Internal search of the evolution matrix in QUasi-Affine TRansformation Evolution (QUATRE) algorithm

Abstract

Optimization demands exist everywhere in the real world especially in science studies and engineering practices, and it is important that the method to deal with intricacy optimization problems should itself be relative simple. Particle Swarm Optimization (PSO) and Differential Evolution (DE) both are simple evolutionary algorithms (EAs) which are proposed for single-objective optimization and both of them have been proved to be efficient methods for optimizing applications, however, there are still some weakness existing within them. A innovative evolutionary method named QUasi-Affine TRansformation Evolutionary (QUATRE) algorithm which is derived from, also tackles some weaknesses of PSO and DE algorithm, and obtains better performance on commonly used test suites. The key characteristic of QUATRE is that an automatically generated matrix named evolution matrix M is implemented in evolutionary process, which is taken as an alternative of employing the crossover rate CR. Here in this paper, we present a novel QUATRE variant, named IS-QUATRE, which can explore the search area in a better way comparison with the previous method, and relatively good optimization ability can be obtained by our proposed IS-QUATRE algorithm under CEC2013 test suit. And the conducted experimental results validate that our proposed IS-QUATRE is competitive with some other famous PSO and DE variants.

Keywords

Internal search QUATRE single-objective optimization stochastic optimization real-parameter optimization

1 Introduction

Optimization demands exist everywhere in the real world especially in science and engineering, and there are many different approaches to tackling such demands, such as Particle swarm optimization (PSO) [1], Differential Evolution (DE) algorithm [2], Monkey King Evolution (MKE) [3], and ebb-tide-fish algorithm [4] etc., mentioned in the literature. Some of the algorithm are known by the easy-coding characteristic and excellent performance, many of these algorithms still have weakness within themselves. Accordingly, some further improvements that can enhance the performance of these algorithm are also advanced after the inception of these algorithm.

Evolutionary Algorithm (EA) can be traced back to the earlier 1950s last century, and they all mainly intimate the bio-evolution of Darwinian priciple “survival of the fittest”. PSO was an EA simulating the forging behavior of bird, and it was first introduced in 1995 [1]. Due to the obvious weakness of PSO, many researchers proposed new variants of it: Shi and Eberhat proposed a novel PSO with an inertia weitht (iw-PSO) [5]. Later, they also proposed a constriction coefficient based particle swarm optimization (ccPSO), and they also compared the performance between iwPSO and ccPSO in [6]. Generally, the key thought of such evolutionary algorithm is to search the domain under some guidance or attraction, such as the global best, or historical best. The introduction of inertia weight or constriction coefficient can definitely enhance the convergence ability of the initial PSO algorithm [4]. Then a standard of the PSO algorithm was defined by Bratton and Kennedy in 2007 [7]. Recent PSO variants also examined some neighborhood topology [8, 9] or comprehensive learning techniques [10, 11], however, these PSO variants still have the “two-step forward, one-step back” weakness.

DE was a powerful EA that achieved great success in the recent optimization competitions [12], and it was first proposed by Price and Storn in [13]. The DE algorithm was actually derived from Genetic Annealing Algorithm, a combined Genetic Algorithm (GA) and Simulated Annealing (SA) algorithm, thus, it had same operations like selection, crossover and mutation [14]. Generally, there were two components have significant influence ont the performance of DE algorithm, i.e. mutation strategy and control parameters [15, 16]. Generally, a convention DE/x/y/z was employed to categorize these different mutation strategies, and three main control parameters, i.e. F, CR, and NP, was employed in the parameter control scheme [17,18, 17,18]. Recent researchers also mainly focus the enhancement of both the mutation strategies and the parameters control techniques [19], however, there still were many weakness existing in these DE variants [20,21, 20,21].

QUasi-Affine TRansformation Evolutionary (QUATRE) algorithm was a new algorithm proposed to tackle the immanent weakness of DE algorithm, and it was first proposed in 2016 [22]. As we know that DE employed a similar exploration behavior with the Genetic Algorithm, and the a cube-manner is employed in exploring the search space. Although the binomial crossover tackled the positional bias still existing in exponential crossover by treating each parameter equally in the crossover operation, it still existed weakness because the vertex was not treated equally in the exploration. Obviously, if there was no previous knowledge obtained during the evolution, the vertex in the high dimensional cube should be equally selected [23]. An affine transformation like form was implemented in the process of evolution in QUATRE. There were also some tips introduced int the literature that can improve the performance of the QUATRE algorithm like the pair-wise competition [24], sort-based techniques [25], and other techniques [26, 27] etc. However, the performance of the QUATRE algorithm can also be improved further, and that’s the reason why we proposed the internal search of the algorithm.

In this paper, A new variant of QUATRE algorithm is introduced, and we name it QUATRE algorithm with internal search. By incorporating the internal search strategy, the canonical QUATRE algorithm can be further improved. The rest of this paper is organized as follows. In Section 2, swarm based evolutionary algorithms are briefly introduced. Section 3, we describes new variant of QUATRE in detail. Section 4, we present the comparison with other algorithm. Finally, Section 5 gives the conclusion.

2 Swarm based evolutionary algorithms

2.1 PSO and its variants

It’s known to all that PSO is a simple but effective algorithm, simulating the behavior of a bird flock. In PSO, particles explore the search space under the guidance both from the historical personal best location and the global best location of the whole population. The evolution procedure of the original PSO is presented as follow:

Initialization: Initialize the locations and velocities of the particles by random values (X = [X₁, X₂, . . . , X_NP] and velocities V = [V₁, V₂, . . . , V_NP]) of the search space: $X_{i} = (x_{i, 1}, x_{i, 2}, . . ., x_{i, D}), i = 1, 2, . . ., NP$ (1) $V_{i} = (v_{i, 1}, v_{i, 2}, . . ., v_{i, D}), i = 1, 2, . . ., NP$ (2) Denotes X_i and V_i denote the location and velocity of the i^th individual respectively. All the particles of the population should make a relative full cover of the whole search space, therefore, the j^th parameter can be initialized according to the following equation: $x_{i, g}^{j} = L^{j} + rand (0, 1) * (U^{j} - L^{j})$ (3) where rand(0,1) is a uniformly distributed random value with the range from 0 to 1. and L^j and U^j are the lower limit and upper limit of the i^th parameter respectively.

