E–02	1.30E–03	0	8.10E–01	6.38E–02	0
	CDGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₄	GWO	8.60E–06	4.51E–06	0	3.05E–01	7.66E–02	0	1.12E+02	2.24E+01	0
	CDGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₅	GWO	2.43E+01	7.87E+00	0	6.57E+01	5.50E+00	0	7.82E+02	3.61E+00	0
	CDGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₆	GWO	1.98E+02	4.10E–01	0	4.98E+02	3.33E–01	0	1.05E+03	2.94E+01	0
	CDGWO	1.98E+02	1.05E–01	0	4.98E+02	7.85E–02	0	9.98E+02	3.17E–01	0
f ₇	GWO	3.53E–17	4.32E–17	0	1.06E–09	1.78E–09	0	7.25E–06	6.92E–06	0
	CDGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₈	GWO	1.54E–02	6.10E–03	0	4.90E–02	1.48E–02	0	1.53E–01	3.77E–02	0
	CDGWO	1.02E–04	8.83E–05	0	5.91E–05	4.42E–05	0	7.14E–05	7.34E–05	0
f ₉	GWO	2.74E–20	1.50E–19	0	1.24E–04	3.74E–04	0	2.30E–03	4.10E–03	0
	CDGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₁₀	GWO	2.29E+01	1.17E+01	0	7.23E+01	2.16E+01	0	2.01E+02	5.07E+01	0
	CGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₁₁	GWO	2.41E–05	6.33E–06	0	1.90E–03	2.33E–04	0	1.84E–02	3.20E–03	0
	CGWO	8.88E–16	0.00E+00	0	8.88E–16	0.00E+00	0	8.88E–16	0.00E+00	0
f ₁₂	GWO	8.60E–03	1.78E–02	0	1.24E–02	3.17E–02	0	7.00E–02	9.80E–02	0
	CGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₁₃	GWO	1.30E–02	3.40E–03	0	7.03E–02	1.31E–02	0	8.67E–01	9.96E–01	0
	CGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₁₄	GWO	1.05E–10	6.87E–11	0	1.12E–06	4.58E–07	0	1.79E–04	3.79E–05	0
	CGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₁₅	GWO	9.98E+02	2.11E+02	0	3.80E+03	4.48E+02	0	7.57E+03	6.69E+02	0
	CGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₁₆	GWO	9.35E+01	8.79E–01	0	2.41E+02	9.31E–01	0	4.88E+02	1.25E+00	0
	CGWO	0.00E+00	0.00E+00	100	1.66 E + 00	3.49E+00	70	1.13E+01	2.36E+01	50
f ₁₇	GWO	1.54E–04	2.27E–05	0	2.71E+01	9.89E–01	0	6.37E+02	1.34E+02	0
	CGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₁₈	GWO	5.43E–01	8.58E–02	0	1.08E+00	1.11E–01	0	1.93E+00	1.19E–01	0
	CGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100
f ₁₉	GWO	1.80E–01	3.09E–02	0	3.67E–01	2.82E–02	0	4.75E–01	7.50E–03	0
	CGWO	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100	0.00E+00	0.00E+00	100

4.3 Comparison of CDGWO with mGWOand EEGWO algorithms

In this section, we investigated the performance of the proposed algorithm and other improved GWO algorithms, i.e., mGWO [52] and EEGWO [13]. The parameter settings and results of mGWO and EEGWO algorithms were taken from reference [13] and the parameter settings of the CDGWO are the same as in the above experiments, and no increase in population size or number of function evaluations was required. In this experiment, “Mean” means the mean of fitness, “St. dev” represents the standard deviation of fitness. The dimensions were set to 30, 60, 500, and 1000. The experimental results are recorded in Table 4, and the best results among the three methods are shown in bold. The statistical significance of the experimental results on 19 test functions is obtained by performing Wilcoxon rank sum test at a 0.05 significance level, and the results are reported in Table 5, where “–” denotes significantly worse, “+” denotes significantly better and “≈” denotes similar; “Dim” denotes dimension.

The CDGWO and EEGWO algorithms achieved the theoretical global optima value (0) on 16 functions (i.e., f₁–f₅, f₇, f₉, f₁₀ and f₁₂–f₁₉) with dimensions 30 and 60, whereas the mGWO algorithm only obtained the theoretical global optima value (0) with 5 functions (i.e., f₇, f₉, f₁₀ and f₁₂ and f₁₄) (Table 4). However, on 3 functions (i.e., f₆, f₈ and f₁₁), neither the CDGWO or EEGWO methods could achieve the global optima solutions under the maximum number of fitness function evaluations under dimensions 30 and 60, but their results are very close to each other. There were 5 functions (i.e., f₁–f₃, f₁₃ and f₁₇) where the mGWO algorithm could find solutions very close to the global optimum. With respect to the CDGWO and EEGWO, mGWO achieved better results for function f₆. Furthermore, we can see that the proposed CDGWO algorithm shows the same stable search ability and result accuracy as the EEGWO for the low dimensional problems, and it is better than the mGWO algorithm (Table 4).

