An enhanced symbiotic organisms search algorithm with perturbed global crossover operator for global optimization

Abstract

Symbiotic organisms search (SOS) algorithm is a nature-inspired meta-heuristic algorithm, which has been successfully applied to solve a wide range of benchmarks and real-world optimization problems. In this paper, an extended version of SOS, namely symbiotic organisms search with perturbed global crossover operator (PGCSOS), is introduced to enhance the performance of the basic SOS. In parasitism phase, an organism can benefit from other organisms that are better than it, and a perturbed crossover scheme is incorporated into the parasitism phase, which aims at maintaining the trade-off between exploration and exploitation effectively, and preventing the current best solution from getting trapped into local optima. The performance of PGCSOS is assessed by solving global optimization functions with different characteristics and real-world problems. Compared to the SOS, modified SOS and other promising heuristic methods, numerical results reveal that the PGCSOS has better optimization performance.

Keywords

Symbiotic organisms search perturbed global crossover operator exploration exploitation

1 Introduction

Many real-life problems in science and engineering can be formulated as optimization problems. In general, any single-objective real parameter optimization problem can be expressed as follows: $min f (x), x = (x_{1}, x_{2}, \dots, x_{D}), x_{i} \in [{lb}_{i}, {ub}_{i}]$ (1) where x is a solution for a problem with dimension D, x_i is the variable, f (x) is the objective function being optimized over x, each variable x_i, i = 1, 2, ⋯ , D, is limited by the upper ub_i and lower boundaries lb_i, respectively.

Over past decades, many popular and well-known nature-inspired meta-heuristic algorithms have received widespread attention to solve the real-life problems and obtained the better performance. Various meta-heuristic algorithms, such as genetic algorithm (GA) [1], particle swarm optimization (PSO) [2], differential evolution (DE) [3], harmony search (HS) [4], artificial bee colony (ABC) [5], teaching learning based optimization (TLBO) [6], grey wolf optimizer (GWO) [7], sine cosine algorithm (SCA) [8], and so on, have been developed to solve a wide range of complex optimization problems.

Symbiotic organisms search (SOS) algorithm [9], which simulates the interactive behavior of organisms in the ecosystem, is a recently proposed meta-heuristic algorithm. It has the advantages of simple principles and powerful search ability, which make it to be a good method in solving real-life optimization problems. Since SOS is introduced in 2014, several versions of the SOS algorithm have been proposed and successfully used in real-world optimization problems. A brief overview of some recently proposed variants and applications of the SOS algorithm is highlighted below.

SOS variants: In order to make SOS algorithm suitable for problems with different characteristics, several variants of the SOS algorithm were developed.

Panda et al. [10] combined SOS with adaptive penalty function to solve multi-objective constrained optimization problems. Al-Sharhan and Omranthe [11] developed simple SOS variant. The best organism in the ecosystem is replaced with randomly chosen one of the top 100p% organisms in the current ecosystem with p ∈ [0, 1]. Vincent et al. [12] developed six discrete SOS algorithms and applied them for the capacitated vehicle routing problem. Panda and Pani [13] introduced augmented lagrange multiplier method into the SOS to solve the constrained optimization problems.

Hybrid SOS [14] was proposed by combining SOS algorithm with simple quadratic interpolation to deal with large scale nonlinear complex optimization problems. The proposed algorithm provided more efficient behavior when dealing with real-world and large scale problems. Guha et al. [15] integrated quasi-oppositional based learning with SOS to solve the load frequency control problem of the power system. Ezugwu et al. [16] combined SOS with simulated annealing to solve traveling salesman problem. Absalom et al. [17] proposed a soft set symbiotic organisms search algorithm for optimizing virtual machine resource selection in cloud computing environment. Miao et al. [18] introduced the modified versions of SOS by incorporating the simplex method in the original SOS algorithm to solve the unmanned combat aerial vehicle path planning problem. Saha et al. [19] presented a reduced SOS integrated with a chaotic local search to improve the solution accuracy and convergence mobility of the basic SOS. The comprehensive and collective description of modifications and hybridization of SOS algorithms can be found in [20, 21].