Update pbest and gbest: Calculate the fitness value for each particle and make a comparison between particle’s fitness value and pbest value (the best fitness value the corresponding particle has ever achieved). If current solution is better than or equal to the corresponding pbest solution, then update the pbest value and the pbest position; Compare fitness value with the overall previous best solution of the whole population, then update the gbest position.

Update of the velocity and position: Velocity and position of particles are renewed according to the following equations respectively:

$\begin{matrix} v_{id} = & v_{id} + c_{1} * rand () * (p_{id} - x_{id}) \\ + c_{2} * rand () * (p_{gd} - x_{id}) \end{matrix}$ (4) $x_{id} = x_{id} + v_{id}$ (5) Loop to the two steps above until a terminal condition is met, mostly a solution with good fitness value or a maximum iteration. And Fig. 2 illustrated the way particles are updated in each generation.

Fig. 1

The way particles are updated in each generation.

Fig. 2

The relationship among target vector X_i,G, donor vector V_i,G and trial vector U_i,g.

The initial PSO performs not very well in convergence property, to enhance the optimization ability of PSO, various reformations have been proposed. In these PSO variants, a modified PSO with the inertia weight, named iwPSO, has been proved to be a simple but powerful PSO variant. iwPSO was first proposed by Shi and Eberhart in 1998, the velocity and position are update by employing a new parameter named inertia weight, and the update equation are presented in Equations 6 and 7. Every particle assigned a new velocity based on its previous velocity, personal best position, and the global best position of the whole population.

$\begin{matrix} v_{id} = & w * v_{id} + c_{1} * rand () * (p_{id} - x_{id}) \\ + c_{2} * rand () * (p_{gd} - x_{id}) \end{matrix}$ (6) $w = w_{initial} - (w_{initial} - w_{\min}) * gen / gen_\max$ (7) inertia weight w is set to be 0.9 at the first generation of the evolution and linearly reduced to 0.4 at the last generation. The variables c₁ and c₂ are acceleration constants that employed to control the distance the particles move in on generation (c₁=c₁=2 in iwPSO).

Another variant of PSO named ccPSO was proposed by Clerc which suggested that employing a constriction factor may be important to the preformance of PSO. In ccPSO, the constriction coefficient is employed to enhance the convergence of PSO. The simplified version of incorporating it presented in Equations 8 and Equation 9. $\begin{matrix} v_{id} = K * [v_{id} + c_{1} * rand () * (p_{id} - x_{id}) . . . \\ . . . + c_{2} * rand () * (p_{gd} - x_{id})] \end{matrix}$ (8) $K = \frac{2}{| 2 - φ - \sqrt{φ^{2} - 4 φ} |}, φ = c_{1} + c_{2}$ (9)φ is set at 4.1, so we can conclude that constant multiplier K ≈ 0.729, c₁ = c₂ = 2.05. Moreover, particles in ccPSO are allowed to fly outside the area restricted by the lower limit X_min and upper limit X_max.

2.2 DE and its variants

Differential Evolution (DE), first presented in 1995, is one of the most effective method for optimizing real-parameter objective functions. The general steps of DE can be separated into initialization and circle of evolution including mutation, crossover and selection. These steps are listed as follows:

Initialization: At the first generation of the population, the individuals is generated according to a uniform distribution between the lower limit X_min and upper X_max limit (X_min = (x_min,1, x_min,2, . . . , x_min,D), X_max = (x_max,1, x_max,2, . . . , x_max,D)) at random. Generally, the initialization of the population X = [X₁, X₂, . . . , X_NP] can be presented as follow: $X_{i} = x_{j, i} = x_{min, j} + {rand}_{j} (0, 1) \cdot (x_{max, j} - x_{min, j})$ (10) Denotes X_i donates the i^th individual of the population. After initialization, a evolutionary cycle, including mutation, crossover, and selection operation, is employed in DE to find the optimal solution.

Mutation: At each generation, mutation strategy is employed in DE to generate donor vectors V_i,G. Several in common use mutation strategies of DE are presented as follow:

DE/rand/1/z $V_{i, g} = X_{r 0, g} + F * (X_{r 1, g} - X_{r 2, g})$ (11)

DE/rand/2/z $\begin{matrix} V_{i, g} = & X_{r 0, g} + F * (X_{r 1, g} - X_{r 2, g}) \\ + F * (X_{r 3, g} - X_{r 4, g}) \end{matrix}$ (12)

DE/best/1/z $V_{i, g} = X_{gbest, g} + F * (X_{r 1, g} - X_{r 2, g})$ (13)

DE/best/2/z $\begin{matrix} V_{i, g} = & X_{gbest, g} + F * (X_{r 1, g} - X_{r 2, g}) \\ + F * (X_{r 3, g} - X_{r 4, g}) \end{matrix}$ (14)

DE/target-to-best/1/z $\begin{matrix} V_{i, g} = & X_{i, g} + F * (X_{gbest, g} - X_{i, g}) \\ + F * (X_{r 1, g} - X_{r 2, g}) \end{matrix}$ (15)

Denote X_i,g as the vector of i^th individual in the G^th generation and V_i,g as the corresponding donor vector. X_gbest,g is the global best target vector of current generation, and X_rk,g, k ∈ {0, 1, 2, 3, 4}, denotes the vector chosen from the population of G^th generation at random.

Crossover: After the mutation operator, crossover is employed in DE to select the components both from donor vector and target vector, and the components picked from two vectors are combined to generate trial vector U_i,g. The usual crossover operator in differential evolution includes two different manners: binomial crossover and exponential crossover. In most advanced DE variants proposed recently, the binomial crossover is employed more frequently.

