A hybrid Aquila optimizer and its K-means clustering optimization

Abstract

Aiming at the defects of the Aquila optimizer (AO) in dealing with some complex optimization problems, such as slow convergence speed, low convergence accuracy, and easy to fall into local optimum, in this paper, a hybrid Aquila optimizer (HAO) algorithm based on Gauss map and crisscross operator is proposed. First, Gauss map is introduced to initialize the Aquila population to improve the quality of the initial population. Then use the crisscross operator to promote the exchange of information within the population and maintain the diversity of the population in each iteration, which not only enhances the ability of the algorithm to jump out of the local optimum but also accelerates the global convergence of the algorithm. The results of experiments using 21 classical benchmark functions indicate that HAO has better global search ability, faster convergence speed, and better stability than AO. The overall optimization performance of HAO in different dimensions is better than particle swarm optimization (PSO) algorithm, gray wolf optimization (GWO) algorithm, whale optimization algorithm (WOA), and crisscross optimization (CSO) algorithm. Finally, the results of K-means clustering optimization on six University of California (UCI) standard data sets demonstrate that HAO has significant advantages over three algorithms that are good at clustering optimization.

Keywords

Optimization swarm intelligence optimization algorithm chaotic map crossover operator

Introduction

The bionics-based swarm intelligence optimization algorithm is a meta-heuristic algorithm that simulates biological behavior and phenomena in nature (Li et al., 2021b). Compared with traditional optimization algorithms, it is easier to implement and has been applied in many fields such as automatic control, route planning, data mining, and image processing (Blum and Groß, 2015; Li et al., 2021a; Nguyen et al., 2020). In order to solve increasingly complex optimization problems, numerous algorithms have been continuously proposed, such as particle swarm optimization (PSO; Marini and Walczak, 2015; Tian and Chen, 2021), gray wolf optimization (GWO; Chrouta et al., 2021; Mirjalili et al., 2014), and crisscross optimization (CSO; Meng et al., 2014), whale optimization algorithm (WOA; Mirjalili and Lewis, 2016; Tian et al., 2021), and so on. They are some representative swarm intelligence algorithms based on social search, which have the characteristics of fast convergence speed and few parameters; but they have the shortcomings of finding local minimum and weak global search ability (Chegini et al., 2018; Javidrad et al., 2018; Mohammed and Rashid, 2020; Nasrollahzadeh et al., 2021). Given these deficiencies, many scholars have carried out improved research on these basic optimization algorithms. For instance, Heidari and Pahlavani (2017) combined Levy flight and greedy selection strategy with an improved hunting stage and proposed an algorithm that embeds Levy flight into GWO (Levy-embedded GWO, LGWO), enhancing the population diversity of GWO. Ghasemi et al. (2019) suggested a new and primary sample for PSO (phasor particle swarm optimization, PPSO); in the proposed PPSO method, all control variables are incorporated in the phase angle, which is generated by the algorithm; this makes the PSO a non-parametric algorithm with more straightforward calculations.

Aquila optimizer (AO) is a swarm intelligence optimization algorithm proposed by Abualigah et al. (2021). It simulates Aquila’s hunting behavior. The optimization search process proposed by AO has four stages, including two search methods. The exploration method reflects its global exploration ability, and the exploitation method reflects its local search ability. Experimental results have proved that AO has stronger local search ability and certain global search advantages compared with the algorithms mentioned above (Abualigah et al., 2021). And AO has been applied to practical engineering problems such as pressure vessel design, welded beam design, cantilever beam design, and reducer. However, AO still suffers from slow convergence speed, low convergence accuracy, and easy to fall into local optimum when dealing with some complex optimization problems.

From the No Free Lunch Theorems (Wolpert and Macready, 1997), it is known that any optimization algorithm has its limitations and cannot be applied to all optimization problems, so improvement of optimization algorithms is particularly important. Combining the advantages of different optimization algorithms has become a common method for improving swarm intelligence optimization algorithms. A hybrid algorithm combining GWO and Crow Search Algorithm can quickly jump out of local optimum and converge quickly, but its limitation lies in improving optimization while sacrificing time advantage (Arora et al., 2019). Korashy et al. (2019) proposed a hybrid algorithm combining WOA and GWO; in the proposed algorithm, the leadership hierarchy of GWO is utilized in the bubble-net attacking strategy of the WOA, which significantly improves the performance of the WOA algorithm; and it can be used to solve the coordination problem of the directional over-current relay. To improve the global search ability and convergence speed of PSO, a hybrid algorithm of PSO and CSO was proposed; CSO is incorporated as an evolutionary catalytic agent that has powerful capability of searching for a solution of high quality, thereby accelerating the global convergence of PSO (Meng et al., 2016). Based on the search direction adjustment mechanism, Liang et al. (2021) adopted two different arithmetic crossover operators to enhance the global search ability of the algorithm; and proposed a stochastic example learning strategy to enhance the convergence precise and the local search ability of the algorithm. However, there are few mentions in literature of integrating AO with CSO for improvement currently.

In view of the superiority and novelty of the AO algorithm, it has certain application value to improve the AO. Inspired by combining the advantages of different algorithms to propose improved strategies, this paper aimed at proposing a hybrid Aquila optimizer (HAO) algorithm that can complement the advantages of Gauss map and CSO. Initially, Gauss map is introduced to initialize the Aquila population to improve the quality of the initial population. Then, crisscross operator is applied to promote the exchange of information within the population and maintain the diversity of the population in each iteration, which not only enhances the ability of the algorithm to jump out of the local optimum but also accelerates the global convergence of the algorithm. The major contributions of this research work are summarized as follows:

A novel hybridization optimizer based on AO and CSO is proposed.

The proposed optimizer is applied to 21 benchmark function optimization problems.

Based on the experimental results, the performance of the proposed optimizer is analyzed from three aspects of optimization performance, convergence speed, and optimization time, and its effectiveness is verified.

The proposed hybrid approach is employed to solve the K-means clustering optimization problem, and the results are validated on six data sets.

The remainder of this paper is organized as follows: In section “AO,” the AO is introduced. The HAO is addressed in section “HAO.” Section “Experimental results and analysis of function optimization” demonstrates the effectiveness of the proposed algorithm. Section “Application of HAO in K-means clustering optimization” shows the application of HAO to K-means clustering optimization. Conclusions and remarks are made in section “Conclusion.”

AO

The Aquila is a carnivorous bird widely distributed in the Northern Hemisphere, and they are one of the most intelligent and strategic hunters in nature. When catching a prey, they can flexibly convert methods of hunting according to different situations. As an example, when hunting squirrels, their capture strategy is to first search for a rough area where prey is high in the sky and then hover over the prey when they find the prey, so that they can find the right time to attack.

AO simulates the behavior of Aquila during hunting. Its optimization process can be summarized into three steps, namely, selecting the search space, exploring in the diverging search space, and exploiting in the convergent search space. In the algorithm, $N$ is the size of the Aquila population, and $D$ denotes to the dimension size of the problem. $UB$ is the upper bound and $LB$ is the lower bound of the given problem. The current number of iterations is $t$ , and the maximum number of iterations is denoted by $T$ . The conversion condition for switching between exploration and exploitation is $t \leq (2 / 3) T$ , and it means AO is in the exploration phase when this condition holds, otherwise AO is in the exploitation phase.

Expanded exploration

At this stage, the Aquila soars high in the sky. It will use its keen vision to observe and identify the area of prey, and then descend vertically to select the best hunting area. The mathematical description of this process above is as follows:

X (t + 1) = X_{best} (t) \times a + (X_{M} (t) - X_{best} (t) * rand)

(1)

a = 1 - \frac{t}{T}

(2)

X_{M} (t) = \frac{1}{N} \sum_{i = 1}^{N} X_{i} (t)

(3)

where $X (t + 1)$ is the position of population in the next iteration, and $X_{best} (t)$ is the current optimal individual, which reflects the approximate position of prey; $rand$ is the random number on $[0, 1]$ ; $a$ is used to control the process of extensive exploration; $X_{M} (t)$ represents the average position of the current population.