SOS applications: SOS and its improved variants have been successfully applied to various practical problem, such as power systems optimization, construction project scheduling, design of engineering structures, and other fields. Verma et al. [22] applied SOS algorithm on modified IEEE 30- and 57-bus test power system for the solution of congestion management problem. Tran et al. [23] presented multi-objective SOS (MSOS) for optimizing multiple work shifts problem. Das et al. [24] applied SOS algorithm to optimize the distributed generation allocation. Ezugwu and Adewumi [25] proposed a discrete SOS algorithm for finding a near optimal solution for the travelling salesman problem. Duman [26] applied SOS algorithm to solve the optimal power flow problem with valve-point effect and prohibited zones. Sadek et al. [27] proposed an improved adaptive fuzzy backstepping control of a magnetic levitation system based on SOS. Talatahari [28] applied SOS to find optimum design of structures. Prayogo et al. [29] presented modified SOS for coping with the resource leveling problem. Do et al. [30] employed modified SOS to solve two optimization problems: buckling and free vibration with various volume constraints. The collective and comprehensive description of applications of SOS algorithms can be found in [20, 21].

Algorithm with global crossover operators: Among the three fundamental operators – selection, crossover, and mutation, the crossover operator is the most important evolution mechanism. To use the advantages of crossover operator and global best in the given algorithm, recently, Ouyang et al. [31] proposed an efficient perturbed scheme to prevent the current best solution from falling into in local optima, in which the perturbed scheme is incorporated into the teacher phase. As well as, the global crossover operator proposed is incorporated into the learning phase, which aims at balancing the exploration and exploitation effectively. The experimental results on a set of functions with different characteristics show that the strategy can significantly enhance the performance of TLBO. Shah et al. [32] proposed global crossover artificial bee colony (GCABC) algorithm for training multilayer perceptron for solving boolean classification problems. The crossover operator is introduced in the global onlooker bee phase to improve solution based on their fitness and balance exploitation process. The simulation results of the proposed algorithm are achievable and efficient results in benchmark boolean function classification. As mentioned above, it can be seen that global crossover operators have improved the performance of the given algorithm.

SOS is currently an active focus of research, which has good exploitation capabilities in the mutualism and commensalism phases by use of best solution and is good at exploration in the parasitism phase. The basic SOS and the previously developed variants of SOS have obtained satisfactory results in solving engineering field problems. Still, there are several shortcomings that can affect the performance of SOS algorithm such as quality of solution, stuck in local optima for solving complex optimization problems. In the basic SOS algorithm, by analyzing the parasitism phase, only part of the information of the organism X_i, or the information of the organism X_j were shared to the next generation organisms. As a result, it may easily get trapped in a local optima because of the lack of diversity among organisms, and offer over exploration on account of the unbalance between exploration and exploitation, meanwhile the best information has not yet been systematically exploited in the parasitism phase. On the other hand, the no free lunch (NFL) theorem [33] has been proved logically that an optimization algorithm could not be used to solve all the optimization problems. In other words, the average performance of an optimization algorithm is the same when taking into account all optimization problems. Hence, it is always necessary that some modifications should be incorporated to make the current optimization algorithm fit for a particular optimization problem. Motivated by this consideration, further improvements based on the work of Ouyang et al. [31] is proposed to improve the optimization performance of the original SOS. This new approach is referred to as an enhanced symbiotic organisms search algorithm with perturbed global crossover operator (PGCSOS). The tests were carried out on well-known optimization problems comprising of unimodal, multimodal, fixed-dimension multimodal functions and real-world optimization problems.

The remainder of this paper is organized as follows. Section 2 describes briefly an overview of SOS algorithm. In Section 3, a novel improved SOS (PGCSOS) is introduced in detail. The experiment results and comparisons are given in Section 4. Finally, conclusion and future scope are described in Section 5.