From Table 4, compared with mGWO, the CDGWO displayed the best performance for 16 functions (i.e., f₁–f₅, f₇–f₉, f₁₁–f₁₃, and f₁₅–f₁₉) in 500 and 1000 dimensions, and it provided similar optimization results for 3 functions (i.e., f₆, f₁₀ and f₁₄). With respect to the EEGWO, the CDGWO exhibits better results for 4 functions, three of which are f₈, f₁₁ and f₁₉ in 500 and 1000 dimensions, and the other is f₈ function in 1000 dimensions. However, compared with the CDGWO, EEGWO performed best on function f₁₆ in 500 and 1000 dimensions. EEGWO offered the theoretical optima value (0) for 14 functions (i.e., f₁–f₅, f₇, f₅9–f₁₀, and f₁₂–f₁₇) in 500 and 1000 dimensions; in addition, with function f₁₈ in 500 dimensions, the theoretical optima value (0) was also obtained. By contrast, CDGWO offered the theoretical optima value (0) for 15 functions (i.e., f₁–f₅, f₇, f₉–f₁₀, f₁₂–f₁ and f₁₇–f₁₉) in 500 and 1000 dimensions.

Table 4
The means and the standard deviations obtained by the CDGWO, EEGWO and mGWO over 30 independent runs on dimensions 30, 60, 500 and 1000

Function Dim mGWO [52] EEGWO [13] CDGWO

Mean±St.dev Mean±St.dev Mean±St.dev

f1 30 5.04E–282±0.00E+00– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 2.27E–198±0.00E+00– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 2.15E–088±2.04E–088– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 3.01E–074±4.67E–074– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f2 30 3.12E–281±0.00E+00– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 8.13E–200±0.00E+00– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 3.37E–086±7.06E–086– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 9.15E–074±5.31E–074– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f3 30 5.87E–167±0.00E+00– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 3.83E–118±4.40E–118– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 1.88E–051±8.76E–052– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 2.88E–042±7.37E–043– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f4 30 5.05E–079±8.62E–079– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 2.56E–040±5.72E–040– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 1.27E+004±1.95E+004– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 1.99E+005±1.08E+005– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f5 30 3.31E–076±7.15E–076– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 7.74E–050±5.47E–050– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 4.98E+001±9.79E+000– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 7.27E+001±2.54E+000– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f6 30 2.60E+001 ± 1.03E– 000+ 2.90E+01±3.76E–03– 2.89E+01±5.40E–02

60 5.68E+001 ± 1.05E–000+ 5.90E+01±1.52E–02– 5.89E+01±4.52E–02

500 4.98E+002 ± 5.02E–01≈ 4.99E+02 ± 3.60E–02≈ 4.98E+02 ± 7.85E–02

1000 9.98E+002 ± 4.89E–02≈ 9.99E+02 ± 2.12E–02≈ 9.98E+02 ± 3.17E–01

f7 30 0.00E+000 ± 0.00E+000≈ 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 0.00E+000 ± 0.00E+000≈ 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 3.02E–148±3.40E–148– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 3.30E–127±6.41E–127– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f8 30 1.15E–004±1.35E–004– 5.08E–05 ± 4.65E–05+ 6.26E–05±7.59E–05

60 1.87E–004±1.12E–004– 5.56E–05 ± 5.53E–05+ 8.22E–05±7.68E–05

500 8.77E–004±2.57E–004– 1.30E–04±8.23E–05– 5.91E–05 ± 4.42E–05

1000 1.53E–003±8.29E–004– 1.51E–04±1.47E–04– 7.14E–05 ± 7.34E–05

f9 30 0.00E+000 ± 0.00E+000≈ 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 0.00E+000 ± 0.00E+000≈ 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 4.91E–035±1.02E–034– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 1.65E–003±3.66E–003– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f10 30 0.00E+000 ± 0.00E+000≈ 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 0.00E+000 ± 0.00E+000≈ 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 0.00E+00 ± 0.00E+00≈ 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 0.00E+00 ± 0.00E+00≈ 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f11 30 7.99E–015±0.00E+000– 8.88E–16 ± 0.00E+00≈ 8.88E–16 ± 0.00E+00