Narrowed exploration

When the area where prey is discovered, the Aquila first hovers above the target prey, then carefully searches the area of the selected target prey, and then prepares to attack the prey. The mathematical description of the behavior in this phase can be written as

X (t + 1) = X_{best} (t) \times Levy (D) + X_{R} (t) + (y - x) * rand

(4)

where $X_{R} (t)$ is a random individual in the current population; $Levy$ is the Levy flight distribution function, it is expressed as

Levy (D) = s \times \frac{u \times σ}{| v |^{\frac{1}{β}}}

(5)

σ = (\frac{Γ (1 + β) \times sine (\frac{π β}{2})}{Γ (\frac{1 + β}{2}) \times β \times 2^{(\frac{β - 1}{2})}})

(6)

where $s$ and $β$ are constants taking the values 0.01 and 1.5, respectively; $u$ and $v$ are random numbers on $[0, 1]$ ; and $y$ and $x$ indicate the shape of the spiral flight in the search, they are represented as follows:

y = r \times \cos (θ)

(7)

x = r \times \sin (θ)

(8)

r = r_{1} + U \times D_{1}

(9)

θ = - ω \times D_{1} + θ_{1}

(10)

θ_{1} = \frac{3 \times π}{2}

(11)

where $r_{1}$ is a fixed number of search periods on $[1, 20]$ ; $U$ and $ω$ are constants taking the values 0.0056 and 0.005, respectively; and $D_{1}$ is an integer on [1, $D$ ].

Expanded exploitation

In this phase, the prey has been pinpointed and Aquila is ready to land and attack. After the Aquila descends vertically and makes the initial attack, the prey’s reaction will be observed. The above behavior is mathematically described as follows:

\begin{matrix} X (t + 1) = (X_{best} (t) - X_{M} (t)) \times α - rand \\ + ((UB - LB) \times rand + LB) \times δ \end{matrix}

(12)

where $α$ and $δ$ is the adjustment parameter, which takes the value of 0.1 here.

Narrowed exploitation

At the final stage, Aquila approaches its prey to capture it precisely according to the random movements made by the prey on land, the mathematical description of the above behavior is as follows:

\begin{matrix} X (t + 1) = QF \times X_{best} (t) - (G_{1} \times X (t) \times rand) \\ - G_{2} \times Levy (D) + rand \times G_{1} \end{matrix}

(13)

where $QF$ is the quality function of the balanced search strategy.

QF (t) = t^{\frac{2 \times rand - 1}{{(1 - T)}^{2}}}

(14)

$G_{1}$ denotes the various movements of Aquila during the tracking of the escaped prey, and it is a linearly decreasing value from 2 to 0, and its expression is as follows:

G_{1} = 2 \times rand - 1

(15)

$G_{2}$ denotes the slope of the flight of the Aquila from the first to the last position during the tracking of the escaped prey, it can be expressed as

G_{2} = 2 \times (1 - \frac{t}{T})

(16)

HAO

All swarm intelligence optimization algorithms have certain limitations, and AO is no exception. The initial population of the standard AO algorithm is generated by random initialization, which leads to the inability of the algorithm to ensure that the initial population covers the entire search space uniformly, and therefore this limits the optimization performance of the algorithm to a certain extent. And the lack of information exchange between population individuals during the iterative process of the algorithm leads to a decrease in diversity of population, which affects the global search ability of the algorithm. The concept of chaos has been applied to recent optimization methods to achieve a convenient tradeoff between exploration and exploitation. In order to solve the problem that the algorithm is prone to premature aging when the optimal individual falls into a local optimum, the standard AO algorithm can be modified by using Gauss map and crisscross operators.

Gauss map

The chaotic map has the characteristics of randomness and regularity, so that the combination of it with different optimization algorithms can improve the performance of algorithms (Wu et al., 2019). Among them, Logistic map and Tent map have been widely used in the population initialization of the swarm intelligence optimization algorithm (Guo et al., 2020; Li et al., 2020). But they both have their own drawbacks. For example, Logistic map has Chebyshev distribution in the range of [0, 1] and large search blind area; Tent map has small and unstable periods and easily fall into fixed point. Koyuncu (2020) optimized PSO using 10 commonly used chaotic maps, and the final experimental results indicated that Gauss map works best. Thus, Gauss map is selected for population initialization in this paper. The initialization process is as follows:

Y_{n + 1} = {\begin{matrix} 0 & Y_{n} = 0, \\ \frac{1}{Y_{n} mod (1)} = \frac{1}{Y_{n}} - [\frac{1}{Y_{n}}] & Y_{n} \neq 0 . \end{matrix}

(17)

X_{i} = Y_{n} \times (UB - LB) + LB, i = 1, 2, \dots, N

(18)

Crisscross operator

In this paper, we optimize on the populations generated during each iteration of AO to exploit the ability of the crisscross operator to search for high-quality optimal individuals, thus enhancing the hybrid algorithm’s ability to jump out of the local optimum and accelerating the global convergence of the hybrid algorithm.

Horizontal crossover operation

Set the probability to $p_{h}$ , and use horizontal crossover operator to perform arithmetic crossover for all the same dimensions between two different individuals in the parent population. This operation can reduce the blind spots that individuals cannot search, and can also allow parent individuals to compete with offspring individuals, so that individuals with better fitness can be preserved. And finally a new population can be formed. The horizontal crossover operation allows different individuals to communicate and learn from each other, thus improving the global search ability of the algorithm. The specific operation is

M_{id}^{hc} = r_{1} \times X_{id} + (1 - r_{1}) \times X_{jd} + c_{1} \times (X_{id} - X_{jd}),

(19)

M_{jd}^{hc} = r_{2} \times X_{jd} + (1 - r_{2}) \times X_{id} + c_{2} \times (X_{jd} - X_{id}) .

(20)

where $d$ belongs to [1, $D$ ]; $r_{1}$ and $r_{2}$ are random numbers on [0, 1]; $c_{1}$ and $c_{2}$ are random numbers on [−1,1]; $X_{id}$ and $X_{jd}$ denote the $d$ th dimension of the $i th$ individual $X_{i}$ and the $j th$ individual $X_{j}$ of the parent, respectively; $M_{id}^{hc}$ and $M_{jd}^{hc}$ represent the children of the parent individuals $X_{i}$ and $X_{j}$ generated by the horizontal crossover in the dimension $d$ , respectively.

According to equations (19) and (20), it can be observed that the horizontal crossover operation consists of two parts: the first part is the exchange of internal information, searching in the hypercube formed by the parent individuals with a high probability; the second part is the exploration of external information, by applying an extension coefficient between the parent individuals, with a low probability to search the outer edge of the hypercube. In addition, the internal search is an equal probability, while the external search gradually decreases with the increase of the relative distance from the parent individual. This search mechanism can effectively reduce search blind spots and improve the global search ability of the HAO.

Vertical crossover operation

Set the probability to $p_{v}$ , and use the vertical crossover operation to generate a new population by arithmetic crossover of two different dimensions of the same individual in the parent population, after which the parent individuals compete with the child individuals to form a new population. The vertical crossover operation allows individuals to communicate and learn from each other in different dimensions, and each operation updates only one dimension of an individual, which helps the algorithm to continue the search in the stalled dimension, but try not to change the information of the normal dimension, thus helping the algorithm to jump out of the local optimum. The specific operation is as follows:

M_{i d_{1}}^{vc} = r \times X_{i d_{1}} + (1 - r) \times X_{i d_{2} .}

(21)

where $d_{1}$ and $d_{2}$ belong to [1, $D$ ]; $r$ is a random number on [0, 1]; $M_{i d_{1}}^{vc}$ denotes the offspring individuals generated by vertical crossover between the $d_{1} th$ and $d_{2} th$ dimensions of the parent individuals $X_{i}$ on the $d_{1} th$ dimension. It can be seen from equation (21) that the first item on the right side of the equation is the information item contained in the $d_{1} th$ dimension itself, and the second item is the influence of the $d_{2} th$ dimension on the $d_{1} th$ dimension. So that the individual generated after the vertical cross-operation contains the information of the two dimensions without destroying the information of the $d_{2} th$ dimension. Vertical crossover can free up dimensions that may be trapped in local optima. Thus, it makes it easier for HAO to jump out of the local optima.