Exponential Crossover: The crossover is conducted in a continuous length l and starts from the j^th variable. both l and j are determined at random. Then, the donor vector V_i,g will exchange its components with the target vector X_i,g from the j^th variable to the (j + l) ^th (s.t. (j + l) ≤ D) variable to generate the corresponding trial vector U_i,g.

Binomial Crossover: the binomial operation is conducted on each dimension according to the crossover rate control parameter CR, and the process of binomial way of crossover operation is showed in Equation 16. $\begin{matrix} U_{i, g} & = u_{j, i, g} \\ = {\begin{matrix} v_{j, i, g} & {rand}_{j} (0, 1) \leq CR or j = j_{rand} \\ x_{j, i, g} & else \end{matrix} \end{matrix}$ (16)

After the crossover operation, a re-initialization will be conducted if the trial vector is out of the default boundary. And Fig. 2 illustrated the relationship among target vector X_i,G, donor vector V_i,G and trial vector U_i,g.

Selection: After the mutation and crossover, a comparison will be conducted between target vector and trial vector (i.e. X_i,g and U_i,g), and the one with better objective function value (fitness value) will be chosen as the member of next generation. Equation 17 shows the process of selection for a minimization problem. $X_{i, g + 1} = {\begin{matrix} U_{i, g} & f (U_{i, g}) \leq f (X_{i, g}) \\ X_{i, g} & else \end{matrix}$ (17)

Although the DE algorithm show high search accuracy, robustness, and good convergence speed, it’s difficult for researchers to select suitable control parameters at different evolution stages. Moreover, DE performance is sensitive to its control parameter [28], different parameters setting have big effect on the convergence result of DE. To further enhance the performance of DE, some empirical guidelines have been introduced for choosing appropriate control parameters in the past decades [29 –31].

jDE is a variant of DE proposed by Janez Brest, the control parameters F and CR can be altered with some probabilities (τ₁ and τ₂) and better control parameters combination will be reserved in the next generations. The kernel idea of jDE is that a better control parameter is more likely to generate a better offspring, and the better control parameter would probably be retained in next generation. The update schemes of the control parameter F and CR in jDE are presented in Equations 18 and 19, respectively. $F_{i, g + 1} = {\begin{matrix} F_{l} + {rand}_{1} * F_{u} if {rand}_{2} < τ_{1} \\ F_{i, g} else \end{matrix}$ (18) ${CR}_{i, g + 1} = {\begin{matrix} {rand}_{3} if {rand}_{4} < τ_{2} \\ {CR}_{i, g} else \end{matrix}$ (19)rand₁ ≠ rand₂ ≠ rand₃ ≠ rand₄ F_l = 0.1, F_u = 0.9

τ₁ and τ₂ denotes the probabilities to alter F and CR respectively. In jDE algorithm, the initial value of F and CR is 0.5 and 0.9, τ₁ = τ₂ = 0.1, F_l = 0.1 and F_u = 0.9 are employed. And the new CR ranges from 0 to 1. By changing the parameter setting adaptively, better performance can be achieved in jDE algorithm.

2.3 QUasi-Affine TRansformation Evolution algorithm

QUATRE is a new optimize algorithm in which a quasi-affine transformation style (X ↦ MX + B) is introduced as an evolution method to achieve preferable individual cooperation in process of the algorithm. The evolution process of QUATRE is presented in Equation 20. Denote M as the evolution matrix, and $\bar{M}$ represents an inverse binary operation of all elements in matrix M (the inverse value of zero elements are one, while non-zero element are ones). ⨂ in Equation 20 is component-wise multiplication, and X = [X₁, X₂, . . . , X_NP] ^T denotes the position of the particles. And Equation 21 shows an example of the transformation process from a lower triangular matrix (initial matrix M_initial) to the evolution matrix M. First, we firstly rearrange the row element of the lower triangular matrix at random (each row is disposed separately), and then, rearrange the row vectors of the matrix at random with row elements remain unchanged. ${\begin{matrix} B = X_{gbest} + F * (X_{r 1} - X_{r 2}) \\ X \mapsto M ⨂ X + \bar{M} ⨂ B \end{matrix}$ (20) $M_{initial} = [\begin{matrix} 1 \\ 1 & 1 \\ \dots \\ 1 & 1 & \dots & 1 \\ ⋮ \\ 1 \\ 1 & 1 \end{matrix}] \sim [\begin{matrix} 1 & 1 \\ \dots \\ 1 & 1 & \dots & 1 \\ 1 \\ ⋮ \\ 1 & 1 \\ 1 & \dots & 1 \end{matrix}] = M$ (21)

3 The QUATRE algorithm with internal search

The good performance of QUATRE owes to the cooperative search matrix, so the particles can explore the search space well. Although the efficiency of QUATRE on optimization problems, the convergence ability of QUATRE algorithm can also enhanced by adjusting the evolution matrix M. In this section, we propose a new version of the cooperative search matrix M named internal-search matrix which show better performance in high dimensions. The evolution operations of the proposed algorithm can be introduced as follows.

The main parameters include F, CR and G_h (range from 0.3 to 1), when nfe > 0.5 * nfe _ max, the population size will start to reduce, and the population size will decrease to (D + 1) at the last generation.

When nfe < G_h * nfe _ max, we creates the cooperative search matrix M by following steps:

M_initial is made up of the fundamental cooperative search matrix M_basic, and the M_basic is formed by a lower triangular matrix and a row vector (the elements of the row vector are equal to 1/2.). Equation 20 shows an example of the transformation.

Transform the M_initial (i.e. same way as QUATRE) so we can get the cooperative search matrix M₁, and $\bar{M_{1}} = {Ones}_{NP \times D} - M_{1}$ .

When Nfe ≥ G_h * nfe _ max, the elements of the last row vector of M_basic have 90 percent change into rand number (range from 0 to 1).

Remaining operations are same as QUATRE.