60 1.30E–014±3.18E–015– 8.88E–16 ± 0.00E+00≈ 8.88E–16 ± 0.00E+00

500 2.36E–014±3.20E–015– 3.02E–15±1.95E–15– 8.88E–16 ± 0.00E+00

1000 2.72E–014±4.78E–015– 3.73E–15±1.59E–15– 8.88E–16 ± 0.00E+00

f12 30 0.00E+000 ± 0.00E+000≈ 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 0.00E+000 ± 0.00E+000≈ 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 1.11E–016±0.00E+00– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 1.33E–016±4.96E–017– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f13 30 1.76E–158±0.00E+000– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 6.50E–17±8.40E–117– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 1.82E–051±7.01E–052– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

10001. 20E–042±3.42E–043– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f14 30 0.00E+000 ± 0.00E+000≈ 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 0.00E+000 ± 0.00E+000≈ 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 0.00E+00 ± 0.00E+00≈ 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 0.00E+00 ± 0.00E+00≈ 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f15 30 1.52E–092±3.39E–092– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 1.47E–039±3.24E–039– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 7.27E+002±6.02E+001– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 2.92E+003±2.49E+002– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f16 30 9.51E+000±1.06E+000– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 2.23E+001±3.95E+000– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 2.28E+002±1.91E+001– 0.00E+00 ± 0.00E+00+ 1.66E+00±3.49E+00

1000 3.84E+002±1.86E+002– 0.00E+00 ± 0.00E+00+ 1.13E+01±2.36E+01

f17 30 3.12E–278±0.00E+00– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 1.02E–194±0.00E+00– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 3.11E–085±4.87E–085– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 3.41E–071±3.92E–071– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

f18 30 9.99E–002±0.00E+00– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 9.99E–002±0.00E+00– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 2.00E–001±0.00E+00– 0.00E+00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

1000 2.60E–001±5.48E–002– 3.00E–01±1.64E–04– 0.00E–00 ± 0.00E+00

f19 30 9.70E–003±0.00E+00– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

60 9.70E–003±0.00E+00– 0.00E–00 ± 0.00E+00≈ 0.00E–00 ± 0.00E+00

500 5.62E–002±4.80E–002– 2.47E–01±2.26E–01– 0.00E–00 ± 0.00E+00

1000 7.16E–002±3.71E–002– 7.78E–03±4.35E–03– 0.00E–00 ± 0.00E+00

Function	Dim	mGWO [52]	EEGWO [13]	CDGWO
		Mean±St.dev	Mean±St.dev	Mean±St.dev
f1	30	5.04E–282±0.00E+00–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	2.27E–198±0.00E+00–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	2.15E–088±2.04E–088–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	3.01E–074±4.67E–074–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f2	30	3.12E–281±0.00E+00–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	8.13E–200±0.00E+00–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	3.37E–086±7.06E–086–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	9.15E–074±5.31E–074–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f3	30	5.87E–167±0.00E+00–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	3.83E–118±4.40E–118–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	1.88E–051±8.76E–052–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	2.88E–042±7.37E–043–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f4	30	5.05E–079±8.62E–079–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	2.56E–040±5.72E–040–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	1.27E+004±1.95E+004–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	1.99E+005±1.08E+005–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f5	30	3.31E–076±7.15E–076–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	7.74E–050±5.47E–050–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	4.98E+001±9.79E+000–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	7.27E+001±2.54E+000–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f6	30	2.60E+001 ± 1.03E–	000+ 2.90E+01±3.76E–03–	2.89E+01±5.40E–02
	60	5.68E+001 ± 1.05E–000+	5.90E+01±1.52E–02–	5.89E+01±4.52E–02
	500	4.98E+002 ± 5.02E–01≈	4.99E+02 ± 3.60E–02≈	4.98E+02 ± 7.85E–02
	1000	9.98E+002 ± 4.89E–02≈	9.99E+02 ± 2.12E–02≈	9.98E+02 ± 3.17E–01
f7	30	0.00E+000 ± 0.00E+000≈	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	0.00E+000 ± 0.00E+000≈	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	3.02E–148±3.40E–148–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	3.30E–127±6.41E–127–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f8	30	1.15E–004±1.35E–004–	5.08E–05 ± 4.65E–05+	6.26E–05±7.59E–05
	60	1.87E–004±1.12E–004–	5.56E–05 ± 5.53E–05+	8.22E–05±7.68E–05
	500	8.77E–004±2.57E–004–	1.30E–04±8.23E–05–	5.91E–05 ± 4.42E–05
	1000	1.53E–003±8.29E–004–	1.51E–04±1.47E–04–	7.14E–05 ± 7.34E–05
f9	30	0.00E+000 ± 0.00E+000≈	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	0.00E+000 ± 0.00E+000≈	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	4.91E–035±1.02E–034–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	1.65E–003±3.66E–003–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f10	30	0.00E+000 ± 0.00E+000≈	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	0.00E+000 ± 0.00E+000≈	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	0.00E+00 ± 0.00E+00≈	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	0.00E+00 ± 0.00E+00≈	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f11	30	7.99E–015±0.00E+000–	8.88E–16 ± 0.00E+00≈	8.88E–16 ± 0.00E+00
	60	1.30E–014±3.18E–015–	8.88E–16 ± 0.00E+00≈	8.88E–16 ± 0.00E+00
	500	2.36E–014±3.20E–015–	3.02E–15±1.95E–15–	8.88E–16 ± 0.00E+00
	1000	2.72E–014±4.78E–015–	3.73E–15±1.59E–15–	8.88E–16 ± 0.00E+00
f12	30	0.00E+000 ± 0.00E+000≈	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	0.00E+000 ± 0.00E+000≈	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	1.11E–016±0.00E+00–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	1.33E–016±4.96E–017–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f13	30	1.76E–158±0.00E+000–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	6.50E–17±8.40E–117–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	1.82E–051±7.01E–052–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	10001.	20E–042±3.42E–043–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f14	30	0.00E+000 ± 0.00E+000≈	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	0.00E+000 ± 0.00E+000≈	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	0.00E+00 ± 0.00E+00≈	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	0.00E+00 ± 0.00E+00≈	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f15	30	1.52E–092±3.39E–092–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	1.47E–039±3.24E–039–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	7.27E+002±6.02E+001–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	2.92E+003±2.49E+002–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f16	30	9.51E+000±1.06E+000–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	2.23E+001±3.95E+000–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	2.28E+002±1.91E+001–	0.00E+00 ± 0.00E+00+	1.66E+00±3.49E+00
	1000	3.84E+002±1.86E+002–	0.00E+00 ± 0.00E+00+	1.13E+01±2.36E+01
f17	30	3.12E–278±0.00E+00–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	1.02E–194±0.00E+00–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	3.11E–085±4.87E–085–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	3.41E–071±3.92E–071–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
f18	30	9.99E–002±0.00E+00–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	9.99E–002±0.00E+00–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	2.00E–001±0.00E+00–	0.00E+00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	1000	2.60E–001±5.48E–002–	3.00E–01±1.64E–04–	0.00E–00 ± 0.00E+00
f19	30	9.70E–003±0.00E+00–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	60	9.70E–003±0.00E+00–	0.00E–00 ± 0.00E+00≈	0.00E–00 ± 0.00E+00
	500	5.62E–002±4.80E–002–	2.47E–01±2.26E–01–	0.00E–00 ± 0.00E+00
	1000	7.16E–002±3.71E–002–	7.78E–03±4.35E–03–	0.00E–00 ± 0.00E+00

