Multi-strategy improved parallel antlion algorithm and applied to feature selection

Abstract

Antlion Optimization Algorithm (ALO) is a promising bionic swarm intelligence algorithm, which has good robustness and convergence, but there are still many areas to be improved and modified. Aiming at the fact that the ALO algorithm is more likely to fall into the local optimum, proposes three strategies to improve the classic ALO algorithm in this paper. First of all, we adopt a parallel idea in the algorithm, through the communication strategy between groups based on Quantum-Behaved to enhance the diversity of the population. Secondly, we adopted two strategies, Opposition Learning, and Gaussian Mutation, to balance the performance of exploration and exploitation during the execution of the algorithm, further formed the MSALO algorithm. The CEC2013 Benchmark function is selected as the standard, and MSALO is compared with other intelligent optimization algorithms. The experimental results show that MSALO has stronger optimization performance compared with other intelligent algorithms. Besides, we applied MSALO to the practical scenarios of feature selection, and use SVM classifiers as training evaluators to improve the accuracy of feature extraction from high-dimensional data.

Keywords

ALO quantum-behaved opposite learning gaussian mutation feature selection

1 Introduction

Optimization refers to the process of finding the best value for some variables or combination schemes of a specific problem based on certain constraints so that a certain performance of the problem can be optimized [38]. In our daily life and work, we will encounter various optimization problems, which are widely present in the fields of mathematical science, intelligent computing, industrial control, distribution and scheduling, and play a central role. Intelligent Computing is divided into three branches: Artificial Neural Network(ANN), Genetic Algorithm(GA) [40, 41], and Fuzzy [51], reasonable optimization algorithms have an increasing influence on intelligent computing. Optimization algorithms can be divided into Exact algorithms and Heuristic algorithms [31], where Exact algorithms such as branch and bound algorithm, background segmentation algorithm, dynamic programming, and other computing more complex, only suitable for small-scale problem solving; Heuristic [26] algorithms are divided into meta-heuristic and specific. Among them, meta-heuristic has individual-based algorithms, such as Simulated Annealing Algorithm (SA) [13], which applies the physical process of solid heating annealing to the algorithm; Tabu Search Algorithm (TS) [19 , 30], which uses flexible memory technology to choose. There are also population-based algorithms, such as Genetic Algorithm (GA) [36], which refers to the mechanism of natural selection and biological genetic evolution; Particle Swarm Optimization (PSO) [18 , 57], which simulates the behavior of migration and clustering in the predation process of birds [12, 32]; Ant Colony Algorithm (AC) [11 , 43], which simulates the behavior of path marking when ants foraging [37]; Differential evolution algorithm (DE) [27] uses the principle of cross mutation in genetics. With the advancement of science and technology, more and more complex high-dimensional optimization problems will be encountered in many modern disciplines and scientific fields. When we use traditional optimization methods to solve, there will always be problems such as high complexity, a large amount of calculation, long calculation time, etc. Especially for some difficult and nonlinear combinatorial optimization problems. Traditional methods need to traverse all search solutions in the optimization process, which further increases the search range and optimization difficulty. Therefore, the improvement of algorithms has become a hot topic in society, and look for more efficient optimization method is particularly important during the development of a variety of algorithms. In recent decades, optimization algorithms have received extensive attention from all walks of life. To solve more complex optimization problems and real scenarios. Research scholars based on the existing classic optimization evolutionary algorithm [5 , 53], combined with a variety of innovative strategies to synchronize the improvement of the algorithm, such as Parallel Strategy [20, 35], Adaptive Parameter Technology [3 , 33], Opposite Learning Strategy, Neural Network Technology [56], Neighborhood Operation [6 , 48], Chaos Mechanism, QUATRE [15 , 34], etc. Combine these strategies organically, learn from each other’s concepts, enabling the algorithm to achieve a more efficient global optimization.

The ALO algorithm is inspired by the foraging behavior of antlions and obtains the global optimal solution by simulating the trap mechanism of antlions. ALO was first proposed by Australian scholar Mirjalili in 2015. As a meta-heuristic algorithm, because of its better robustness, convergence, and fewer adjustable parameters, and is not bound by the object, at this point, many scholars have proposed a variety of improvement strategies and applied the improved algorithms to multiple fields. For example, Zhang and others applied the Antlion algorithm to the problem of air material configuration optimization; Zhao et al. optimized the SVM parameters with the improved ALO [55]; Chen et al. Applied the ALO algorithm to the optimization of PID controller parameters; Xu and others applied ALO improved by the hybrid strategy to the optimization of wireless sensor network coverage. First of all, we divide the population into several groups using parallelism and communicate between groups based on based on quantum-behaved. Besides, we combine The Opposition Learning Strategy and The Gaussian Mutation Strategy [7] to improve the Antlion algorithm to form MSALO. We selected 28 test functions of CEC2013 as the standard, including single-mode functions, basic multi-mode functions, to test the performance of the algorithm, and compared with other algorithms in terms of optimization capabilities. Experiments show that MSALO is superior to other algorithms in optimization ability, and is superior to the original ALO.