Fig. 3 shows an example of 3-D generation scheme of candidate solutions in our proposed algorithm. The potential positions of X in next generation are the positions A, A₁, C, C₁, D, D₁ and E in Fig. 3, and the evolution matrix M is employed to select the solutions of X in next generation.

Fig. 3

Candidates in IS-QUATRE algorithm

4 Experiment results

It is not easy to predict the performance of optimization method, because of the limited theory, so benchmark functions are important for analyzing the performance of optimization algorithm. In this paper, 28 benchmark functions for real-parameter single objective optimization in CEC2013 test suit is employed to verify the new algorithm. Benchmark functions in CEC2013 can be divided into unimodal functions (f₁ - f₅), multimodal functions (f₆ - f₂₀) composition functions (f₂₁ - f₂₈), separable functions (f₁,f₅,f₁₁), non-separable functions (f₂ - f₄,f₆ - f₁₀,f₁₂ - f₂₀) and asymmetrical functions (f₄, f₇-f₉,f₁₁-f₁₆,f₁₈,f₂₀). Especially, the fitness errors $△ f = f_{i} - f_{i}^{*}$ of these benchmark functions are collected, compared and analyzed in conducted experiment.

To study the influence of the control parameter G_h in presented algorithm, we tested the performance of the proposed algorithm under different G_h values. In the conducted experiment, maximum number of function evaluations nfe _ max and initial population size NP are set to nfe _ max = 10000 * D and NP = 10 * D for fair performance comparison. And the performance of 51 independent runs under 30-D shown in Table 1. Especially, the best ones in the comparisons are marked in BOLDFACE way. Obviously, We can figure out that G_h = 0.4 may a appropriate value for G_h. Therefore, G_h = 0.4 is employed as the recommended value in the proposed IS-QUATRE.

Table 1
Comparison of different G_h value

30D G_h = 0.2 G_h = 0.4 G_h = 0.6 G_h = 0.8

No. Mean/Std. Mean/Std. Mean/Std. Mean/Std.