From the Wilcoxon rank sum test statistical results listed in Table 5, the CDGWO was the best performing algorithm compared to the mGWO and EEGWO algorithms, and the EEGWO was the second best. The CDGWO outperformed the mGWO in 13 cases in 30 and 60 dimensions, and it gave the same performance in 5 cases. However, for the high dimensional problems (Dim = 500 and 1000), the optimization performance of mGWO hardly outperformed CDGWO, with the exception of 3 cases. EEGWO was likely to provide very competitive results under low dimensions (Dim = 30 and 60) compared to CDGWO. However, with the increase of dimensions, the optimization performance of CDGWO is better than EEGWO.

Table 5

Wilcoxon rank sum test statistical results

CDGWO Vs	Dim=30			Dim=60			Dim=500			Dim=1000
	–	+	≈	–	+	≈	–	+	≈	–	+	≈
mGWO	13	1	5	13	1	5	16	0	3	16	0	3
EEGWO	1	1	17	1	1	17	3	1	15	4	1	14

4.4 Comparison of the CDGWO with other state-of-the-art algorithms

In this subsection, the CDGWO algorithm was compared with other state-of-the-art optimization methods that have been proposed recently. The comprehensive learning particle swarm optimizer (CLPSO) [54], chaotic dynamic weight particle swarm optimization (CDWPSO) [15], composite differential evolution (CoDE) [55], memory-based hybrid dragonfly algorithm (MHDA) [56], firefly algorithm with neighborhood attraction (NaFA) [57], improved harmony search (LHS) [10], and the multistrategy ensemble artificial bee colony (MEABC) [58] were compared with the proposed method. For the CDGWO algorithm, the number of function evaluations and the experimental parameter settings were kept the same as those used in the previous subsection. The experimental parameters and the number of function evaluations of the other seven algorithms were consistent with their original papers. The functions used to benchmark the performance of the seven state-of-the-art optimization algorithms were all the classical test functions listed in Table 1. In this experiment, the dimension of the seven classical test functions is set as 30, the mean value (Mean) and the standard variance (St. dev) of the best fitness value are shown in Table 6.