Feature Selection [14 , 52] is an important method for dimensionality reduction of high-dimensional big data. It is a process of screening data features and obtains a set of optimal feature subsets by reducing irrelevant features and redundant features, And this feature subset does not lose important data but preserves the true meaning of the data set. We use the wrapper method combined with our MSALO algorithm for feature selection and specify the SVM classifier as our learning algorithm to train the evaluator on the initial feature set, to select a subset of features that can achieve high performance for real-world application problems, to reduce data dimensions and improve algorithm optimization efficiency without affecting data accuracy.

Section II will briefly introduce several optimization algorithms, such as PSO and the original ALO algorithm, and our strategy to improve ALO. Section III introduces MSALO in detail and how to apply it to feature selection. The results of The Section IV performance demonstrates MSALO CEC2013 Benchmark suite, and compared with other algorithms. Finally, in Section V, the experimental results of the comparison and analysis of the fourth quarter, and draw conclusions.

2 Related works

2.1 ALO algorithm

2.1.1 Ants walk randomly and fall into traps

ALO algorithm [2] simulates the trap mechanism used by antlions when preying. The antlion will dig a cone-shaped sandpit, then hide at the bottom of the sandpit and wait for the ants. When the ant falls into the sandpit, it will slide to the bottom of the pit. Recorrect the pit to prepare for the next hunting. Among them, the ant represents the attempted solution; the antlion represents the local optimal solution; the elite antlion represents the global optimal solution. Ants will walk randomly within the search range, and the formula for moving is $\begin{matrix} X (t) = [0, cumsum (2 r (t_{1}) - 1), cumsum (2 r (t_{2} -) \\ - 1), \dots, cumsum (2 r (t_{n}) - 1)], \end{matrix}$ (1) Where r (t) ∈[0,1], cumsum is a matrix column accumulation function, and n is the number of ant population, t is the current numbe of iteration. In addition, to ensure that the updated position of the ant does not exceed the search space, we use Equ.(2) to normalize the position. $X_{i}^{t} = \frac{(X_{i}^{t} - a_{i}) \times (d_{i}^{t} - c_{i}^{t})}{(b_{i} - a_{i})} + c_{i}^{t},$ (2) Where a_i and b_i represent the maximum and minimum valves of the i-th ant, and $c_{i}^{t}$ and $d_{i}^{t}$ represent the maximum and minimum valves of the ant in the t-th iteration.

2.1.2 AntLion builds trap to prey on ants

Antlions will build traps proportional to their fitness. We use mathematical methods to simulate the process of ants sliding down the trap, and adaptively reduce the radius of the trap, that is, the swimming radius of the ant. $c^{t} = \frac{c^{t}}{I}, d^{t} = \frac{d^{t}}{I},$ (3) $I = 10^{ω} \frac{t}{T},$ (4) $ω = {\begin{matrix} 1, & t \leq 0.1 T \\ 2, & 0.1 T < t \leq 0.5 T \\ 3, & 0.5 T < t \leq 0.75 T \\ 4, & 0.75 T < t \leq 0.9 T \\ 5, & 0.9 T < t \leq 0.95 T \\ 6, & t \geq 0.95 T \end{matrix}$ (5) Where c^t and d^t represent the minimum and maximum values of all variables in the t-th iteration, respectively. The value of I is as Equation (4). As the antlion’s trap changes, the random walk of the surrounding ants will also be affected, as shown in Equ.(6). $c_{i}^{t} = {Antlion}_{j}^{t} + c^{t}, d_{i}^{t} = {Antlion}_{j}^{t} + d^{t},$ (6) Where c^t and d^t respectively represent the minimum and maximum values of all variables at the t-th iteration, and $c_{i}^{t}$ , $d_{i}^{t}$ respectively represent the minimum and maximum values of the i-th ant at the t-th iteration. ${Antlion}_{j}^{t}$ represents the position of the j-th ant in the t-th iteration. When the ant has a higher adaptability than its corresponding antlion, it means that the antlion has successfully preyed on the ant, and the position of the ant hunted is updated to the latest position of the antlion to facilitate the next hunting. ${Antlion}_{j}^{t} = {Ant}_{i}^{t}, iff ({Ant}_{i}^{j}) > f ({Antlion}_{j}^{t}),$ (7) In addition, ALO algorithm saves the best antlion obtained in each iteration as an Elite Antlion. When the ants are around the trap, they randomly walk according to the roulette antlion and the Elite Antlion. Where $R_{A}^{t}$ represents the ant lion chosen by roulette, $R_{E}^{t}$ represents the elite antlion. ${Ant}_{i}^{t} = \frac{R_{A}^{t} + R_{E}^{t}}{2}$ (8)