f ₁ 4.5475E-14/9.2825E-14 3.6380E-14/8.5075E-14 5.4570E-14/9.9110E-14 4.5475E-14/9.2825E-14

f ₂ 3.9442E+05/1.8406E+05 2.6524E+05/1.0110E+05 1.9869E+05/9.7063E+04 1.7764E+05/8.8993E+04

f ₃ 1.4349E+06/1.8128E+06 1.3047E+06/2.8055E+06 3.2190E+06/6.1379E+06 1.2534E+06/3.5325E+06

f ₄ 6.0366E+01/3.1046E+01 5.2509E+00/3.5805E+00 5.9837E-01/4.5555E-01 7.0346E-02/9.0209E-02

f ₅ 1.1369E-13/0.0000E+00 1.1369E-13/0.0000E+00 1.1369E-13/0.0000E+00 1.1369E-13/0.0000E+00

f ₆ 1.1413E+01/9.8261E+00 7.6027E+00/4.3924E+00 7.8634E+00/8.5465E+00 6.0121E+00/6.4532E+00

f ₇ 1.1501E+01/1.0915E+01 1.1569E+01/1.0399E+01 9.2305E+00/8.2083E+00 1.1512E+01/9.7882E+00

f ₈ 2.0922E+01/6.6448E-02 2.0892E+01/8.6441E-02 2.0915E+01/6.6209E-02 2.0933E+01/8.5110E-02

f ₉ 1.0908E+01/3.9705E+00 1.1732E+01/4.3729E+00 1.2902E+01/4.7009E+00 1.3254E+01/4.6084E+00

f ₁₀ 1.3989E-02/1.1115E-02 2.1776E-02/1.4818E-02 1.9998E-02/1.4330E-02 2.0976E-02/1.7349E-02

f ₁₁ 1.7710E+01/5.9003E+00 1.3890E+01/3.7392E+00 1.7113E+01/6.0248E+00 1.8148E+01/5.5641E+00

f ₁₂ 5.3456E+01/1.7412E+01 5.2947E+01/1.7133E+01 4.4961E+01/1.5364E+01 4.8284E+01/1.5070E+01

f ₁₃ 1.0969E+02/3.2292E+01 9.7971E+01/2.8737E+01 1.0635E+02/3.5272E+01 9.4458E+01/1.6698E+01

f ₁₄ 6.1744E+02/2.3860E+02 4.8426E+02/1.9793E+02 4.7112E+02/2.2648E+02 5.8808E+02/2.3437E+02

f ₁₅ 4.3408E+03/9.2411E+02 4.2630E+03/8.9796E+02 4.3215E+03/6.6079E+02 4.2891E+03/7.0681E+02

f ₁₆ 1.3873E+00/5.0141E-01 1.2571E+00/5.1022E-01 1.3519E+00/6.1130E-01 1.5049E+00/6.7241E-01

f ₁₇ 4.1511E+01/2.5551E+00 4.1048E+01/2.4989E+00 4.1454E+01/3.6396E+00 4.0734E+01/2.3097E+00

f ₁₈ 1.2206E+02/2.2164E+01 1.2655E+02/2.6099E+01 1.2872E+02/2.0288E+01 1.2126E+02/2.5957E+01

f ₁₉ 2.5140E+00/5.7772E-01 2.4106E+00/5.1376E-01 2.5565E+00/6.2601E-01 2.6035E+00/7.3808E-01

f ₂₀ 1.1469E+01/5.4877E-01 1.1465E+01/5.7222E-01 1.1342E+01/5.6842E-01 1.1249E+01/6.3658E-01

f ₂₁ 3.0645E+02/9.4116E+01 3.0697E+02/7.1065E+01 3.1845E+02/8.0395E+01 3.2019E+02/9.2205E+01

f ₂₂ 5.7581E+02/3.0963E+02 5.7133E+02/2.5101E+02 5.6614E+02/2.0846E+02 5.7999E+02/1.9825E+02

f ₂₃ 4.7285E+03/6.3757E+02 4.3213E+03/8.2847E+02 4.7187E+03/6.4468E+02 4.2697E+03/6.3644E+02

f ₂₄ 2.2279E+02/1.4740E+01 2.2547E+02/1.2926E+01 2.2774E+02/1.1076E+01 2.2744E+02/1.0711E+01

f ₂₅ 2.5002E+02/6.8552E+00 2.5129E+02/7.5814E+00 2.5210E+02/7.1397E+00 2.5187E+02/6.8460E+00

f ₂₆ 2.1979E+02/4.6262E+01 2.2964E+02/5.3999E+01 2.1093E+02/3.7826E+01 2.3545E+02/5.8073E+01

f ₂₇ 5.8790E+02/8.2306E+01 5.4398E+02/9.2331E+01 5.9809E+02/8.9535E+01 5.7739E+02/8.9692E+01

f ₂₈ 3.4137E+02/2.0683E+02 3.0000E+02/2.1269E-13 3.7568E+02/2.9476E+02 3.7496E+02/2.9230E+02