Table 6
Mean and the standard deviations of the best fitness value (Dim=30)

Function Criteria CLPSO CDWPSO CoDE MHDA NaFA LHS MEABC CDGWO

f ₁ Mean 3.90E–03 0.00E+00 2.36E–10 4.07E–42 4.43E–29 0.00E+00 4.85E–40 0.00E+00

St.dev 1.50E–03 0.00E+00 1.69E–10 2.22E–41 4.06E–30 0.00E+00 2.31E–40 0.00E+00

f ₃ Mean 9.00E–03 0.00E+00 6.64E–06 6.62E–15 2.98E–15 5.51E–260 1.25E–21 0.00E+00

St.dev 1.60E–03 0.00E+00 3.41E–06 3.61E–14 2.80E–15 0.00E+00 3.56E–21 0.00E+00

f ₄ Mean 6.27E+03 0.00E+00 8.10E–01 2.55E–50 2.60E–28 7.73E–66 9.81E+03 0.00E+00

St.dev 1.24E+03 0.00E+00 5.39E–01 1.30E–49 2.37E–29 4.18E–65 2.49E+03 0.00E+00

f ₅ Mean 21.3532 0.00E+00 2.95E–02 4.99E–05 3.43E–15 1.30E–143 4.89E+00 0.00E+00

St.dev 1.8724 0.00E+00 1.10E–02 2.73E–04 2.89E–17 7.04E–143 1.37E+00 0.00E+00

f ₆ Mean 1.94E+02 2.81E+01 27.8691 3.34E–22 2.39E+01 2.68E+01 2.86E–01 2.89E+01

St.dev 4.82E+01 1.88E–01 17.4895 5.67E–22 8.96E–01 5.76E–01 3.48E–01 5.40E–02

f ₈ Mean 2.55E–02 0.00E+00 1.38E–02 5.25E–05 2.91E–02 2.83E–04 2.29E–02 6.26E–05

St.dev 5.40E–03 0.00E+00 5.70E–03 5.02E–05 1.56E–02 1.82E–04 1.38E–02 7.59E–05

f ₁₀ Mean 4.6733 0.00E+00 34.1225 5.90E–07 2.09E+01 0.00E+00 0.00E+00 0.00E+00

St.dev 1.8074 0.00E+00 5.6981 3.23E–06 6.96E+00 0.00E+00 0.00E+00 0.00E+00

f ₁₁ Mean 3.54E–02 8.88E–16 3.89E–06 6.34E–15 3.02E–14 3.08E–15 2.90E–14 8.88E–16

St.dev 8.70E–03 0.00E+00 1.51E–06 2.72E–14 8.94E–15 1.23E–15 1.32E–14 0.00E+00

f ₁₂ Mean 1.95E–02 0.00E+00 5.14E–05 2.40E–04 0.00E+00 0.00E+00 0.00E+00 0.00E+00

St.dev 7.70E–03 0.00E+00 2.81E–04 2.25E–02 0.00E+00 0.00E+00 0.00E+00 0.00E+00

Function	Criteria	CLPSO	CDWPSO	CoDE	MHDA	NaFA	LHS	MEABC	CDGWO
f ₁	Mean	3.90E–03	0.00E+00	2.36E–10	4.07E–42	4.43E–29	0.00E+00	4.85E–40	0.00E+00
	St.dev	1.50E–03	0.00E+00	1.69E–10	2.22E–41	4.06E–30	0.00E+00	2.31E–40	0.00E+00
f ₃	Mean	9.00E–03	0.00E+00	6.64E–06	6.62E–15	2.98E–15	5.51E–260	1.25E–21	0.00E+00
	St.dev	1.60E–03	0.00E+00	3.41E–06	3.61E–14	2.80E–15	0.00E+00	3.56E–21	0.00E+00
f ₄	Mean	6.27E+03	0.00E+00	8.10E–01	2.55E–50	2.60E–28	7.73E–66	9.81E+03	0.00E+00
	St.dev	1.24E+03	0.00E+00	5.39E–01	1.30E–49	2.37E–29	4.18E–65	2.49E+03	0.00E+00
f ₅	Mean	21.3532	0.00E+00	2.95E–02	4.99E–05	3.43E–15	1.30E–143	4.89E+00	0.00E+00
	St.dev	1.8724	0.00E+00	1.10E–02	2.73E–04	2.89E–17	7.04E–143	1.37E+00	0.00E+00
f ₆	Mean	1.94E+02	2.81E+01	27.8691	3.34E–22	2.39E+01	2.68E+01	2.86E–01	2.89E+01
	St.dev	4.82E+01	1.88E–01	17.4895	5.67E–22	8.96E–01	5.76E–01	3.48E–01	5.40E–02
f ₈	Mean	2.55E–02	0.00E+00	1.38E–02	5.25E–05	2.91E–02	2.83E–04	2.29E–02	6.26E–05
	St.dev	5.40E–03	0.00E+00	5.70E–03	5.02E–05	1.56E–02	1.82E–04	1.38E–02	7.59E–05
f ₁₀	Mean	4.6733	0.00E+00	34.1225	5.90E–07	2.09E+01	0.00E+00	0.00E+00	0.00E+00
	St.dev	1.8074	0.00E+00	5.6981	3.23E–06	6.96E+00	0.00E+00	0.00E+00	0.00E+00
f ₁₁	Mean	3.54E–02	8.88E–16	3.89E–06	6.34E–15	3.02E–14	3.08E–15	2.90E–14	8.88E–16
	St.dev	8.70E–03	0.00E+00	1.51E–06	2.72E–14	8.94E–15	1.23E–15	1.32E–14	0.00E+00
f ₁₂	Mean	1.95E–02	0.00E+00	5.14E–05	2.40E–04	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	St.dev	7.70E–03	0.00E+00	2.81E–04	2.25E–02	0.00E+00	0.00E+00	0.00E+00	0.00E+00