2.2 Quantum-behaved

The Individual learning process is similar to the Quantum-Behaved [9], and every particle in quantum space can simulate an individual. According to the Aggregation properties of swarm intelligence and the movement of particles in quantum space, we know that particles will constantly approach the point p_i. There is potential energy at point pi that attracts the particles to reach a bound state, which is the particle in the bound state. There will be a certain probability to appear at any point in space so that it can search in quantum space under the attraction of potential energy. Based on Quantum behavior, the evolution equation is derived by the Monte Carlo method as follows, ${Apbest}_{i} = \frac{1}{M} \times \sum_{t = 1}^{M} {Pbest}_{i}^{t},$ (9) Where ${pbest}_{i}^{t}$ is the optimal value searched by the i-th particle in the t-th iteration, representing individual experience. M is the number of current iterations, and Apbest_i represents the optimal average value of the i-th individual performing the current M iterations. $α = \frac{0.5 + 0.5 \times T - t}{T},$ (10) Where T is the maximum number of iterations, t is the current number of iterations, and α represents a parameter of quantum behavior communication learning. ${\begin{matrix} P_{i} = φ \times P_{id} + (1 - φ) \times P_{gd}, \\ φ = \frac{c_{1} r_{1}}{c_{1} r_{1} + c_{2} r_{2}}, \end{matrix}$ (11) Here p_i is the point with attractive potential energy, where φ is an inertia weight coefficient. The best solution that the i-th particle remembers by itself is recorded as p_id; the best position the entire population has experienced, which is the optimal solution currently searched, denoted as p_gd.

In MSALO, we use the idea of quantum-behaved. Every 11 iterations of the algorithm, we select several better ones at the same time Point, use Equ.(12) to update the better antlion, and select a few weak points from the antlion population, use Equ.(13) to perform the position of the poor antlion. The difference between Equ.(12) and Equ.(13) is that Equ.(12) needs to judge the updated antlion after updating the better antlion. Whether the antlion is more than the antlion before the update, if we can make the better antlion better, we will replace it. while Equ.(13) directly updates the position to make the poor antlion better [9]; This is a greedy strategy in which based on quantum-behaved, the evolution equation is derived by Monte Carlo method as follows, ${\begin{matrix} X_{inew} = P_{i} \pm α \times | X_{i} - {Abest}_{i} | \times \ln \frac{1}{u}, \\ X_{i} = X_{inew}, if f (X_{inew}) < f (X)_{i}, \end{matrix}$ (12) ${\begin{matrix} X_{inew} = P_{i} \pm α \times | X_{i} - {Abest}_{i} | \times \ln \frac{1}{u}, \\ X_{i} = X_{inew}, \end{matrix}$ (13) Where X_i is the position of the current individual, X_inew is the updated position, A is an average value of the i-th individual’s historical and historical best, and u is a random number from 0 to 1.

2.3 Quasi-opposing learning

When the optimization algorithm is performing optimization, the random solution in the search space may be far from the optimal solution, and it takes a long time to converge. Quasi-Opposite Learning [44] is an optimization strategy that increases the diversity of the population by solving the opposite solution of the current solution, and then selects the best of the population and the opposite population. ${\begin{matrix} {ox}_{i} = (ub + lb) - x_{i}, \\ {qox}_{i} = rand (\frac{ub + lb}{2}, {ox}_{i}), \end{matrix}$ (14) Where ox_i is the opposite solution of the i-th ant; ub, lb is the upper and lower bounds of the feasible region, qox_irepresents the quasi-opposite individual of the i-th ant, and rand represents a random number between $\frac{ub + lb}{2}$ and ox_i.

2.4 Gaussian mutation

Gaussian Mutation [17] is to add the disturbance term of the normal distribution to the original solution vector. During the mutation, the direction is guided by the change of the elite antlion, which not only increases the diversity of the population but does not affect the local development.