The population generated by horizontal crossover has to compete with the population generated by AO, and the population generated by vertical crossover has to compete with the population generated by horizontal crossover. This mutually reinforcing competition mechanism makes Aquila individuals move quickly to the search area with better fitness and speeds up the convergence to the global optimum.

Steps of the HAO algorithm

The iterative process of HAO proposed in this paper starts from generating a set of feasible individuals to be selected. In each iteration, the vertical and horizontal crossover operation is performed first after searching through the mentioned four search strategies. As shown in Figure 1, the detailed steps of the HAO algorithm proposed in this paper are as follows. It is worth noting that, in general, the vertical crossover probability is set to be less than 1. A large number of experimental results show that when the $p_{v} = 0.8$ value is within the range of $[0.2, 0.8]$ , better results will be generated (Meng et al., 2014).

Step 1: Setting the population size $N$ , individual dimension $D$ , maximum number of iterations $T$ , horizontal crossover probability $p_{h} = 1$ , and vertical crossover probability $p_{v} = 0.8$ , then initializing the algorithm parameters.

Step 2: Referring to equation (18), initializing populations with Gauss map.

Step 3: Calculating the fitness of all individuals in the population and finding the optimal individual $X_{best} (t)$ .

Step 4: To determine whether the condition $t \leq (2 / 3) T$ is valid, if so, enter the exploration phase and update the population with equal probability using equation (1) or (4), otherwise enter the exploitation phase and update the population with equal probability using equation (12) or (13).

Step 5: Cross the population horizontally, and all individuals in the population are randomly divided into pairs of $N / 2$ without repetition. If $rand < p_{h}$ is true, use equations (19) and (20) to generate progeny populations. After the cross-border processing, the new population after the competition enters the next step.

Step 6: Cross the population vertically and horizontally, and all dimensions in the population are randomly and non-repeatedly divided into $D / 2$ pairs. If $rand < p_{v}$ it is true, use equation (21) to generate progeny populations. After cross-border processing and competition, the new population enters the next step.

Step 7: Determine whether condition $t \leq T$ holds, if it does, return to Step 3 and continue, otherwise output the optimal individual $X_{best} (t)$ and its fitness $f (X_{best} (t))$ .

Figure 1.

The flowchart of HAO.

Experimental results and analysis of function optimization

In order to fully verify the effectiveness and superiority of HAO proposed in this paper, PSO (Marini and Walczak, 2015), GWO (Chrouta et al., 2021; Mirjalili et al., 2014), WOA CSO (Meng et al., 2014; Mirjalili and Lewis, 2016) and standard AO (Abualigah et al., 2021) algorithm are selected as the comparison algorithms. Twenty-one classical benchmark functions are used to test the performance of the algorithm. The unimodal functions f1–f7 in Table 1 test the local search ability of the algorithm, the multimodal functions f8–f13 in Table 2 test the global search ability of the algorithm, and the fixed-dimension multimodal functions f14–f21 in Table 3 test the ability of the algorithm to handle complex problems. In order to compare the above six algorithms fairly, on the same experimental platform, set the parameters N = 30, T = 500, and the number of runs of the algorithm 30 times. The other parameters of the comparison algorithm are consistent with the original literature. All algorithms in this paper are programmed using Matlab R2020a, computer operating system is Windows 10, and the processor used is AMD R5 4600H 16 GB.

Table 1.

Unimodal benchmark functions.

Function	Dimension	Range	Optimal solution
$f_{1} (x) = \sum_{i = 1}^{n} x_{i}^{2}$	–	[−100, 100]	0
$f_{2} (x) = \sum_{i = 0}^{n} \| x_{i} \| + Π_{i = 0}^{n} \| x_{i} \|$	–	[−10, 10]	0
$f_{3} (x) = \sum_{i = 1}^{d} {(\sum_{j = 1}^{i} x_{j})}^{2}$	–	[−100, 100]	0
$f_{4} (x) = \max {\| x_{i} \|, 1 \leq i \leq n}$	–	[−100, 100]	0
$f_{5} (x) = \sum_{i = 1}^{n - 1} [100 {(x_{i}^{2} - x_{i + 1})}^{2} + {(1 - x_{i})}^{2}]$	–	[−30, 30]
$f_{6} (x) = \sum_{i = 1}^{n} {([x_{i} + 0.5])}^{2}$	–	[−100, 100]	0
$f_{7} (x) = \sum_{i = 0}^{n} {ix}_{i}^{4} + random [0, 1)$	–	[−128, 128]	0

Table 2.

Multimodal benchmark functions.

Function	Dimension	Range	Optimal solution
$f_{8} (x) = \sum_{i = 1}^{n} (- x_{i} \sin (\sqrt{\| x_{i} \|}))$	–	[−500, 500]	−418.9829*n
$f_{9} (x) = \sum_{i = 1}^{n} [x_{i}^{2} - 10 \cos (2 π x_{i}) + 10]$	–	[−5.12, 5.12]	0
$f_{10} (x) = - 20 \exp (- 0.2 \sqrt{\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{2}}) - \exp (\frac{1}{n} \sum_{i = 1}^{n \sum^{(2 π x_{i})}} \cos (2 π x_{i})) + 20 + e$	–	[−32, 32]	0
$f_{11} (x) = 1 + \frac{1}{4000} \sum_{i = 1}^{n} x_{i}^{2} - Π_{i = 1}^{n} \cos (\frac{x_{i}}{\sqrt{i}})$	–	[−600, 600]	0
$\begin{matrix} f_{12} (x) = \frac{π}{n} {10 \sin (π y_{1})} \\ + \sum_{i = 1}^{n - 1} {(y_{i} - 1)}^{2} [1 + 10 si n^{2} (π y_{i + 1}) + \sum_{i = 1}^{n} u (x_{i}, 10, 100, 4)] \end{matrix}$ $where y_{i} = 1 + \frac{x_{i} + 1}{4},$ $u (x_{i}, a, k, m) {\begin{matrix} K {(x_{i} - a)}^{m} if x_{i} > a \\ 0 - a \leq x_{i} \geq a \\ K {(- x_{i} - a)}^{m} - a \leq x_{i} \end{matrix}$	–	[−50, 50]	0
$\begin{matrix} f_{13} (x) = 0.1 (si n^{2} (3 π x_{1}) + \sum_{i = 1}^{n} {(x_{i} - 1)}^{2} [1 + si n^{2} (3 π x_{i} + 1)] + {(x_{n} - 1)}^{2} + si n^{2} (2 π x_{n})) \\ + \sum_{i = 1}^{n} u (x_{i}, 5, 100, 4) \end{matrix}$	–	[−50,50]	0

Table 3.

Fixed-dimension multimodal benchmark functions.