As shown in Table 6, the CDGWO algorithm offered higher accuracy for 3 functions (f₃–f₅) than all of the other seven algorithms. CDGWO found the theoretical global optima values for 6 functions (f₁, f₃, f₄, f₅, f₁₀ and f₁₂). LHS obtained the theoretical global optima values for 3 functions (f₁, f₁₀ and f₁₂), and with the f₃ and f₅ functions, their results were much closer to the global optimum with the high accuracy of 1e-260 and 1e-143, respectively. With respect to MHDA, the CDGWO obtained better results for all functions except for f₆ and f₈. Compared to CLPSO and CoDE, the CDGWO exhibited better results for all functions except for function f₆. For function f₁₁, the CDGWO obtained the same results as the CDWPSO and is superior to the other six algorithms. With respect to NaFA, the CDGWO offered better and similar results for seven functions (f₁, f₃, f₄, f₅, f₈ f₁₀ and f₁₁) and one function (f₁₂), respectively, but performed a little worse with function f₆. Compared to the MEABC algorithm, the CDGWO, respectively, obtained better results for six functions (f₁, f₃, f₄, f₅, f₈ and f₁₁) and similar results for two functions (f₁₀ and f₁₂), whereas MEABC obtained a little better results for function f₆. With respect to LHS, the CDGWO found better results for five functions (f₃, f₄, f₅, f₈ and f₁₁) and similar results for two functions (f₁ and f₁₁), but a little worse results for function f₆. It should be emphasized that the CDWPSO results, which are below 10^–50 (error rate) are assumed to be the theoretical global optima values [15]. Thus, we can consider that the theoretical global optima values obtained by CDWPSO are worse than that obtained by CDGWO. With respect to the CDWPSO, the CDGWO achieved better results for seven functions (f₁, f₃, f₄, f₅, f₁₀, f₁₁ and f₁₂) and worse results for two functions (f₆ and f₈).

4.5 The effectiveness of the two improvement strategies in the CDGWO

As can be seen from Section 3, two improvements (i.e., the modified position-updated equation and the nonlinear control parameter strategy) were introduced into the standard GWO algorithm to propose the CDGWO algorithm. A large number of experiments that were performed on different comparison algorithms above have shown that the CDGWO is very effective for complex and high dimensional optimization problems. However, exactly which kind of CDGWO improvement methods improved the performance of the standard GWO algorithm is still unknown. Therefore, it is necessary to investigate the effectiveness of these two improvement methods. In this subsection, two meaningful experiments were carried out on 19 low-dimensional (Dim = 30) benchmark functions.

In the first experiment, the CDGWO only adopted the modified position-updated equation (i.e., Equation (9)) and the linear control parameter of the GWO was used to replace the nonlinear control parameter strategy of the CDGWO (denoted as CDGWO-1). And then, in the second experiment, CDGWO only adopted the nonlinear control parameter strategy (i.e., Equation (13)) and its modified position-updated equation was replaced by that of the standard GWO algorithm (i.e., Equation (6)) (denoted as CDGWO-2). The maximum number of iterations and the population size were set as 500 and 30, respectively. It should be pointed out that each test function was independently carried out 30 times and the typical results (i.e., “Mean” and “St. dev”) are listed in the Table 7. The statistical significance of the experimental results of the CDGWO–1, CDGWO–2, and CDGWO were obtained by performing the Wilcoxon rank sum test at a 0.05 significance level, and the results are reported in Table 7, where “–” denotes significantly worse, “+” denotes significantly better, “≈” denotes similar, and “Dim” denotes the dimension; the best results are marked in bold.