We select an improvement rate γ as the standard, select the best fitness antlion of the current iteration minus the best fitness antlion of the previous tenth iteration, and divide the absolute value of the difference by the best fitness of the current iteration, When the value is less than the improvement rate γ, the gaussian mutation is performed, and the mutation formula is as Equ.(15). ${\begin{matrix} if \frac{| fitness [g_{best} (t)] - fitness [g_{best} (t - 10)] |}{fitness [g_{best} (t)]} \leq γ, \\ g_{best} (t) = g_{best} (t) + β N (0, 1) \end{matrix}$ (15) Where β is the coefficient of variation, we take the golden section value β=0.618, and N (0, 1) is the standard normal distribution.

2.5 Feature selection

In the data to be processed in real life, an object has multiple attributes, and data with too many attributes will cause dimensionality disasters when applied. The characteristics of the data can be divided into related features, irrelevant features, and redundant features. The factors that cause dimensional disasters are irrelevant features and redundant features. Feature selection is very important in the field of machine learning and big data.

In addition, another reason for feature selection is: when we need a large number of features, too many or too few features will have a great impact. When the features are insufficient, data overlap is prone to occur, in which case any classifier will fail. As shown in Fig. 1, relying only on the x-axis or y-axis cannot distinguish the two types of data. Adding features can be understood as mapping to a high-dimensional space. When this "dimensionality" is too high, it is easy to cause similar data to be far away in the space and become sparse, which also easily makes many classification algorithms invalid. As shown in Fig. 1, only the x-axis can be divided into features, but the introduction of the y-axis makes the same category no longer clustered. Furthermore, we know that there is a method similar to feature selection, that is, data dimensionality reduction. The effect of the two is the same, that is, to reduce the features in the feature data set. The difference between simple dimensionality reduction and feature selection is that the former is combined through the linear relationship between multiple features to become a new feature, which changes the space of the original feature set; while The latter is to select beneficial features as feature subset. The advantage is to remove features that don’t affect the results of the learning algorithm, do not change the space of the original feature data set, and maintain the authenticity and validity of the feature data set.

Fig. 1

Two phenomena caused by the number of features.

Fig. 2 shows the four processes of feature selection. First, the Generation process is to search for combined data to generate feature subsets; The second is the Evaluation function, which evaluates the generated feature subsets; The third is the Stop condition to determine the stop of feature selection; The fourth is the Verification process to verify whether the selected feature subset is valid.

Fig. 2

Feature selection process.

In this paper, we use the Wrapper and select the SVM classifier for feature selection.

3 The MSALO proposed in this paper and the application in feature selection

3.1 MSALO algorithm

In terms of improving the ALO algorithm, we first adopted a parallel idea, dividing ants and antlions into groups for optimization, and comparing and exchanging information between groups based on quantum-behaved in the communication between groups. Besides, we use two strategies, opposition learning, and Gaussian mutation, to increase the diversity of the population and prevent the algorithm from stagnating to the local optimum. The steps of the algorithm are as follows: The first step is to initialize variables and population; In the second step, the ant go through the process from Equ.(1) to Equ.(8), but the process of ant learning from the opposite is added; The third step is to iterate continuously, every 11 iterations will find the global optimal value, and use the communication strategy based on quantum-behaved to update the Antlion position. Among them, if there is little change between the current optimal iteration and the first ten iterations optimal, that is, when Equ.(15) is less than the improvement rate, Gaussian mutation is performed.

Algorithm 1 Shows the Pseudo Code of MSALO

Algorithm.

1: //Initialization

Initialize the values of the maximum ub and minimum lb of

the ant search space boundary,the total population ps,the

number of grouping groups gs,Inter-group communication

step cs,the dimension D of the population search space,the

maximum number of iterations T_max and the current

number of iterations T are 1.

2: //MainLoop

subps = ps/gs; //divide the ant and antlion populations

into gs groups

3: Randomly initialize the positions of ants and antlions,

evaluate the fitness value of antlions, and sort and select

the elite antlions and the global optimal antlions of

each group.

4: while (T < T_max) do

5: if (T%cs == 0) then //Every cs iterations, the

communication strategy based on quantum behavior will

communicate in each group of search solutions

6: for (g = 1;g < = gs;g + +) do

7: /*step 1*/

8: Select a few poor position in the population to

communicate between groups and update with Equ.(12).

9: /*step 2*/

10: Select a few better position in the population to

communicate between groups and update with Equ.(13).

11: end for

12: end if

13: for (g = 1;g < = gs;g + +) do

14: /*Random walk phase of ants*/

15: Affected by roulette antlion and Elite antlion, the

random walk of ants is simulated by Equ.(8) and the ant

of the opposition learning Equ.(14) update and normalized

by Equ.(2).