2 SOS

The SOS algorithm is a new population-based algorithm proposed by Cheng et al. [9] inspired from the cooperating behaviour among species in the society. In this optimization algorithm, a group of organisms (individuals) in an ecosystem is considered a population, and each organism are considered candidate solution to the given optimization problem. Each organism in the ecosystem is associated with the objective function value of the optimization problem. The SOS process is divided into three stages, the mutualism phase, commensalism phase and parasitism phase. The detailed description of SOS can be found in [9]. these three phases are briefly explained in the subsequent subsections.

2.1 Mutualism phase

This phase is the first part of the algorithm. In this part, two different organisms get benefits from each other mutually. An organism, X_j, is randomly selected from the ecosystem to mutually interact with X_i where X_i ≠ X_j. The new organism $X_{i}^{*}$ and $X_{j}^{*}$ in the ecosystem for each of X_i and X_j are generated according to Equation (2) and (3). $X_{i}^{*} = X_{i} + U (0, 1) * (X_{best} - MV \times {BF}_{1})$ (2) $X_{j}^{*} = X_{j} + U (0, 1) * (X_{best} - MV \times {BF}_{2})$ (3) where X_best is the best organism obtained so far with the best objective function value in the ecosystem and $MV = \frac{X_{i} + X_{j}}{2}$ is a mutual vector which represents the relationship characteristic between X_i and X_j, U (0, 1) is a uniform random number between 0 and 1, and it is used to guide the direction of exploration. BF₁ and BF₂ are defined as beneficial factors which are randomly selected as either 1 or 2 [9]. $X_{i}^{*}$ and $X_{j}^{*}$ are however, accepted if they give a better objective function values than them.

2.2 Commensalism phase

This phase is the second part of the algorithm, similar to mutualism phase, an organism, X_j, is randomly chosen to interact with X_i. However, in this case, X_i tries to benefit by interaction while X_j neither gains nor loses from the relationship. The new organism $X_{i}^{*}$ in the ecosystem is generated according to Equation (4). $X_{i}^{*} = X_{i} + U (- 1, 1) * (X_{best} - X_{j})$ (4) where U (-1, 1) represents a uniform random number between -1 and 1. It is used to intensify the exploration. After that, the organism $X_{i}^{*}$ replaces X_i if it is fitter.

2.3 Parasitism phase

This phase is the third part of the algorithm, an artificial parasite vector, X_pv, is created in the search space by duplicating organism X_i and then modifying randomly selected dimensions using a random number. Another organism, X_j, is randomly chosen from the ecosystem to serve as a host to the X_pv. The X_pv replaces X_j in the ecosystem if it is fitter.

The algorithm steps of SOS are listed in Algorithm 1.

Algorithm 1 SOS Algorithm [9]

Input: Set population size N, create population of organisms X_i, i = 1, 2, ⋯ , N, initialize X_i, Set stopping criteria.

Output: Optimal solution

1: Identify the best organism, X_best

2: while stopping criteria is not met

3: for i = 1 to N do

4: Mutualism Phase

5: $MV = \frac{X_{i} + X_{j}}{2}$

6: BF₁ = round (1 + U (0, 1))

7: BF₂ = round (1 + U (0, 1))

8: $X_{i}^{*} = X_{i} + U (0, 1) * (X_{best} - MV \times {BF}_{1})$

9: $X_{j}^{*} = X_{j} + U (0, 1) * (X_{best} - MV \times {BF}_{2})$

10: Calculate $f (X_{i}^{*})$ and $f (X_{j}^{*})$

11: if $f (X_{i}^{*}) < f (X_{i})$ then

12: $X_{i} = X_{i}^{*}$

13: end if

14: if $f (X_{j}^{*}) < f (X_{j})$ then

15: $X_{j} = X_{j}^{*}$

16: end if

17: Commensalism Phase

18: $X_{i}^{*} = X_{i} + U (- 1, 1) * (X_{best} - X_{j})$

19: Calculate $f (X_{i}^{*})$

20: if $f (X_{i}^{*}) < f (X_{i})$ then

21: $X_{i} = X_{i}^{*}$

22: end if

23: Parasitism Phase

24: Create X_pv

25: Calculate f (X_pv)

26: if f (X_pv) < f (X_j) then

27: X_j = X_pv

28: end if

29: Identify the best organism, X_best

30: end for

31: end while

3 Proposed algorithm

SOS has good exploitation capabilities in the mutualism and commensalism phases by use of best solution, and is good at exploration by duplicating organism and modifying randomly selected dimensions in the parasitism phase.