30D	G_h = 0.2	G_h = 0.4	G_h = 0.6	G_h = 0.8
f ₁	4.5475E-14/9.2825E-14	3.6380E-14/8.5075E-14	5.4570E-14/9.9110E-14	4.5475E-14/9.2825E-14
f ₂	3.9442E+05/1.8406E+05	2.6524E+05/1.0110E+05	1.9869E+05/9.7063E+04	1.7764E+05/8.8993E+04
f ₃	1.4349E+06/1.8128E+06	1.3047E+06/2.8055E+06	3.2190E+06/6.1379E+06	1.2534E+06/3.5325E+06
f ₄	6.0366E+01/3.1046E+01	5.2509E+00/3.5805E+00	5.9837E-01/4.5555E-01	7.0346E-02/9.0209E-02
f ₅	1.1369E-13/0.0000E+00	1.1369E-13/0.0000E+00	1.1369E-13/0.0000E+00	1.1369E-13/0.0000E+00
f ₆	1.1413E+01/9.8261E+00	7.6027E+00/4.3924E+00	7.8634E+00/8.5465E+00	6.0121E+00/6.4532E+00
f ₇	1.1501E+01/1.0915E+01	1.1569E+01/1.0399E+01	9.2305E+00/8.2083E+00	1.1512E+01/9.7882E+00
f ₈	2.0922E+01/6.6448E-02	2.0892E+01/8.6441E-02	2.0915E+01/6.6209E-02	2.0933E+01/8.5110E-02
f ₉	1.0908E+01/3.9705E+00	1.1732E+01/4.3729E+00	1.2902E+01/4.7009E+00	1.3254E+01/4.6084E+00
f ₁₀	1.3989E-02/1.1115E-02	2.1776E-02/1.4818E-02	1.9998E-02/1.4330E-02	2.0976E-02/1.7349E-02
f ₁₁	1.7710E+01/5.9003E+00	1.3890E+01/3.7392E+00	1.7113E+01/6.0248E+00	1.8148E+01/5.5641E+00
f ₁₂	5.3456E+01/1.7412E+01	5.2947E+01/1.7133E+01	4.4961E+01/1.5364E+01	4.8284E+01/1.5070E+01
f ₁₃	1.0969E+02/3.2292E+01	9.7971E+01/2.8737E+01	1.0635E+02/3.5272E+01	9.4458E+01/1.6698E+01
f ₁₄	6.1744E+02/2.3860E+02	4.8426E+02/1.9793E+02	4.7112E+02/2.2648E+02	5.8808E+02/2.3437E+02
f ₁₅	4.3408E+03/9.2411E+02	4.2630E+03/8.9796E+02	4.3215E+03/6.6079E+02	4.2891E+03/7.0681E+02
f ₁₆	1.3873E+00/5.0141E-01	1.2571E+00/5.1022E-01	1.3519E+00/6.1130E-01	1.5049E+00/6.7241E-01
f ₁₇	4.1511E+01/2.5551E+00	4.1048E+01/2.4989E+00	4.1454E+01/3.6396E+00	4.0734E+01/2.3097E+00
f ₁₈	1.2206E+02/2.2164E+01	1.2655E+02/2.6099E+01	1.2872E+02/2.0288E+01	1.2126E+02/2.5957E+01
f ₁₉	2.5140E+00/5.7772E-01	2.4106E+00/5.1376E-01	2.5565E+00/6.2601E-01	2.6035E+00/7.3808E-01
f ₂₀	1.1469E+01/5.4877E-01	1.1465E+01/5.7222E-01	1.1342E+01/5.6842E-01	1.1249E+01/6.3658E-01
f ₂₁	3.0645E+02/9.4116E+01	3.0697E+02/7.1065E+01	3.1845E+02/8.0395E+01	3.2019E+02/9.2205E+01
f ₂₂	5.7581E+02/3.0963E+02	5.7133E+02/2.5101E+02	5.6614E+02/2.0846E+02	5.7999E+02/1.9825E+02
f ₂₃	4.7285E+03/6.3757E+02	4.3213E+03/8.2847E+02	4.7187E+03/6.4468E+02	4.2697E+03/6.3644E+02
f ₂₄	2.2279E+02/1.4740E+01	2.2547E+02/1.2926E+01	2.2774E+02/1.1076E+01	2.2744E+02/1.0711E+01
f ₂₅	2.5002E+02/6.8552E+00	2.5129E+02/7.5814E+00	2.5210E+02/7.1397E+00	2.5187E+02/6.8460E+00
f ₂₆	2.1979E+02/4.6262E+01	2.2964E+02/5.3999E+01	2.1093E+02/3.7826E+01	2.3545E+02/5.8073E+01
f ₂₇	5.8790E+02/8.2306E+01	5.4398E+02/9.2331E+01	5.9809E+02/8.9535E+01	5.7739E+02/8.9692E+01
f ₂₈	3.4137E+02/2.0683E+02	3.0000E+02/2.1269E-13	3.7568E+02/2.9476E+02	3.7496E+02/2.9230E+02