16: /*Ants in the trap stage*/

17: Use Equ.(3) to Equ.(6) to simulate

ants falling into a trap and adjust the size of the trap

18: /*Antlion eats ants stage*/

19: In Equ.(7), when the ant becomes stronger

than the antlion, the ant is eaten, and the antlion position

has been updated to the current ant position

20: /*Select the Elite antlion stage*/

21: Each group chooses the best as the Elite antlion in

the group

22: Compare each group of elite antlions with the global

elite antlions, and select the global elite antlions

23: end for

24: end while

3.2 The application of MSALO in feature selection

In this paper, we choose Wrapper for feature selection, use LIBSVM [1, 4] as the evaluation criterion for feature subsets, use recursive feature elimination (RFECV) to evolve the training set and test set in SVM [16], and combine the optimization process of MSALO to select The best feature subset.

3.2.1 SVM and LIBSVM

SVM is essentially a two-class model, which can not only learn the characteristics of known samples but also predict unknown samples. Its basic principle is to find an optimal classification hyperplane in the n-dimensional data space. The hyperplane equation is $ω^{T} x + b = 0$ (16) Where ω is the weight vector and b is the classification threshold. As shown in Fig. 3, there are two different types of data: a Pentagram and a circle. You can use a straight line to separate the two types of data. This straight line is equivalent to a hyperplane, we can use the classification function f (x) = w^Tx + b means. And some points closest to the hyperplane are Support

Fig. 3

Two-dimensional classification example.

Vectors, which determine the position of the line.

When f (x) is equal to 0, x is a point on the hyperplane, and the point with f (x) greater than 0 corresponds to the data class represented by the circle, and the point with f (x) less than 0 corresponds to the data class represented by the Pentagram.

We need to find the optimal value of weight ω and threshold b to minimize the weight cost function Equ.(17). $min φ (x) = \frac{1}{2} ‖ ω ‖^{2},$ (17)

For the problem of linear inseparability, SVM maps the samples in the n-dimensional space to the higher-dimensional space, so that the samples in the high-dimensional space are linearly separable. Choose different kernel functions according to the nature of the problem and the size of the data.

In the paper, we choose a simple, easy-to-use, fast, and effective software package LIBSVM developed and designed by Professor Lin for SVM pattern recognition and regression, and it provides many default parameters and can interact. Inspection is currently the most widely used SVM library in China. The definition of the RBF kernel function we used is as follows. $K (x, y) = \exp (- y ‖ x - y ‖^{2}), y > 0$ (18)

3.2.2 The main steps of MSALO for feature selection

In this article, we choose Wrapper for feature selection, use LIBSVM as the feature subset evaluation function, use recursive feature elimination (RFECV) to evolve the training set and test set in SVM, and combine the optimization process of MSALO to select The best feature subset. The main steps of MSALO applied to feature selection are as follows and the detailed description is shown in Fig. 4.

Fig. 4

Flowchart of feature selection by MSALO.

Fig. 5

omparison of the best ?tness for functions f₁ - f₄ with 30D optimization.

Fig. 6

Comparison of the best ?tness for functions f₅ - f₂₈ with 30D optimization.

Step1, Initialization parameters, and population;

Step2, Normalize the data set and divide the data set into 10 points, of which 1/10 is used as the test set;

Step3, Use libsvm for 10 times of training, and verify on the test set, and calculate the error rate of all subsets through LIBSVM;

Step4, Use MSALO to optimize the feature subset and find the feature subset with the smallest error.

4 Experimental analysis

In this section, we choose CEC2013 BenchMark [23] to test the performance of our improved algorithm and introduce the results of the MSALO application in feature selection.

4.1 Simulation results of CEC2013 benchmark test of MSALO

CEC2013 BenchMark contains 28 benchmark functions, including 5 Unimodal Functions(f₁ - f₅), 15 Basic Multimodal Functions(f₆ - f₂₀), and 8 Composition Functions(f₂₁ - f₂₈). In order to further verify its performance, we compare the proposed minimum value of MSALO with ALO, PSO, PPSO, SOS [9], BA. The spatial dimension of these algorithms for optimization is 30, the search range is [-100, 100], and the parallel algorithms that need to be grouped are divided into 4 groups.