In this section, an improved version of SOS called enhanced SOS algorithm with perturbed global crossover operator (PGCSOS) will be introduced. PGCSOS brings in two improvements to the original SOS: interaction information strategy and perturbed global crossover operator.

In the parasitism phase, to balance local and global searching effectively and prevent the current best solution from getting trapped in local optima, useful interaction information of organisms and perturbed global crossover strategy are introduced into SOS to generate the artificial parasite vector X_pv. Thus, the performance of the SOS can be improved. In addition, the proposed changes are comparatively simple and do not need an extra parameter setting for PGCSOS. Here it is noted that any extra objective function evaluation efforts are not incorporated in the algorithm. So the function evaluation remains same in both the algorithms. These improvements are described in detail as follows:

In the parasitism phase, firstly, organism can randomly interact with each other through communications to further improve its performance. For organism, X_i, it obtains useful information from another organism X_j randomly that is better than it. The formula of communication operation is introduced according to Equation (5). $X_{i}^{*} = {\begin{matrix} X_{i} + U (0, 1) * (X_{i} - X_{j}), if f (X_{i}) < f (X_{j}) \\ X_{i} + U (0, 1) * (X_{j} - X_{i}), otherwise \end{matrix}$ (5) where $X_{i}^{*}$ means the trail vector of organism X_i, X_j is randomly selected from the population, and j ≠ i, U (0, 1) is a uniform random number between 0 and 1. Randomly selected organism X_j is employed to maintain the population diversity. Meanwhile, when the organism X_i is better than organism X_j, organism X_j is attracted toward the better organism X_i. By use of useful interaction information of organisms, Equation (5) might help in exploiting the neighborhood solutions of the current better solution. Hence this operation not only can increase the diversity of solutions in the search process, but can also strengthen the ability to converge at the global optimum; secondly, for the sake of enhancing the capability of exploitation and using the information of the current population, a perturbed global crossover operation is proposed here. The perturbed scheme is used to exploit the neighborhood solutions of the best solution in the current population and prevent the current best solution from falling into local optima. The detail expression is shown as below: $U_{i, k} = {\begin{matrix} X_{i, k}^{*}, if r_{1} \leq r_{2} or k = D_{k} \\ V_{i, k}, otherwise \end{matrix}$ (6) $\begin{matrix} V_{i, k} = \\ {\begin{matrix} X_{best, k} + G (0, 1) * (X_{best, k} - X_{j, k}), if r_{3} \leq r_{4} \\ X_{i, k}, otherwise \end{matrix} \end{matrix}$ (7) where i = 1, 2, ⋯ , N, k = 1, 2, ⋯ , D, G (0, 1) is a Gaussian random real number with a mean of 0 and a standard deviation of 1. r ∈ {1, 2, ⋯ , N} is a randomly generated integer number and r ≠ best, r ≠ i, r₁, r₂, r₃ and r₄ are uniform random numbers between 0 and 1. The perturbed global crossover operator is used as a guide to search the region around the best individual by Gaussian distribution and thus improve the exploitation capability. The exploration capability is enhanced by the interaction information. This operation can balance the exploration and exploitation abilities effectively and avoid sticking in local optima. Once a new parasite vector is formed, a greedy selection is applied between parasite vector X_pv and X_j according to their objective function value.

The pseudo code for the implementation of the modified parasitism phase is illustrated in Algorithm 2. It can be seen that the difference between PGCSOS and SOS lies in generation of the artificial parasite vector, X_pv. Lines 6-17 state the procedures of the proposed perturbed global crossover operator, and other codes are (Lines 1-5) communication operation procedures. Combining the interaction and the perturbed global crossover operator can achieve a better trade-off between exploration and exploitation. It does not increase any algorithm-specific adjusting parameters.

Algorithm 2 The Modified Parasitism Phase

Input: organism X_i and X_j, j ≠ i.