Table 7
Mean and the standard deviations of CDGWO— 1, CDGWO— 2, and CDGWO on 19 functions (Dim=30) with 15,000 FEs

Function CDGWO–1 (Mean±St. dev) CDGWO–2 (Mean±St. dev) CDGWO (Mean±St. dev)

f ₁ 1.24E–322±0.00E–000– 1.37E–033±2.86E–033– 0.00E-00 ± 0.00E+00

f ₂ 8.40E–323±0.00E–000– 9.59E–035±1.33E–034– 0.00E-00 ± 0.00E+00

f ₃ 4.73E–164±0.00E–000– 1.69E–020±1.51E–020– 0.00E-00 ± 0.00E+00

f ₄ 3.27E–320±0.00E–000– 1.25E–032±2.52E–032– 0.00E-00 ± 0.00E+00

f ₅ 7.96E–155±2.07E–154– 8.45E–002±2.67E–001– 0.00E-00 ± 0.00E+00

f ₆ 2.85E+001±3.63E–001≈ 2.59E+001 ± 4.87E–000+ 2.89E+01±5.40E-02

f ₇ 0.00E+00 ± 0.00E+000≈ 3.73E–061±1.31E–060– 0.00E-00 ± 0.00E+00

f ₈ 1.39E–004±1.32E–004– 1.40E–003±9.08E–004– 6.26E-05 ± 7.59E-05

f ₉ 0.00E+00 ± 0.00E+000≈ 4.90E–118±2.64E–117– 0.00E-00 ± 0.00E+00

f ₁₀ 0.00E+00 ± 0.00E+000≈ 2.71E+001±5.41E+001– 0.00E-00 ± 0.00E+00

f ₁₁ 8.88E–016 ± 0.00E–000≈ 1.35E+000±5.15E+000– 8.88E-16 ± 0.00E+00

f ₁₂ 0.00E+00 ± 0.00E+000≈ 0.00E+00 ± 0.00E+000≈ 0.00E-00 ± 0.00E+00

f ₁₃ 9.09E–070±4.98E–069– 4.14E+000±1.08E+001– 0.00E-00 ± 0.00E+00

f ₁₄ 0.00E+00 ± 0.00E+000≈ 0.00E+00 ± 0.00E+000≈ 0.00E-00 ± 0.00E+00

f ₁₅ 1.67E–296±0.00E–000– 6.11E–010±2.02E–009– 0.00E-00 ± 0.00E+00

f ₁₆ 2.67E+000±1.31E–000– 5.37E+000±4.36E+000– 0.00E-00 ± 0.00E+00

f ₁₇ 1.93E–315±0.00E–000– 2.62E–030±5.21E–030– 0.00E-00 ± 0.00E+00

f ₁₈ 1.20E–104±6.55E–104– 1.53E–001±5.07E–002– 0.00E-00 ± 0.00E+00

f ₁₉ 0.00E+00 ± 0.00E+000≈ 2.62E–002±1.37E–002– 0.00E-00 ± 0.00E+00

+ 0 1

– 11 16

≈ 8 2

Function	CDGWO–1 (Mean±St. dev)	CDGWO–2 (Mean±St. dev)	CDGWO (Mean±St. dev)
f ₁	1.24E–322±0.00E–000–	1.37E–033±2.86E–033–	0.00E-00 ± 0.00E+00
f ₂	8.40E–323±0.00E–000–	9.59E–035±1.33E–034–	0.00E-00 ± 0.00E+00
f ₃	4.73E–164±0.00E–000–	1.69E–020±1.51E–020–	0.00E-00 ± 0.00E+00
f ₄	3.27E–320±0.00E–000–	1.25E–032±2.52E–032–	0.00E-00 ± 0.00E+00
f ₅	7.96E–155±2.07E–154–	8.45E–002±2.67E–001–	0.00E-00 ± 0.00E+00
f ₆	2.85E+001±3.63E–001≈	2.59E+001 ± 4.87E–000+	2.89E+01±5.40E-02
f ₇	0.00E+00 ± 0.00E+000≈	3.73E–061±1.31E–060–	0.00E-00 ± 0.00E+00
f ₈	1.39E–004±1.32E–004–	1.40E–003±9.08E–004–	6.26E-05 ± 7.59E-05
f ₉	0.00E+00 ± 0.00E+000≈	4.90E–118±2.64E–117–	0.00E-00 ± 0.00E+00
f ₁₀	0.00E+00 ± 0.00E+000≈	2.71E+001±5.41E+001–	0.00E-00 ± 0.00E+00
f ₁₁	8.88E–016 ± 0.00E–000≈	1.35E+000±5.15E+000–	8.88E-16 ± 0.00E+00
f ₁₂	0.00E+00 ± 0.00E+000≈	0.00E+00 ± 0.00E+000≈	0.00E-00 ± 0.00E+00
f ₁₃	9.09E–070±4.98E–069–	4.14E+000±1.08E+001–	0.00E-00 ± 0.00E+00
f ₁₄	0.00E+00 ± 0.00E+000≈	0.00E+00 ± 0.00E+000≈	0.00E-00 ± 0.00E+00
f ₁₅	1.67E–296±0.00E–000–	6.11E–010±2.02E–009–	0.00E-00 ± 0.00E+00
f ₁₆	2.67E+000±1.31E–000–	5.37E+000±4.36E+000–	0.00E-00 ± 0.00E+00
f ₁₇	1.93E–315±0.00E–000–	2.62E–030±5.21E–030–	0.00E-00 ± 0.00E+00
f ₁₈	1.20E–104±6.55E–104–	1.53E–001±5.07E–002–	0.00E-00 ± 0.00E+00
f ₁₉	0.00E+00 ± 0.00E+000≈	2.62E–002±1.37E–002–	0.00E-00 ± 0.00E+00
+	0	1
–	11	16
≈	8	2