As shown in Table 1, it is the Mean and Standard deviation of the optimal value result of each algorithm compared in the CEC2013 benchmark function test, where the value in bold is the minimum fitness value for the comparison of the six algorithms. The MSALO algorithm is better than other algorithms in 13 benchmark functions of f₆, f₈, f₉, f₁₁, f₁₄, f₁₅, f₁₇, f₁₈, f₁₉, f₂₂, f₂₃, f₂₅, and f₂₈, and can reach the optimal fitness value; The ALO algorithm is tested in the 5 benchmark functions of f₁, f₅, f₁₀, f₁₆, f₂₁, the PPSO in the 3 benchmark functions of f₂, f₄, and f₁₂, and the SOS algorithm in the 6 benchmark functions of f₇, f₁₃, f₂₀, f₂₄, f₂₆, f₂₇, the BA algorithm in the benchmark function of f₃, the optimization performance is better than other algorithms. Among the 28 benchmark functions, the PSO algorithm does not get the optimal value in comparison with other algorithms.

Table 1
The mean and standard deviation of 20 runs of 28 functions

30D MSALO ALO PSO PPSO SOS BA

Mean Std Mean Std Mean Std Mean Std Mean Std Mean Std

f ₁ -1399.84 143.97 -1400 1.04E-08 1459.21 1.79E+03 -1399.9 8.71E-02 325.8604 1.49E+03 -1395.04 7.53E-01

f ₂ 8943518 4E+06 7223690 1768902 5.5E+07 19038349 3E+06 878831.3 31288099 13667182 3077421 951496.8

f ₃ 1.12E+09 4E+08 4.5E+08 5.14E+08 7.8E+10 7.03E+10 1.9E+09 4.13E+09 1.591E+10 6.56E+09 2.6E+08 3.24E+08

f ₄ 45798.46 10331 52820.4 20769.27 8190.48 3883.788 13015 4794.198 7806.5925 2338.317 36316.53 11708.16

f ₅ -999.862 0.127 -1000 0.001965 67.6194 1353.703 -996.65 10.46965 -602.0156 158.4442 -998.28 0.161904

f ₆ -871.59 23.682 -836.21 26.4413 -658.73 246.4008 -852.11 21.76103 -680.0632 71.12041 -864.085 33.28373

f ₇ -673.942 42.118 -671.94 27.25038 -553.93 746.6258 -657.31 37.08864 -701.786 18.88621 30657.01 41689.08

f ₈ -679.12 0.0552 -679.02 0.056025 -679.02 0.045242 -679.05 0.054406 -679.0405 0.073791 -678.971 0.053074

f ₉ -574.72 4.0939 -572.92 4.027204 -569.73 3.216596 -570.6 3.333288 -573.2947 3.090826 -562.236 3.240121

f ₁₀ -498.712 258.63 -498.9 0.835135 172.508 364.0625 -496.99 2.189041 -55.20328 174.8831 -498.048 0.243248

f ₁₁ -358.43 91.909 -232.94 72.69976 -126.56 71.49529 -244.46 52.02539 -257.1812 31.25475 292.4035 89.39463

f ₁₂ -113.302 60.85 -121.11 44.83254 -53.969 125.6014 -151.8 40.45248 -144.7172 34.83593 326.8223 104.3978

f ₁₃ 49.1048 55.178 90.9537 43.74626 123.369 78.79651 40.7025 52.09909 27.60386 39.78359 630.5248 144.0065

f ₁₄ 1592.73 983.36 3782.58 597.115 4115.55 848.7205 4087.45 909.1047 5097.8425 1001.482 4978.858 748.6605

f ₁₅ 3588.36 576.24 4197.33 796.738 5003.3 588.5505 4541.59 777.2011 7362.397 321.1915 5043.61 805.7863

f ₁₆ 201.0108 0.3595 200.74 0.27279 201.477 0.427599 201.227 0.439237 202.45834 0.343175 202.4066 0.951042

f ₁₇ 371.631 63.125 519.351 54.13623 614.056 67.46381 542.588 48.43852 488.64239 46.58357 1488.876 217.5064

f ₁₈ 629.459 55.533 647.642 54.43727 730.805 58.94401 632.324 72.24282 673.42379 52.49569 1572.73 267.8297

f ₁₉ 502.966 2.0148 511.426 3.609335 3877.47 5599.916 517.966 6.214634 963.03113 307.0156 536.156 5.073259

f ₂₀ 614.3299 0.5159 613.769 1.167616 613.188 0.803796 613.975 1.104805 612.146 0.717677 614.9482 0.126587

f ₂₁ 1017.326 86.554 994.36 93.5405 1116.34 250.1778 1027.75 106.5219 1971.1052 341.7407 1052.301 60.94011