Function	Dimension	Range	Optimal solution
$f_{14} (x) = {(\frac{1}{500} + \sum_{j = 1}^{25} \frac{1}{j + \sum_{i = 1}^{2} (x_{j} - a_{ij})})}^{- 1}$	2	[−65, 65]	1
$f_{15} (x) = \sum_{i = 1}^{11} {[a_{i} - \frac{x_{1} (b_{i}^{2} + b_{i} x_{2})}{b_{i}^{2} + b_{i} x_{3} + x_{4}}]}^{2}$	4	[−5, 5]	0.00030
$f_{16} (x) = {(x_{2} - \frac{5.1}{4 π^{2}} x_{1}^{2} + \frac{5}{π} x_{1} - 6)}^{2} + 10 (1 - \frac{1}{8 π}) \cos x_{1} + 10$	2	[−5, 5]	0.398
$f_{17} (x) = [1 + {(x_{1} + x_{2} + 1)}^{2} (19 - 14 x_{1} + 3 x_{1}^{2} - 14 x_{2} + 6 x_{1} x_{2} + 3 x_{2}^{2})] \times$ $[30 + {(2 x_{1} - 3 x_{2})}^{2} \times (18 - 32 x_{i} + 12 x_{1}^{2} + 48 x_{2} - 36 x_{1} x_{2} + 27 x_{2}^{2})]$	2	[−2, 2]	3
$f_{18} (x) = - \sum_{i = 1}^{4} c_{i} \exp (- \sum_{i = 1}^{3} a_{ij} {(x_{j} - p_{ij})}^{2})$	3	[−1, 2]	−3.86
$f_{19} (x) = - \sum_{i = 1}^{4} c_{i} \exp (- \sum_{i = 1}^{6} a_{ij} {(x_{j} - p_{ij})}^{2})$	6	[0, 1]	−3.32
$f_{20} (x) = - \sum_{i = 1}^{5} {[(x - a_{i}) {(x - a_{i})}^{T} + c_{i}]}^{- 1}$	4	[0, 1]	−10.1532
$f_{21} (x) = - \sum_{i = 1}^{7} {[(x - a_{i}) {(x - a_{i})}^{T} + c_{i}]}^{- 1} 4$	4	[0, 1]	−10.4028

In order to verify the performance of HAO, this section adopts three sets of experiments: Experiment 1 examines the impact of different improvement strategies on AO, and verifies the effectiveness of the proposed improvement strategy; Experiment 2 examines the performance of HAO in solving low-dimensional function optimization problems; Experiment 3 examines the performance of HAO in solving different dimensional function optimization problems.

Analysis of the simulation results of the impact of different improvement strategies on AO

In Experiment 1, to analyze the difference between the performance of the algorithms improved using different improvement strategies, the experimental results of HAO, AO, AO1, CSAO1, and CSAO2 on functions f14∼f21 and 30 dimensional functions f1∼ f13 are compared, in which AO1 is improved with only Gauss map, CSAO1 is improved with only horizontal crossover, the CSAO2 is improved with only vertical crossover. The results are shown in Table 4, where the smaller mean value represented by mean indicates the higher solution accuracy of the algorithm, and the smaller standard deviation represented by Std indicates the better stability of the algorithm.

Table 4.

Optimization performance of different improved AO algorithms.

Function		AO	AO1	CSAO1	CSAO2	HAO
f1	Mean	9.0177E–104	1.3075E–115	0	3.1730E–110	0
f1	Std	4.9231E–103	7.1610E–115	0	1.5633E–109	0
f2	Mean	1.3490E–60	1.3531E–67	8.1413E–138	1.7140E–60	9.9424E–139
f2	Std	7.3890E–60	7.4106E–67	3.1789E–137	9.3873E–60	5.4456E–138
f3	Mean	6.5544E–106	4.7672E–106	0	4.0501E–101	0
f3	Std	3.5760E–105	2.6087E–105	0	2.2140E–100	0
f4	Mean	6.3448E–63	1.2879E–52	2.7828E–139	9.3295E–53	2.0404E–144
f4	Std	3.4752E–62	7.0544E–52	8.2587E–139	5.1099E–52	1.0430E–143
f5	Mean	1.9246E–03	4.8714E–03	1.4859E–05	1.0799E–03	4.7957E–06
f5	Std	2.2121E–03	1.1901E–02	5.9169E–05	7.3238E–03	1.8131E–05
f6	Mean	1.5402E–04	4.9130E–05	1.9959E–07	2.7565E–05	6.5604E–14
f6	Std	3.4724E–04	6.9493E–05	1.0202E–06	1.0383E–04	1.5297E–13
f7	Mean	1.0601E–04	1.0341E–04	1.3063E–04	9.3411E–05	8.1562E–05
f7	Std	1.0940E–04	9.6005E–05	1.1460E–04	6.4963E–05	8.7709E–05
f8	Mean	−7.2744E03	−8.1040E03	−7.6471E03	−1.1208E04	−1.1496E04
f8	Std	3.4823E03	3.7477E03	2.4964E03	9.2395E02	1.0477E03
f9	Mean	0	0	0	0	0
f9	Std	0	0	0	0	0
f10	Mean	8.8818E–16	8.8818E–16	8.8818E–16	8.8818E–16	8.8818E–16
f10	Std	0	0	0	0	0
f11	Mean	0	0	0	0	0
f11	Std	0	0	0	0	0
f12	Mean	6.5808E–06	2.1473E–06	8.1452E–13	2.3213E–07	8.4136E–14
f12	Std	1.7605E–05	3.7259E–06	4.3243E–12	1.0801E–06	4.2262E–13
f13	Mean	9.4924E–06	3.0439E–05	7.2318E–09	1.9598E–06	3.2290E–13
f13	Std	1.5904E–05	6.6830E–05	3.9045E–08	6.6558E–06	7.3858E–13
f14	Mean	3.1500E00	3.3510E00	1.1634E00	1.3871E00	9.9800E–01
f14	Std	3.8757E00	3.6330E00	5.2656E–01	2.1311E00	0
f15	Mean	4.9689E–04	4.5122E–04	3.6498E–04	3.2977E–04	3.6044E–04
f15	Std	1.1063E–04	7.3524E–05	9.8686E–05	1.4630E–05	2.4411E–04
f16	Mean	3.9807E–01	3.9809E–01	3.9789E–01	3.9790E–01	3.9789E–01
f16	Std	2.4603E–04	2.6858E–04	5.8356E–11	1.5432E–05	2.6021E–08
f17	Mean	3.0344E00	3.0289E00	3.0000E00	3.0034E00	3
f17	Std	4.5004E–02	3.0571E–02	2.1010E–07	6.5103E–03	9.4745E–16
f18	Mean	−3.8538E00	−3.8554E00	−3.8599E00	−3.8620E00	−3.8614E00
f18	Std	8.5520E–03	5.1550E–03	3.7988E–03	8.3964E–04	3.2413E–03
f19	Mean	−3.1593E00	−3.1434E00	−3.2180E00	−3.2665E00	−3.2824E00
f19	Std	9.7652E–02	1.1051E–01	1.0871E–01	5.2764E–02	4.7820E–02
f20	Mean	−1.0144E01	−1.0145E01	−1.0153E01	−1.0153E01	−1.0153E01
f20	Std	2.6253E–02	1.4452E–02	1.2915E–04	2.4894E–06	6.8481E–15
f21	Mean	−1.0384E01	−1.0393E01	−1.0403E01	−1.0403E01	−1.0403E01
f21	Std	3.4624E–02	1.1592E–02	4.3304E–05	5.2484E–05	1.1302E–07

AO: Aquila optimizer; HAO: hybrid Aquila optimizer.

As can be seen from Table 4, AO1 improves over the mean and standard deviation of AO on the 12 benchmark functions, indicating that the initial population generated by the Gauss map improves the algorithm’s search performance to some extent. The experimental results on functions f1–f6 show that the algorithm performance of CSAO1 and HAO are close, while CSAO2 is inferior to CSAO1. These results suggest that the horizontal crossover substantially improves the performance of the algorithm on the unimodal function problem. The experimental results of the algorithm on functions f12–f21 show that both CSAO1 and CSAO2 perform less well than HAO but better than AO, which indicates that the crisscross crossover operators improve the algorithm’s performance in finding the optimal performance on multimodal and complex problems.

Figure 2 shows the average convergence curves of the above five algorithms after running for 30 times, respectively. Due to limited space, only representative algorithms are given.

Figure 2.

Comparison of convergence curves of improved algorithms: (a) f1, (b) f6, (c) f8, (d) f12, (e) f14, and (f) f18.