The comparison was also made among jDE, iwPSO, ccPSO, QUATRE, and proposed algorithm, In the experiment, Max-number for evaluation is set to 10000 * D for fair performance comparison. Especially, in our experiment, we employed w_min = 0.4, w_max = 0.9, c₁ = c₂ = 2.0 and NP = 10 * D for iwPSO; φ = 4.1, K = 0.729 and NP = 10 * D for ccPSO; F_l = 0.1, F_u = 0.9, τ₁ = τ₂ = 0.1 and NP = 10 * D for jDE; F = 0.7 and NP = 10 * D for QUATRE algorithm; The decision for using these values is based on proposed articles. And the results of 51 independent runs under 10-D and 30-D are summarized in Table 2 and Table 3 respectively. The total performances are also summarized at the bottom of the table. Symbols “ > ", “ = " or “ < " in the table means that proposed algorithm is obviously better than, not obviously different from or obviously worse than the compared algorithm on the corresponding function. And the result shows that our approach have better performance compared with jDE, iwPSO, ccPSO and QUATRE algorithm.

Table 2

Mean/Std. of 51-run fitness error comparisons between proposed algorithm and other swarm based evolutionary algorithms with the same number of function evaluations. The overall performance are given in the last three rows of the table

10D	jDE	iwPSO	ccPSO	QUATRE	Proposed Algorithm
NO.	Mean/Std.	Mean/Std.	Mean/Std.	Mean/Std.	Mean/Std.
f ₁	0.0000E+00/0.0000E+00(=)	2.9871E-13/1.9554E-13(<)	1.0147E-01/4.3416E-01(<)	0.0000E+00/0.0000E+00(=)	0.0000E+00/0.0000E+00
f ₂	9.8120E-05/6.9101E-04(>)	4.2213E+03/9.9337E+03(<)	1.4342E+05/4.0988E+05(<)	2.8090E-03/7.4303E-03(<)	4.5788E+01/2.2741E+02
f ₃	1.2076E+00/1.9231E+00(<)	3.6454E+07/8.9562E+07(<)	4.0680E+08/6.1695E+08(<)	7.0159E-01/1.7431E+00(<)	2.0877E-01/6.7391E-01
f ₄	1.1891E-09/5.7925E-09(<)	2.3335E+00/1.4185E+01(<)	5.5857E+00/1.9975E+01(<)	2.3444E-07/2.7773E-07(<)	1.9099E-13/1.8190E-13
f ₅	0.0000E+00/0.0000E+00(=)	1.9669E-03/3.1981E-03(<)	5.0924E+00/1.8897E+01(<)	0.0000E+00/0.0000E+00(=)	0.0000E+00/0.0000E+00
f ₆	6.5416E+00/4.6716E+00(<)	1.5437E+00/1.8341E+00(>)	1.3434E+01/2.0390E+01(<)	4.6552E+00/4.9002E+00(>)	6.2415E+00/4.6076E+00
f ₇	1.3542E-03/1.7744E-03(>)	4.8765E+01/2.1982E+01(<)	6.8684E+01/3.5356E+01(<)	5.5157E-01/1.3446E+00(>)	6.8364E-01/1.5677E+00
f ₈	2.0345E+01/6.6203E-02(<)	2.0364E+01/6.8423E-02(<)	2.0359E+01/6.9690E-02(<)	2.0473E+01/7.9287E-02(<)	2.0326E+01/1.3110E-01
f ₉	6.5840E+01/1.1042E+00(<)	6.7549E+00/1.4608E+00(<)	6.2577E+00/1.8474E+00(<)	1.9333E+00/1.5909E+00(<)	1.9037E+00/1.1352E+00
f ₁₀	4.1770E-02/2.7096E-02(>)	1.0384E+00/1.0706E+00(<)	2.9812E+00/4.5299E+00(<)	1.9522E-01/1.1245E-01(<)	1.1745E-01/8.8082E-02
f ₁₁	0.0000E+00/0.0000E+00(>)	3.4004E+01/1.6486E+01(<)	4.4306E+01/1.9684E+01(<)	2.7907E+00/1.2823E+00(<)	2.7461E+00/1.2280E+00
f ₁₂	1.0777E+01/2.7534E+00(>)	3.1292E+01/1.5971E+01(<)	4.4777E+01/1.6814E+01(<)	1.4162E+01/4.3454E+00(<)	1.2210E+01/6.1605E+00
f ₁₃	1.2348E+01/3.9807E+00(>)	4.3791E+01/1.2259E+01(<)	7.2258E+01/2.2571E+01(<)	2.0593E+01/8.8592E+00(<)	1.8660E+01/8.3969E+00
f ₁₄	1.7833E-14/1.2735E-13(>)	1.0168E+03/3.0966E+02(<)	8.7944E+02/3.2166E+02(<)	9.9196E+01/8.6298E+01(>)	1.0296E+02/8.8073E+01
f ₁₅	1.0291E+03/1.9002E+02(<)	1.0552E+03/2.9865E+02(<)	1.0309E+03/2.9187E+02(<)	9.8706E+02/2.6895E+02(<)	7.6897E+02/2.2535E+02
f ₁₆	1.1223E+00/2.2000E-01(<)	1.0221E+00/2.3480E-01(<)	8.2164E-01/3.0182E-01(<)	1.3281E+00/3.1977E-01(<)	7.0339E-01/4.2239E-01
f ₁₇	1.0122E+01/2.9291E-11(>)	3.7772E+01/1.2815E+01(<)	6.5629E+01/2.2194E+01(<)	1.1398E+01/2.4695E+00(<)	1.0544E+01/3.0864E+00
f ₁₈	3.3256E+01/3.8784E+00(<)	3.8238E+01/1.2646E+01(<)	6.4103E+01/2.9412E+01(<)	3.2840E+01/8.2368E+00(<)	2.6724E+01/9.4142E+00
f ₁₉	4.2922E-01/6.7585E-02(>)	2.3147E+00/1.4233E+00(<)	2.5857E+00/1.9118E+00(<)	6.6208E-01/1.7348E-01(<)	5.7264E-01/1.9166E-01
f ₂₀	2.6337E+00/2.2787E-01(=)	3.5441E+00/4.2570E-01(<)	3.5991E+00/3.5193E-01(<)	2.8775E+00/5.9721E-01(<)	2.6339E+00/6.8932E-01
f ₂₁	3.7861E+02/6.7329E+01(<)	3.7271E+02/7.2392E+01(=)	3.4544E+02/8.9946E+01(>)	3.9043E+02/4.3659E+01(<)	3.7219E+02/4.0039E+01
f ₂₂	1.5689E+01/1.3574E+01(<)	1.1488E+03/2.8697E+02(<)	1.0224E+03/3.8182E+02(<)	1.7248E+02/1.1221E+02(<)	1.4142E+02/9.5288E+01
f ₂₃	9.3453E+02/1.7203E+02(<)	1.2034E+03/3.0506E+02(<)	1.1641E+03/3.0871E+02(<)	9.9183E+02/3.1367E+02(<)	8.5453E+02/4.0035E+02
f ₂₄	2.0791E+02/1.5726E+01(<)	2.1730E+02/3.5873E+00(<)	2.1782E+02/1.7493E+01(<)	2.0509E+02/4.9993E+00(>)	2.0774E+02/5.4598E+00
f ₂₅	1.9862E+02/1.1866E+01(>)	2.1662E+02/3.2979E+00(<)	2.1647E+02/1.5859E+01(<)	2.0302E+02/1.4745E+01(>)	2.0582E+02/5.5118E+00
f ₂₆	1.1884E+02/2.1263E+01(>)	1.9615E+02/4.0090E+01(<)	2.2182E+02/7.3283E+01(<)	1.6859E+02/4.2001E+01(>)	1.7564E+02/5.0561E+01
f ₂₇	3.0000E+02/1.1389E-03(>)	4.7187E+02/3.3573E+01(<)	5.8475E+02/3.8659E+01(<)	3.4533E+02/8.2157E+01(>)	3.7173E+02/9.8043E+01
f ₂₈	2.7647E+02/6.5079E+01(<)	5.2092E+02/2.2960E+02(<)	5.4979E+02/2.5001E+02(<)	2.8249E+02/8.1701E+01(<)	2.7582E+02/8.3392E+01
win	12	1	1	7	–
draw	3	1	0	2	–
lose	13	26	27	19	–