f ₂₂ 2755.79 980 5274.73 932.8862 5145.31 791.1509 5255.41 783.0436 6028.8816 771.9126 6879.54 956.9165

f ₂₃ 5402.46 748.91 6061.53 904.732 5902.59 546.9681 5907.68 568.7574 8264.8598 963.8973 6917.945 691.4021

f ₂₄ 1292.14 7.9214 1285.75 10.639 1298.21 9.469519 1293.79 8.573211 1280.995 9.51687 1341.251 31.96404

f ₂₅ 1390.08 7.7333 1426.81 16.73573 1408 11.38885 1405.67 8.508627 1401.2516 8.870783 1394.409 9.848061

f ₂₆ 1460.005 74.89 1503.6 84.48532 1528.12 81.13354 1499.45 88.3673 1401.718 1.008625 1540.525 97.06036

f ₂₇ 2368.243 124.4 2405.07 120.2363 2442.17 84.92531 2420.03 89.9925 2331.231 85.92371 2644.304 78.54568

f ₂₈ 1886.63 479.71 2456.59 1124.305 3718.98 560.4602 2151.19 456.6188 3379.2136 431.2116 6856.505 835.1362

30D	MSALO	ALO	PSO	PPSO	SOS	BA
	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std
f ₁	-1399.84	143.97	-1400	1.04E-08	1459.21	1.79E+03	-1399.9	8.71E-02	325.8604	1.49E+03	-1395.04	7.53E-01
f ₂	8943518	4E+06	7223690	1768902	5.5E+07	19038349	3E+06	878831.3	31288099	13667182	3077421	951496.8
f ₃	1.12E+09	4E+08	4.5E+08	5.14E+08	7.8E+10	7.03E+10	1.9E+09	4.13E+09	1.591E+10	6.56E+09	2.6E+08	3.24E+08
f ₄	45798.46	10331	52820.4	20769.27	8190.48	3883.788	13015	4794.198	7806.5925	2338.317	36316.53	11708.16
f ₅	-999.862	0.127	-1000	0.001965	67.6194	1353.703	-996.65	10.46965	-602.0156	158.4442	-998.28	0.161904
f ₆	-871.59	23.682	-836.21	26.4413	-658.73	246.4008	-852.11	21.76103	-680.0632	71.12041	-864.085	33.28373
f ₇	-673.942	42.118	-671.94	27.25038	-553.93	746.6258	-657.31	37.08864	-701.786	18.88621	30657.01	41689.08
f ₈	-679.12	0.0552	-679.02	0.056025	-679.02	0.045242	-679.05	0.054406	-679.0405	0.073791	-678.971	0.053074
f ₉	-574.72	4.0939	-572.92	4.027204	-569.73	3.216596	-570.6	3.333288	-573.2947	3.090826	-562.236	3.240121
f ₁₀	-498.712	258.63	-498.9	0.835135	172.508	364.0625	-496.99	2.189041	-55.20328	174.8831	-498.048	0.243248
f ₁₁	-358.43	91.909	-232.94	72.69976	-126.56	71.49529	-244.46	52.02539	-257.1812	31.25475	292.4035	89.39463
f ₁₂	-113.302	60.85	-121.11	44.83254	-53.969	125.6014	-151.8	40.45248	-144.7172	34.83593	326.8223	104.3978
f ₁₃	49.1048	55.178	90.9537	43.74626	123.369	78.79651	40.7025	52.09909	27.60386	39.78359	630.5248	144.0065
f ₁₄	1592.73	983.36	3782.58	597.115	4115.55	848.7205	4087.45	909.1047	5097.8425	1001.482	4978.858	748.6605
f ₁₅	3588.36	576.24	4197.33	796.738	5003.3	588.5505	4541.59	777.2011	7362.397	321.1915	5043.61	805.7863
f ₁₆	201.0108	0.3595	200.74	0.27279	201.477	0.427599	201.227	0.439237	202.45834	0.343175	202.4066	0.951042
f ₁₇	371.631	63.125	519.351	54.13623	614.056	67.46381	542.588	48.43852	488.64239	46.58357	1488.876	217.5064
f ₁₈	629.459	55.533	647.642	54.43727	730.805	58.94401	632.324	72.24282	673.42379	52.49569	1572.73	267.8297
f ₁₉	502.966	2.0148	511.426	3.609335	3877.47	5599.916	517.966	6.214634	963.03113	307.0156	536.156	5.073259
f ₂₀	614.3299	0.5159	613.769	1.167616	613.188	0.803796	613.975	1.104805	612.146	0.717677	614.9482	0.126587
f ₂₁	1017.326	86.554	994.36	93.5405	1116.34	250.1778	1027.75	106.5219	1971.1052	341.7407	1052.301	60.94011
f ₂₂	2755.79	980	5274.73	932.8862	5145.31	791.1509	5255.41	783.0436	6028.8816	771.9126	6879.54	956.9165
f ₂₃	5402.46	748.91	6061.53	904.732	5902.59	546.9681	5907.68	568.7574	8264.8598	963.8973	6917.945	691.4021
f ₂₄	1292.14	7.9214	1285.75	10.639	1298.21	9.469519	1293.79	8.573211	1280.995	9.51687	1341.251	31.96404
f ₂₅	1390.08	7.7333	1426.81	16.73573	1408	11.38885	1405.67	8.508627	1401.2516	8.870783	1394.409	9.848061
f ₂₆	1460.005	74.89	1503.6	84.48532	1528.12	81.13354	1499.45	88.3673	1401.718	1.008625	1540.525	97.06036
f ₂₇	2368.243	124.4	2405.07	120.2363	2442.17	84.92531	2420.03	89.9925	2331.231	85.92371	2644.304	78.54568
f ₂₈	1886.63	479.71	2456.59	1124.305	3718.98	560.4602	2151.19	456.6188	3379.2136	431.2116	6856.505	835.1362