From Figure 2(a), it can be seen that HAO and CSAO1 reach convergence with the lowest number of iterations and the highest accuracy, indicating that horizontal crossover contributes to the convergence performance of the algorithm. As can be seen from Figure 2(b), the convergence accuracy of AO1 in the early and middle stages of the iteration is several orders of magnitude higher than that of AO, indicating that the quality of the initial population generated by Gauss map is higher. From the Figure 2(c), we can find that CSAO2 can quickly jump out of the local optimum in the middle and late stages of the iteration, and it can achieve high convergence accuracy, which shows that the vertical crossover improves the ability of the algorithm to jump out of the local optimum. From the remaining sets of curves, we can infer that both the Gauss map and the crisscross operator contribute to the HAO search performance.

Performance analysis of HAO in low-dimensional case

Optimization performance analysis in low-dimensional case

The purpose of experiment 2 is to examine the performance of HAO in low dimensions. Run six algorithms 30 times independently on functions f14–f21 and 10 dimensional functions f1∼f13, and the experimental results are shown in Table 5. The algorithm ranking is indicated by Rank, and the name of the algorithm ranked No. 1 is bolded; total indicates the total number of algorithms ranked No. 1. Analysis carried out in Table 5 shows: when testing single-peaked functions, HAO achieved the first place seven times and the theoretical optimal value with a standard deviation of 0 on the functions f1, f3, and f6, indicating that it has strong exploring capacity; when testing multimodal functions, HAO obtained the first place six times and the theoretical optimal value on the functions f9 and f11, its accuracy and stability are the best among the six algorithms; when testing fixed low-dimensional multimodal function, HAO achieved the first place seven times and the theoretical optimal value of the average value on the functions f17 and f20, these indicate that HAO has the strongest ability to handle complex optimization problems among the six algorithms. Overall, HAO has an advantage over the remaining five algorithms in terms of its performance in finding the optimal value in the low-dimensional benchmark functions.

Table 5.

Comparison of optimization results of low-dimensional benchmark functions.

Function		PSO	GWO	WOA	CSO	AO	HAO
	Mean	4.8696E–21	2.392E57	3.3386E–77	4.7562E–32	1.2104E–107	0
f1	Std	1.1205E–20	6.6381E–57	1.629E–76	1.3879E–31	6.6296E–107	0
	Rank	6	5	3	4	2	1
	Mean	9.1826E–12	5.3243E–33	2.3551E–51	3.0158E–22	9.6451E–67	2.2029E–141
f2	Std	2.0761E–11	8.5020E–33	5.7321E–51	1.2687E–21	3.6025E–66	1.1315E–140
	Rank	6	4	3	5	2	1
	Mean	2.9368E–06	9.5199E–23	1.0148E+02	1.0752E+00	5.6935E–109	0
f3	Std	3.6848E–06	5.1711E–22	1.5689E+02	2.1748E+00	2.4459E–108	0
	Rank	4	3	6	5	2	1
	Mean	9.9717E–06	5.1641E–18	5.4453E+00	6.1052E–03	5.8421E–53	7.3430E–139
f4	Std	1.1417E–05	1.2980E–17	1.0178E+01	2.3061E–02	3.0098E–52	2.1690E–138
	Rank	4	3	6	5	2	1
	Mean	5.0497E00	8.9354E00	5.7666E–01	4.8489E00	1.2002E–03	8.5709E–08
f5	Std	3.4726E00	1.2818E01	5.5208E–02	3.7726E00	1.8171E–03	2.5430E–07
	Rank	5	6	3	4	2	1
	Mean	3.4178E–21	8.3459E–03	9.9174E–03	5.1358E–34	2.9858E–05	0
f6	Std	6.8879E–21	4.5694E–02	4.1858E–02	2.8130E–33	5.0231E–05	0
	Rank	3	5	6	2	4	1
	Mean	7.4652E–03	6.6309E–04	1.8988E–03	6.6936 E–04	1.2667E–04	1.1270E–04
f7	Std	3.8288E–03	5.8959E–04	3.1652E–03	4.4067E–04	1.1617E–04	1.2424E–04
	Rank	6	3	5	4	2	1
	Mean	−2.2814E03	−2.7424E03	−3.2836E03	−4.1503E03	−3.4975E03	–4.1898E03
f8	Std	4.11261E02	3.5828E02	5.5940E02	2.1712E01	8.0050E02	2.1624E02
	Rank	6	5	4	2	3	1
	Mean	5.4062E+00	8.7369E–01	9.4739E–16	0	0	0
f9	Std	2.8346E+00	2.4541E+00	5.1891E–15	0	0	0
	Rank	4	3	2	1	1	1
	Mean	4.9841E–11	7.2831E–15	3.4935E–15	7.0462E–15	8.8818E–16	8.8818E–16
f10	Std	5.5962E–11	1.4454E–15	2.4567E–15	1.8504E–15	0	0
	Rank	5	4	2	3	1	1
	Mean	2.0290E–01	2.7588E–02	6.4739E–02	1.1203E–01	0	0
f11	Std	1.4561E–01	2.9614E–02	1.4079E–01	8.9600E–02	0	0
	Rank	5	2	3	4	1	1
	Mean	3.0647E–23	6.6569E–03	8.4190E–03	1.4045E–30	7.5083E–06	8.7736E–31
f12	Std	6.8290E–23	9.5749E–03	9.8508E–03	7.4343E–30	1.1893E–05	4.5443E–30
	Rank	3	5	6	2	4	1
	Mean	1.1646E–20	2.3420E–02	4.1415E–02	6.8963E–30	1.7808E–05	5.6919E–31
f13	Std	4.9371E–20	5.7319E–02	6.1417E–02	3.4684E–29	2.4388E–05	3.0007E–30
	Rank	3	5	6	2	4	1
	Mean	2.6758E00	5.5273E00	3.5135E00	1.4142E00	2.2397E00	9.9800E–01
f14	Std	2.2247E00	4.6948E00	3.7641E00	1.5051E00	2.9123E00	0
	Rank	2	6	5	2	3	1
	Mean	8.8713E–04	4.4786 E–03	6.5868 E–04	2.1176 E–03	5.6064 E–04	3.5179 E–04
f15	Std	1.2577 E–04	8.0838 E–03	4.7722 E–04	3.4421 E–03	1.5011 E–04	2.3474 E–04
	Rank	4	6	3	5	2	1
	Mean	3.9789E–01	3.9789E–01	3.9789E–01	4.1202E–01	3.9808E–01	3.9789E–01
f16	Std	0	1.4538E–06	1.1565E–05	6.9774E–02	2.5434E–04	0
	Rank	1	2	3	5	4	1
	Mean	3	5.7000E+00	3.9026E+00	3.9297E+00	3.0240E+00	3
f17	Std	0	1.4789E+01	4.9294E+00	4.9246E+00	2.7255E–02	1.2064E–15
	Rank	1	6	4	5	3	2
	Mean	−3.8628E00	−3.8613E00	−3.8454E00	−3.8351E00	−3.8563E00	–3.8617E00
f18	Std	2.6684e–15	2.4291E–03	2.9352E–02	3.0935E–02	4.4967E–03	2.1682E–03
	Rank	5	3	2	6	4	1
	Mean	−3.2744E+00	−3.2419E+00	−3.2051E+00	−3.2519E+00	−3.1426E+00	–3.2813E+00
f19	Std	5.9241E–02	8.6312E–02	1.2658E–01	6.8424E–02	9.1119E–02	4.9891E–02
	Rank	2	4	5	3	6	1
	Mean	−8.1432E+00	−8.9666E+00	−7.5958E+00	−9.2378E+00	−1.0143E+01	–1.0153E+01
f20	Std	2.9755E+00	2.1842E+00	2.8210E+00	2.4055E+00	1.4198E–02	6.9584E–15
	Rank	5	4	6	3	2	1
	Mean	−8.3769E00	−1.0401E01	−6.7438E00	−8.1419E00	−1.0395E01	–1.0403E01
f21	Std	3.2035E00	9.0337E–04	3.1730E00	3.3169E00	1.1266E–02	2.5929E–08
	Rank	5	2	6	4	3	1
Total		2	0	0	1	3	20

PSO: particle swarm optimization; GWO: gray wolf optimization; WOA: whale optimization algorithm; CSO: crisscross optimization; AO: Aquila optimizer; HAO: hybrid Aquila optimizer.