Table 3

30D	jDE	iwPSO	ccPSO	QUATRE	Proposed Algorithm
NO.	Mean/Std.	Mean/Std.	Mean/Std.	Mean/Std.	Mean/Std.
f ₁	0.0000E+00/0.0000E+00(>)	5.3741E+02/4.2224E+02(<)	2.9018E+03/1.8211E+03(<)	4.5475E-14/9.2825E-14(=)	4.5475E-14/9.2825E-14
f ₂	1.6457E+05/1.1068E+05(>)	1.0157E+07/5.6994E+06(<)	1.3742E+07/1.0376E+07(<)	3.3950E+05/1.6276E+05(<)	2.6524E+05/1.0110E+05
f ₃	1.3588E+06/2.4539E+06(<)	2.3262E+10/1.3833E+10(<)	3.7434E+10/1.9221E+10(<)	8.2725E+05/2.9222E+06(>)	1.3047E+06/2.8055E+06
f ₄	2.7151E+01/3.2782E+01(<)	2.2279E+04/9.4499E+03(<)	7.3745E+02/3.8114E+02(<)	1.8833E+01/1.3024E+01(<)	5.2509E+00/3.5805E+00
f ₅	1.1369E-13/0.0000E+00(=)	2.7947E+02/9.6432E+01(<)	4.7906E+02/4.1248E+02(<)	1.0719E-13/2.6774E-14(>)	1.1369E-13/0.0000E+00
f ₆	1.2096E+01/2.3786E+00(<)	1.3375E+02/4.8695E+01(<)	3.1155E+02/2.7496E+02(<)	7.6444E+00/1.1228E+01(<)	7.6027E+00/4.3924E+00
f ₇	1.5534E+00/1.2984E+00(>)	2.7408E+02/2.1405E+02(<)	1.7979E+02/5.7485E+01(<)	2.2932E+01/1.6666E+01(<)	1.1569E+01/1.0399E+01
f ₈	2.0947E+01/5.6431E-02(<)	2.0855E+01/7.1420E-02(>)	2.0855E+01/4.7568E-02(>)	2.1011E+01/5.0164E-02(<)	2.0892E+01/8.6441E-02
f ₉	2.3895E+01/5.2581E+00(<)	3.6787E+01/3.1688E+00(<)	3.3745E+01/2.7956E+00(<)	1.5578E+01/6.2844E+00(<)	1.1732E+01/4.3729E+00
f ₁₀	3.7189E-02/1.9446E-02(<)	1.6741E+02/8.0945E+01(<)	5.1829E+02/2.7751E+02(<)	2.5199E-02/1.9867E-02(<)	2.1776E-02/1.4818E-02
f ₁₁	1.1146E-15/7.9597E-15(>)	3.6360E+02/7.7695E+01(<)	3.2137E+02/7.7733E+01(<)	2.6506E+01/8.6723E+00(<)	1.3890E+01/3.7392E+00
f ₁₂	5.7324E+01/1.0560E+01(<)	3.3436E+02/8.9860E+01(<)	3.0211E+02/7.3708E+01(<)	8.1980E+01/2.8084E+01(<)	5.2947E+01/1.7133E+01
f ₁₃	8.8714E+01/1.8697E+01(>)	4.2641E+02/8.0113E+01(<)	4.6924E+02/7.5580E+01(<)	1.2709E+02/3.2273E+01(<)	9.7971E+01/2.8737E+01
f ₁₄	2.0411E-03/7.5121E-03(>)	4.8500E+03/8.9220E+02(<)	3.7494E+03/6.5118E+02(<)	8.6696E+02/2.2570E+02(<)	4.8426E+02/1.9793E+02
f ₁₅	5.1911E+03/3.5276E+02(<)	5.0093E+03/7.1307E+02(<)	4.4501E+03/7.3292E+02(<)	5.2308E+03/7.1241E+02(<)	4.2630E+03/8.9796E+02
f ₁₆	2.3937E+00/2.5616E-01(<)	4.9098E-01/9.5814E-01(>)	1.8420E+00/5.2740E-01(<)	2.4093E+00/4.6661E-01(<)	1.2571E+00/5.1022E-01
f ₁₇	3.0434E+01/6.4922E-05(>)	4.5540E+02/9.9515E+01(<)	5.1284E+02/1.0468E+02(<)	5.7248E+01/7.6418E+00(<)	4.1048E+01/2.4989E+00
f ₁₈	1.5817E+02/1.4834E+01(<)	4.0376E+02/8.7484E+01(<)	5.1307E+02/1.0273E+02(<)	1.6477E+02/2.2476E+01(<)	1.2655E+02/2.6099E+01
f ₁₉	1.6224E+00/1.6841E-01(>)	1.2212E+02/7.6962E+01(<)	1.3377E+02/1.5942E+02(<)	3.6773E+00/9.4073E-01(<)	2.4106E+00/5.1376E-01
f ₂₀	1.1810E+01/3.2168E-01(<)	1.4192E+01/5.8565E-01(<)	1.3898E+01/9.2687E-01(<)	1.1805E+01/6.0029E-01(<)	1.1465E+01/5.7222E-01
f ₂₁	2.8099E+02/6.0562E+01(>)	7.4495E+02/2.2902E+02(<)	9.0896E+02/2.5377E+02(<)	2.8336E+02/8.1048E+01(<)	3.0697E+02/7.1065E+01
f ₂₂	1.1977E+02/2.4286E+01(>)	5.5199E+03/7.4584E+02(<)	4.1467E+03/6.5896E+02(<)	8.7539E+02/2.6680E+02(<)	5.7133E+02/2.5101E+02
f ₂₃	5.0200E+03/4.0623E+02(<)	5.4948E+03/9.2863E+02(<)	5.4612E+03/9.4617E+02(<)	5.1909E+03/8.2268E+02(<)	4.3213E+03/8.2847E+02
f ₂₄	2.0591E+02/6.2410E+00(>)	2.9180E+02/4.7474E+00(<)	2.9194E+02/8.6552E+00(<)	2.3671E+02/1.3108E+01(<)	2.2547E+02/1.2926E+01
f ₂₅	2.6200E+02/1.4586E+01(<)	2.9272E+02/4.8132E+00(<)	2.9655E+02/8.3876E+00(<)	2.5536E+02/8.8836E+00(<)	2.5129E+02/7.5814E+00
f ₂₆	2.0231E+02/1.6432E+01(>)	3.2397E+02/9.2078E+01(<)	3.1183E+02/9.0035E+01(<)	2.4610E+02/6.5088E+01(<)	2.2964E+02/5.3999E+01
f ₂₇	5.5237E+02/2.1786E+02(<)	1.1890E+03/6.2922E+01(<)	1.1673E+03/8.7224E+01(<)	6.8653E+02/8.1850E+01(<)	5.4398E+02/9.2331E+01
f ₂₈	3.0000E+02/0.0000E+00(=)	3.4061E+03/6.6272E+02(<)	2.4306E+03/5.6044E+02(<)	3.8624E+02/3.0847E+02(<)	3.0000E+02/2.1269E-13
win	12	2	1	2	-
draw	2	0	0	1	-
lose	14	26	27	25	-

From the Table 2 and Table 3, we can find that our proposed IS-QUATRE algorithm reveals a relative better performance than the contrasted 4 other famous optimization methods. For 10-D optimization, IS-QUATRE algorithm obtains either similar or better performance improvement 16 out of 28 benchmark functions in contrast with jDE, 27 out of 28 functions in contrast with iwPSO, 27 out of 28 in contrast with ccPSO, and 21 out of 28 functions in contrast with classical QUATRE. For 30-D optimization, proposed IS-QUATRE algorithm achieves at least similar or even better performance enhancement 16 out of 28 benchmark functions in contrast with jDE, 26 out of 28 functions in contrast with iwPSO, 27 out of 28 in contrast with ccPSO, and 26 out of 28 functions in contrast with classical QUATRE.

The convergence rate of these optimization algorithms are also compared by drawing the convergence curves of median values in Fig. 4. Especially, we employed three different kinds of benchmark function, including unimodal function, multimodal function and composition function, to contrast the convergence speed of different algorithm. From the figure, we can see that the proposed algorithm have relatively good convergence speed in three different kinds of benchmark function.

Fig. 4

Convergence rate comparisons are given here by utilizing the median value of the 51 independent runs for each algorithm on different kinds of benchmark function for 30-D optimization.

5 Conclusion

An new QUATRE variant with internal search strategy was proposed in this paper, and this algorithm is compared with four famous evolution algorithms. From the experiment results we can see that our algorithm with internal search strategy achieved better performance in comparison with these contrasted algorithm. Moreover, by combining the self-adaptive control parameter and the novel internal search strategy of the evolution matrix, our algorithm improved the search capability of the population in comparison with the canonical QUATRE algorithm. Some optimization methods [32, 33] many be employed to further improve the efficiency of the proposed scheme, and we also attempt to use the novel algorithm in tackling the problems mentioned in [34, 35] in the near future.

Footnotes

Acknowledgments

This research is supported by the National Natural Science Foundation of China with the Grant No. 61906042 and Scientific Research Startup Foundation of Fujian University of Technology (GY–Z19013).

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