Table 2

The error rate of the algorithm for feature selection

Algorithm	MSALO-FS	ALO-FS	PSO-FS	PPSO-FS	SOS-FS	BA-FS
Error Rate	0.1838	0.2647	0.2774	0.219	0.2647	0.2609

The global optimal iterative process of the 6 algorithms based on 28 benchmark functions is shown in the Figs. 10 and 11. We can see that in the process of optimizing the MSALO algorithm and other algorithms on 28 functions, the convergence speed and convergence results of the MSALO algorithm are better in most functions. According to the optimization results and the optimization process, the MSALO algorithm has better optimization capabilities than other algorithms and faster convergence.

4.2 Simulation results of feature selection by MSALO

In this section, we show the results of the feature selection of our proposed MSALO, and compare them with the results of feature selection of ALO, PSO, PPSO, SOS, BA to verify the performance of MSALO in practical applications. This application experiment is to optimize the error rate with a single objective. The data set is Parkinson (used to detect and classify Parkinson’s disease), and the optimized result is the error rate. The table summarizes the comparison results of the algorithm for feature selection, and the figure shows the error rate of the algorithm for application. From the table, we can see that in the environment of 30-dimensional space, 100 individuals, and 1000 times, the error rate of the MSALO algorithm for feature selection is 0.1838, which is much smaller than the error rate of other algorithms, and the error is small. Therefore, MSALO has better accuracy in single target feature selection.

5 Conclusions

As a new intelligent optimization algorithm, ALO has been applied in many fields such as 5G [47, 50] and smart grid [49], and achieved results. However, it may still fall into a local optimum. For this, we propose an improved ALO with multiple strategies based on quantum-behaved, called MSALO. MSALO adopts a parallel idea, divide the population into several subgroups for independent optimization, and then communicates between the subgroups based on quantum-behaved so that the better solutions in the population become better, and the poor solutions are replaced with good points, Always maintain the quality of the population, thereby improving the convergence speed of the algorithm. In addition, MSALO combines the Gaussian mutation strategy and the quasi-adversarial learning strategy to enhance the diversity of the population and prevent the algorithm from falling into a local optimal stagnation. Use 5 Unimodal Functions, 15 Basic Multimodal Functions, and 8 Composition Functions to compare the optimization process and results of MSALO with several optimization algorithms. The comparison result shows that MSALO is better than ALO, PSO, PPSO, SOS, BA intelligent algorithms. In addition, we apply MSALO to feature selection and use Wrapper to optimize the error rate of feature selection. The experiment shows that MSALO can be effectively applied to feature selection for data dimensionality reduction. In future work, we will further study more efficient and comprehensive optimization algorithms, innovative communication strategies, and improvement strategies to better improve algorithm efficiency and performance. For the application of the algorithm, there are many aspects to discuss, for example, use different search strategies and evaluation functions for feature selection; feature selection in binary and non-binary forms; try to apply the algorithm to multi-objective feature selection. Through further research and improvement, more efficient feature selection will be applied to scenarios such as document processing, drug diagnosis, and gene selection.

Footnotes

Acknowledgment

This work was support edinpart by National Natural Science Foundation of China under Grant No. 619761-26, Shandong Natural Science Foundation under Grant No. ZR2019MF003, No. ZR2017MF054.

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