Aquila optimizer convergence performance analysis in low-dimensional case

In order to examine the convergence performance of the algorithms, the average convergence curves of the six algorithms run 30 times in Experiment 2 are calculated in this paper, and only the representative convergence plots are given due to space limitation, as in Figure 3. From Figure 3(a), it can be seen that HAO convergence performance is best when testing single-peaked functions f1 and f5; the convergence performance of HAO and AO is not much different when testing function f7, but both are superior to the remaining four algorithms. HAO and CSO convergence performance is not much different when testing multimodal functions f8 and f13, but better than the remaining four algorithms; HAO and AO have the best convergence accuracy, but the convergence speed of AO is inferior to HAO and GWO. From Figure 3(g) and (h), we can see that HAO has a strong ability to jump out of the local optimal solution, high convergence accuracy and fast convergence speed, while the convergence performance of AO is inferior; from Figure 3(i), we can see that the difference between the convergence performance of HAO and AO is not much, and both are better than the remaining four algorithms. Overall, the convergence performance of HAO is better than the remaining five algorithms.

Figure 3.

Comparison of convergence curves in low dimensions: (a) f1, (b) f5, (c) f7, (d) f8, (e) f10, (f) f13, (g) f14, (h) f18, and (i) f21.

Running time analysis of the algorithm

In order to inspect the running speed of the algorithm program, the running time of the algorithm was tested in Experiment 2. The results are shown in Table 6, and the unit of data in the table is $S$ . The ascending ranking of average running time of six algorithms can be obtained from the table, which is PSO, WOA, GWO, CSO, AO, and HAO. It is obvious that the long running time of HAO is caused by the introduction of Gauss map initialization and crisscross operator, which expand the diversity of populations and information exchange between populations but increase the computational complexity of the algorithm. But in general, compared with the improvement of convergence speed, convergence accuracy, and stability, the time consumption of HAO algorithm is within a reasonable range.

Table 6.

Time consumption of algorithms.

Function	PSO	GWO	WOA	CSO	AO	HAO
f1	4.3229E–02	5.5729E–02	3.7500E–02	6.5625E–02	1.1146E–01	1.9740E–01
f2	3.3854E–02	6.6667E–02	4.0104E–02	7.8125E–02	1.1927E–01	2.2604E–01
f3	1.0052E–01	1.0104E–01	8.7500E–02	1.8385E–01	2.2448E–01	4.3021E–01
f4	3.2292E–02	6.6146E–02	3.9583E–02	7.5000E–02	1.2031E–01	2.0208E–01
f5	4.7917E–02	6.6667E–02	5.4688E–02	1.1667E–01	1.5417E–01	3.3229E–01
f6	3.6458E–02	5.1562E–02	3.7500E–02	6.7187E–02	1.1042E–01	1.9115E–01
f7	5.8854E–02	7.0313E–02	5.9375E–02	1.2292E–01	1.5990E–01	3.1250E–01
f8	5.0521E–02	6.3542E–02	4.4271E–02	8.9583E–02	1.3073E–01	2.5417E–01
f9	3.2813E–02	5.5208E–02	4.4792E–02	7.6042E–02	1.1771E–01	2.0260E–01
f10	3.5417E–02	5.6250E–02	4.3229E–02	7.5000E–02	1.1875E–01	2.1927E–01
f11	5.3646E–02	7.1354E–02	5.4688E–02	1.0469E–01	1.4010E–01	2.9167E–01
f12	1.3750E–01	1.4531E–01	1.3125E–01	3.1146E–01	3.0156E–01	6.6563E–01
f13	1.2812E–01	1.4688E–01	1.3125E–01	3.0625E–01	2.9427E–01	6.5781E–01
f14	3.8854E–01	3.8646E–01	3.9323E–01	1.0042E+00	8.0573E–01	1.8938E+00
f15	3.1771E–02	4.9479E–02	4.1146E–02	8.1250E–02	1.1510E–01	2.0677E–01
f16	3.0729E–02	4.8438E–02	3.8021E–02	5.9375E–02	9.8437E–02	1.8073E–01
f17	2.3958E–02	4.1667E–02	3.5417E–02	5.3125E–02	9.4792E–02	1.6979E–01
f18	3.6458E–02	4.4792E–02	4.1667E–02	8.3333E–02	1.1615E–01	2.2708E–01
f19	4.1667E–02	5.1562E–02	5.7292E–02	8.8542E–02	1.5156E–01	2.5469E–01
f20	5.3646E–02	5.7813E–02	5.4167E–02	1.1094E–01	1.4063E–01	3.0521E–01
f21	6.0417E–02	6.4583E–02	5.9896E–02	1.2552E–01	1.5781E–01	3.2240E–01

PSO: particle swarm optimization; GWO: gray wolf optimization; WOA: whale optimization algorithm; CSO: crisscross optimization; AO: Aquila optimizer; HAO: hybrid Aquila optimizer.

Wilcoxon rank-sum test

To exclude the influence on the mean results of the algorithm due to chance factors, this paper chose to further test the significant difference between HAO and the rest of the algorithms using the Wilcoxon rank sum test (Derrac et al., 2011) on non-parametric statistics. The results of Experiment 2 are taken as samples and the confidence level is set to 0.05. When $p < 0.05$ , $h = 1$ , it means that there is a significant difference between the two algorithms; when $p > 0.05$ , $h = 0$ , it means that the overall performance of the two algorithms is the same.

Table 7 shows the Wilcoxon rank sum test results between HAO and other algorithms, $p << 0.05$ is always marked. According to the analysis, the number of significant differences between HAO and comparison algorithm in optimizing 21 benchmark functions are 19, 20, 19, 16, and 17, respectively. Statistical analysis shows that HAO is significantly different from the other five algorithms in most of the benchmark functions, which further verifies the effectiveness of HAO.

Table 7.

Comparison of Wilcoxon rank sum test results.

Function	PSO	GWO	WOA	CSO	AO
f1	1.2118E–12	1.2118E–12	1.2118E–12	1.2118E–12	1.2118E–12
f2	3.0199E–11	3.0199E–11	3.0199E–11	3.0199E–11	3.0199E–11
f3	1.2118E–12	1.2118E–12	1.2118E–12	1.2118E–12	1.2118E–12
f4	3.0199E–11	3.0199E–11	3.0199E–11	3.0199E–11	3.0199E–11
f5	3.0199E–11	3.0199E–11	3.0199E–11	3.0199E–11	3.6897E–11
f6	1.2118E–12	1.2118E–12	1.2118E–12	6.0767E–06	1.2118E–12
f7	3.0199E–11	2.8314E–08	9.8329E–08	7.1186E–09	8.4180E–01
f8	1.5394E–11	1.5394E–11	1.5394E–11	1.3160E–01	1.5394E–11
f9	1.2118E–12	4.1926E–02	3.3371E–01	NaN	3.3371E–01
f10	1.2118E–12	1.1772E–13	1.2328E–09	5.0193E–13	NaN
f11	1.2118E–12	4.7884E–08	1.4552E–04	4.5736E–12	NaN
f12	4.1040E–11	1.7203E–12	1.7203E–12	3.3371E–01	1.7203E–12
f13	9.4001E–12	9.4001E–12	9.4001E–12	1.2785E–01	9.4001E–12
f14	1.6585E–08	1.2118E–12	1.2118E–12	3.1282E–04	1.2118E–12
f15	3.0199E–11	5.5727E–10	8.1527E–11	7.3891E–11	3.0199E–11
f16	NaN	5.5727E–12	1.2118E–12	5.7720E–11	1.2118E–12
f17	2.4453E–01	1.0752E–11	1.0752E–11	1.9813E–11	1.0752E–11
f18	5.0362E–12	7.0127E–02	2.5721E–07	2.5721E–07	6.5277E–08
f19	1.2238E–03	9.8834E–03	3.3285E–01	3.4029E–01	2.0283E–07
f20	2.1412E–06	7.5741E–12	7.5741E–12	1.8940E–03	7.5741E–12
f21	2.6037E–02	3.0199E–11	3.0199E–11	6.7362E–06	3.0199E–11

PSO: particle swarm optimization; GWO: gray wolf optimization; WOA: whale optimization algorithm; CSO: crisscross optimization; AO: Aquila optimizer; HAO: hybrid Aquila optimizer.