From Table 7 which concerns the optimization performance of the three algorithms, it can be observed that the proposed algorithm competes very favorably with the CDGWO–1 and CDGWO–2 in both indicators of “Mean” and “St. dev”. With respect to the CDGWO, CDGWO–1 achieved similar and worse solutions for 8 functions (f₆–f₇, f₉–f₁₂, f₁₄ and f₁₉) and 11 functions (f₁–f₅, f₈, f₁₃, and f₁₅–f₁₈), respectively. In addition, it should be pointed out that the fitness values of the CDGWO–1 for functions f₁, f₃, f₄, f₅, f₅, f₁ and f₁₈ were smaller than the order of le-322, le-323, le-164, le-320, le-155, le-296, le-315, and le-104, which were much closer to the CDGWO. Therefore, the optimization performance of the CDGWO–1 is marginally worse than the CDGWO algorithm, and there were no significant differences in optimization performance between the CDGWO and CDGWO-1.

As we can see from Table 7, the CDGWO performed better and similar results than CDGWO-2 for 16 functions (f₁–f₅, f₇–f₁₁, f₁₃ and f₁₅–f₁₉) and 2 functions (f₁₂ and f₁₄), respectively. However, for function f₆, compared to the CDGWO, CDGWO-2 obtained a better average fitness value and worse standard deviation. Therefore, we can draw the conclusions that the comprehensive optimization performance of the CDGWO algorithm was significantly better than that of CDGWO-2. According to [13], the nonlinear control parameter strategy was not used dependently in the search process, which may be the main factor that led to the poor performance of the CDGWO-2.

Figure 4 presents a comparison of the CDGWO, CDGWO-1, and CDGWO-2 performance with four typical test functions. From Fig. 4, we observe that the CDGWO achieved the best convergence speed and accuracy for the four functions, and CDGWO-1 has the same convergence accuracy to the CDGWO except for the convergence speed. Therefore, we suggest that the CDGWO-2 algorithm improved the performance of CDGWO-1 by raising its convergence speed.

Fig.4

Comparison of performance of three algorithms on four typical test functions.

5 Conclusions

The CDGWO algorithm is proposed for solving continuous large-scale numerical optimization problems. Two main improvement components were introduced into the GWO algorithm to enhance the search performance of the GWO and to achieve good global optimal ability. A modified position-updated equation based on the chaotic map was proposed to enhance the exploration ability of the GWO algorithm and increase its diversity of potential solutions. Second, the nonlinear control parameter strategy and the nonlinear dynamic weight were introduced to balance the exploration and exploitation of the candidate individual search and convergence speed. Simulation results from nineteen benchmark functions indicate that the CDGWO utilizing the modified position-updated equation and the nonlinear control parameter strategy offered better performance than their counterparts, which lacked the modified position-updated equation or the nonlinear control parameter strategy. In addition, the proposed CDGWO algorithm obtained highly competitive results compared to the standard GWO, other GWO variants and other state-of-the-art algorithms in almost all classical benchmark functions tested, and it is more stable in most of the functions. In summary, the proposed CDGWO algorithm is effective, has high capability to avoid local optima and is suitable for application in high-dimensional optimization problems. We intend to employ the proposed CDGWO algorithm to solve discrete and multiobjective, and combinatorial optimization problems in our future work.

Footnotes

Acknowledgments

This project was supported by the National Social Science Foundation of China (Grant No. 16BJY078), the Soft Science Foundation of Heilongjiang Province (Grant No.GC16D102), and the Philosophy and Social Science Research Planning Program of Heilongjiang Province (Grant No. 17JYH49). In addition, we are grateful to the anonymous reviewers for their valuable comments that helped us improve this paper.

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