Performance analysis of HAO in different dimensional case

In experiment 3, to verify the optimization performance of HAO on different dimensional benchmark functions, run six algorithms independently 30 times on 50 and 500 dimensional benchmark functions f1–f13, respectively. The experimental results are shown in Tables 8 and 9. Combining the analysis of the experimental results on three different dimensions shows that the mean and standard deviation of PSO, GWO, WOA, and CSO on most of the functions become significantly larger as the increase of the dimension of the benchmark functions, which indicates that these four algorithms have poor optimization performance on high-dimensional benchmark functions. The average value and standard deviation of AO and HAO on most functions do not change much, indicating that AO and HAO can still maintain the overall optimization performance on the high-dimensional benchmark function.

Table 8.

Comparison of optimization results on 50 dimensional benchmark functions.

Function		PSO	GWO	WOA	CSO	AO	HAO
	Mean	2.0948E–01	6.8628E–20	3.8569E–71	4.8986E–03	4.2239E–99	0
f1	Std	3.1988E–01	1.1673E–19	1.7886E–70	6.4381E–03	2.3135E–98	0
	Rank	6	4	3	5	2	1
	Mean	1.5219E+00	2.5770E–12	1.1757E–50	1.3185E–03	1.3643E–53	1.7479E–134
f2	Std	7.2707E–01	1.3347E–12	3.2791E–50	6.1166E–04	7.3508E–53	7.1397E–134
	Rank	6	4	3	5	2	1
	Mean	1.5951E+03	3.2148E–01	1.9848E+05	6.7363E+03	3.2341E–110	0
f3	Std	3.4931E+02	8.8080E–01	3.6585E+04	3.2202E+03	1.7714E–109	0
	Rank	4	3	6	5	2	1
	Mean	3.6010E+00	3.7260E–04	6.5350E+01	5.2970E+00	3.8626E–58	1.9969E–137
f4	Std	7.2392E–01	2.8417E–04	2.4662E+01	1.0411E+00	2.1156E–57	1.0586E–136
	Rank	4	3	6	5	2	1
	Mean	3.9330E02	4.7498E01	4.8126E01	1.5534E01	1.1893E–02	1.1628E–04
f5	Std	2.1136E02	8.7569E–01	3.9295E–01	5.8130E01	1.9115E–02	4.8457E–04
	Rank	6	4	3	5	2	1
	Mean	1.7966E–01	2.7979E+00	1.2694E+00	2.9653E–04	6.9104E–04	1.9807E–10
f6	Std	1.1945E–01	7.8736E–01	5.5244E–01	3.3389E–04	2.5377E–03	4.4102E–10
	Rank	4	6	5	2	3	1
	Mean	3.7847E+00	3.1036E–03	3.3304E–03	1.6125E–02	1.0467E–04	8.8959E–05
f7	Std	6.6120E+00	1.4956E–03	3.3403E–03	6.9563E–03	8.5919E–05	8.0314E–05
	Rank	6	3	4	5	2	1
	Mean	−7.0149E03	−9.0324E03	−1.6612E04	–1.8753E04	−1.0798E04	−1.5464E04
f8	Std	2.6780E03	1.4617E03	2.9353E03	8.412E03	6.2361E03	4.1692E03
	Rank	6	5	2	1	4	3
	Mean	1.4023E+02	5.7363E+00	1.8948E–15	4.3943E+00	0	6.0208E–09
f9	Std	2.7759E+01	6.2258E+00	1.0378E–14	2.3992E+00	0	1.8331E–08
	Rank	6	5	2	4	1	3
	Mean	1.5093E+00	3.3378E–11	4.4409E–15	2.5705E–03	8.8818E–16	8.8818E–16
f10	Std	4.5517E–01	1.8135E–11	2.2853E–15	1.3266E–03	0	0
	Rank	5	3	2	4	1	1
	Mean	1.3283E–02	3.7007E–17	1.0097E–02	1.6444E–01	0	0
f11	Std	1.2205E–02	6.0693E–17	5.5301E–02	2.7310E–01	0	0
	Rank	4	2	3	5	1	1
	Mean	1.4777E–01	9.6263E–02	3.3112E–02	1.5302E–06	2.3495E–06	7.4301E–18
f12	Std	2.6569E–01	3.4579E–02	1.7974E–02	1.6053E–06	6.0522E–06	3.0489E–12
	Rank	6	5	4	3	2	1
	Mean	0.1.7502E–01	2.2272E00	1.3329E00	9.3887E–04	2.4835E–05	3.2271E–09
f13	Std	0.1.0854E–01	3.4600E–01	3.8851E–01	1.8787E–03	2.7552E–05	1.3976E–08
	Rank	4	6	5	3	2	1
Total Rank		0	0	0	1	3	11

PSO: particle swarm optimization; GWO: gray wolf optimization; WOA: whale optimization algorithm; CSO: crisscross optimization; AO: Aquila optimizer; HAO: hybrid Aquila optimizer.

Table 9.

Comparison of optimization results on 500 dimensional benchmark functions.

Function		PSO	GWO	WOA	CSO	AO	HAO
	Mean	5.8686E03	1.6696E–03	1.6052E–69	3.0005E04	2.2383E–108	0
f1	Std	4.2839E02	6.2170E–04	6.5906E–69	7.9219E03	1.2260E–107	0
	Rank	5	4	3	6	2	1
	Mean	1.7881E+44	1.0832E–02	5.3855E–49	2.5193E+02	6.2900E–57	8.4639E–128
f2	Std	9.7937E+44	2.0349E–03	1.1340E–48	2.4267E+01	3.4452E–56	4.6359E–127
	Rank	6	4	3	5	2	1
	Mean	5.6269E+05	2.9100E+05	3.2081E+07	8.2629E+05	2.9194E–96	0
f3	Std	1.5710E+05	7.9076E+04	9.8794E+06	3.3651E+05	1.5990E–95	0
	Rank	4	3	6	5	2	1
	Mean	2.7756E+01	6.6067E+01	8.1636E+01	4.3961E+01	4.1475E–52	2.2593E–126
f4	Std	1.5426E+00	6.4000E+00	1.9013E+01	5.4512E+00	2.1480E–51	1.2374E–125
	Rank	3	5	6	4	2	1
	Mean	2.9132E07	4.9813E02	4.9638E02	4.8846E05	8.0167E–02	1.3662E–05
f5	Std	3.9088E06	2.3990E–01	4.1589E–01	2.44428E05	1.4199E–01	7.0697E–05
	Rank	6	3	4	5	2	1
	Mean	5.8886E+03	9.1269E+01	3.1404E+01	2.3372E+04	8.0731E–04	1.1576E–06
f6	Std	4.1303E+02	2.5331E+00	7.2306E+00	5.3297E+03	1.2739E–03	3.0323E–06
	Rank	5	4	3	6	2	1
	Mean	5.7569E+04	5.0034E–02	4.5008E–03	4.1589E+01	9.7108E–05	9.8601E–05
f7	Std	2.7223E+03	1.3939E–02	3.7044E–03	1.1904E+01	9.3425E–05	7.2433E–05
	Rank	6	4	3	5	2	1
	Mean	−2.2675E04	−5.7899E04	–1.6972E05	−9.7092E04	−4.2894E04	−8.8085E04
f8	Std	7.6417E03	3.4258E03	2.6207E04	1.2402E04	1.5548E04	2.0864E04
	Rank	6	4	1	2	5	3
	Mean	6.2613E+03	7.1059E+01	0	1.4378E+03	6.0633E–14	1.8339E–03
f9	Std	2.4205E+02	3.0201E+01	0	1.1318E+02	2.3075E–13	4.0466E–03
	Rank	6	4	1	5	2	3
	Mean	1.2008E+01	2.0000E–03	4.3225E–15	8.9671E+00	8.8818E–16	8.8818E–16
f10	Std	3.7817E–01	2.6480E–04	2.7174E–15	7.6800E–01	0	0
	Rank	5	3	2	4	1	1
	Mean	7.7025E+01	6.2896E–03	0	2.1065E+02	0	0
f11	Std	7.2690E+00	2.3283E–02	0	5.4835E+01	0	0
	Rank	3	2	1	4	1	1
	Mean	2.3909E+05	7.5548E–01	9.8903E–02	5.1356E+01	8.3723E–07	1.0864E–11
f12	Std	1.1622E+05	5.1671E–02	4.4027E–02	1.5731E+02	1.4382E–06	5.6561E–11
	Rank	6	4	3	5	2	1
	Mean	4.2657E56	5.0511E01	1.6899E01	8.69245E00	4.6837E–04	2.6055E–07
f13	Std	7.4123E05	1.2815E00	5.2504E00	8.5495E05	6.7211E–04	9.3870E–07
	Rank	6	4	3	5	2	1
Total Rank		0	0	3	0	2	11

PSO: particle swarm optimization; GWO: gray wolf optimization; WOA: whale optimization algorithm; CSO: crisscross optimization; AO: Aquila optimizer; HAO: hybrid Aquila optimizer.

Figure 4 shows the partial convergence curve, and the analysis combined with Figure 3 shows that the convergence performance of HAO on the benchmark functions f1, f5, and f13 can still maintain the advantage as the dimensions increase, and the convergence performance on f8 is inferior to that of WOA and CSO. Overall, the test results of different dimensional benchmark functions in low and high dimensions show that HAO performance has a significant advantage over the remaining five algorithms.

Figure 4.

Comparison of convergence curves of benchmark functions of different dimensions: (a) f1 (50D), (b) f1 (500D), (c) f5 (50D),(d) f5 (500D), (e) f8 (50D), (f) f8 (500D), (g) f13 (50D), and (h) f13 (500D).

Application of HAO in K-means clustering optimization

Clustering is to partition a data set into different classes or clusters according to some specific criteria, so that the similarity of data objects within the same cluster is as large as possible, while the difference of data objects not in the same cluster is as large as possible (Berkhin, 2006). Among the common clustering algorithms, K-means clustering algorithm has the advantages of simple principle and high efficiency, but it also has the disadvantages of being sensitive to initial points and easily trapped in local optima. In recent years, methods to improve K-means using swarm intelligence optimization algorithms have been proposed with good results (Das et al., 2018; Li et al., 2015).

In this paper, the K-means clustering problem is viewed as a minimization problem to solve the objective function $E$ . Each Aquila individual represents a candidate cluster center and then the optimal individual corresponding to the cluster center produces the smallest distance and the best clustering effect. The objective function $E$ of K-means clustering is as follows:

E = \sum_{k = 1}^{K} \sum_{X_{i} \in C_{k}} ‖ X_{i} - C_{k} ‖

(22)

where $E$ is the sum of Euclidean distances from all data points to the cluster centers to which they belong, $K$ is the number of cluster centers, $C_{k}$ is the $k$ th cluster center, and $X_{i}$ is the data points in the cluster to which the cluster center $C_{k}$ belongs.

This paper used six University of California (UCI) data sets to verify the effectiveness of HAO on K-means clustering optimization. Aquila optimizer, improved genetic algorithm (IGA), interactive particle swarm optimization (IPSO), and improved differential evolution (IDE) were selected as the comparison algorithms, and the number of populations was set to 30, the maximum number of iterations was 500, and each algorithm run 20 times independently. To avoid the situation that some attribute characteristics are ignored due to the large difference in order of magnitude between different dimensions in the data set, the data set is normalized before the start of the experiment.

Table 10 below shows the mean, standard deviation and average accuracy of clustering for the optimal individual fitness obtained after 20 experiments run independently by six algorithms. The algorithm ranked first in each evaluating indicator in the table has been bolded, and the number of samples, attributes, and classes are shown in parentheses for each data set in the table. Analyzing the data in the table, we can see that HAO achieves the first ranking six times in terms of mean, five times in terms of standard deviation, and four times in terms of accuracy. These results show that the clustering accuracy, stability, and accuracy of HAO are significantly ahead of the comparison algorithms. In summary, HAO is effective and superior in K-means clustering optimization problems.

Table 10.

Comparison of the effect of HAO on the application of K-means clustering optimization.

DS		IGA	IPSO	IDE	AO	HAO
Seeds (210 3 7)	Mean	63.7365	64.8621	65.1324	86.2053	63.5458
	Std	2.5334E00	5.2310E00	1.7029E00	5.1216E00	3.14709E–01
	Acc	0.8736	0.8623	0.8902	0.7274	0.8919
Wine (178 3 13)	Mean	90.0020	91.3998	91.8106	119.2307	89.6162
	Std	4.3052E00	5.8032E00	3.8841E00	4.9085E00	5.9604E–01
	Acc	0.9202	0.8907	0.9371	0.6837	0.9508
Vehicle (846 4 18)	Mean	468.9091	466.1243	507.1425	690.0057	453.9239
	Std	2.2538E01	2.5999E01	2.7470E01	2.6837E01	1.2288E01
	Acc	0.3624	0.3676	0.3702	0.34687	0.3700
Glass (214 6 9)	Mean	56.9865	56.8005	58.4445	89.9613	55.2349
	Std	3.7750E00	4.1539E00	3.5255E00	3.4644E00	2.6544E00
	Acc	0.3185	0.3229	0.3023	0.3591	0.3063
Aggregation (788 7 2)	Mean	93.2772	93.4795	95.7088	106.4229	92.7976
	Std	1.6310E00	2.2044E00	8.1238E–01	4.2774E00	1.2708E00
	Acc	0.7549	0.7556	0.7782	0.7602	0.8006
Ecoli (336 8 8)	Mean	83.4898	85.3458	84.7650	130.7205	78.7379
	Std	7.7662E00	8.4997E00	2.4335E00	1.2553E01	2.3555E00
	Acc	0.7417	0.7387	0.65818	0.57936	0.69856

AO: Aquila optimizer; HAO: hybrid Aquila optimizer.

Conclusion

In this paper, a hybrid Aquila optimizer (HAO) algorithm based on Gauss map and crisscross operator is proposed to solve the problems of slow convergence speed, low convergence accuracy, easy to fall into local optimum and other disadvantages of AO algorithm in dealing with certain complex optimization problems. The Gauss map is used to initialize the population so that the initial population is uniformly spread throughout the search space. At the same time, to avoid the situation that the algorithm falls into local optimum due to the reduction of population diversity during the iterative process, the crisscross operator is used to search for high-quality individuals.

The HAO algorithm is compared with the basic AO and four representative swarm intelligence optimization algorithms on 21 benchmark functions. Analysis of the experimental results show that the improved strategy proposed in this paper is effective. HAO ranked first 20 times when dealing with low-dimensional function optimization problems, and 11 times when optimizing 13 high-dimensional benchmark functions. This indicates that HAO can show excellent optimization performance in solving both low-dimensional and high-dimensional function optimization problems. Finally, the experimental results of the practical application of HAO characterize that HAO can effectively deal with the K-means clustering optimization problem. The superiority of HAO makes it applicable to many scenarios, including trajectory planning in robotics, feature selection in data processing, resource scheduling and allocation in energy, intelligent manufacturing in industry, and disease detection in medicine.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Cheng Huang

Jinglin Huang

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