Output: organism X_j

1: if f (X_i) < f (X_j) then

2: $X_{i}^{*} = X_{i} + U (0, 1) * (X_{i} - X_{j})$

3: else

4: $X_{i}^{*} = X_{i} + U (0, 1) * (X_{j} - X_{i})$

5: end if

6: for k = 1 to D do

7: m = randint (1, D)

8: if r₁ < r₂ or k = m then

9: $U_{i, k} = X_{i, k}^{*}$

10: else

11: if r₃ < r₄ then

12: U_i,k = X_best,k + G (0, 1) * (X_best,k - X_j,k)

13: else

14: U_i,k = X_i,k

15: end if

16: end if

17: end for

18: Calculate f (U_i)

19: if f (U_i) < f (X_j) then

20: X_j = U_i

21: end if

4 Experimental results and discussion

To evaluate the performance of the proposed algorithm, in this section, various simulations were carried out. The first is comparison on 23 classic benchmark functions [34, 35], and the second one is comparison on two real-world problems. These problems have been widely used in the literature. All algorithms are implemented in Matlab (version R2015b) and executed on HP machine (Core Xeon(R), 2.53 GHz, 8GB RAM).

4.1 Comparison on classic benchmark functions

In this subsection, the proposed algorithm is applied to 23 well-known benchmark functions with different characteristics which have been extensively solved with different algorithms in the literature in order to test their performance. These functions are given in Tables 1, 2, and 3 in three groups, respectively, where D indicates dimension of the function, Range is the boundary of the function’s search space, and f_min is the optimum. Functions f₁ to f₇ are unimodal functions, f₈ to f₁₃ are multimodal functions, and f₁₄ to f₂₃ are finally fixed dimension (low-dimensional) multimodal functions.

Table 1
Unimodal benchmark functions

Function D Range f _min

$f_{1} (x) = \sum_{i = 1}^{D} x_{i}^{2}$ 30,50 [-100, 100] 0

$f_{2} (x) = \sum_{i = 1}^{D} | x_{i} | + \prod_{i = 1}^{D} | x_{i} |$ 30,50 [-10, 10] 0

$f_{3} (x) = \sum_{i = 1}^{D} (\sum_{j = 1}^{i} x_{j})^{2}$ 30,50 [-100, 100] 0

f₄ (x) = max {|x_i|, 1 ≤ i ≤ D} 30,50 [-100, 100] 0

$f_{5} (x) = \sum_{i = 1}^{D - 1} [100 (x_{i + 1} - x_{i}^{2})^{2} + (x_{i} - 1)^{2}]$ 30,50 [-30, 30] 0

$f_{6} (x) = \sum_{i = 1}^{D} (| x_{i} + 0.5 |)^{2}$ 30,50 [-100, 100] 0

$f_{7} (x) = \sum_{i = 1}^{D} i x_{i}^{4} + rand [0, 1)$ 30,50 [-1 . 28, 1 . 28] 0

Function	D	Range
$f_{1} (x) = \sum_{i = 1}^{D} x_{i}^{2}$	30,50	[-100, 100]
$f_{2} (x) = \sum_{i = 1}^{D} \| x_{i} \| + \prod_{i = 1}^{D} \| x_{i} \|$	30,50	[-10, 10]
$f_{3} (x) = \sum_{i = 1}^{D} (\sum_{j = 1}^{i} x_{j})^{2}$	30,50	[-100, 100]
f₄ (x) = max {\|x_i\|, 1 ≤ i ≤ D}	30,50	[-100, 100]
$f_{5} (x) = \sum_{i = 1}^{D - 1} [100 (x_{i + 1} - x_{i}^{2})^{2} + (x_{i} - 1)^{2}]$	30,50	[-30, 30]
$f_{6} (x) = \sum_{i = 1}^{D} (\| x_{i} + 0.5 \|)^{2}$	30,50	[-100, 100]
$f_{7} (x) = \sum_{i = 1}^{D} i x_{i}^{4} + rand [0, 1)$	30,50	[-1 . 28, 1 . 28]

Table 2

Multimodal benchmark functions

Function	D	Range	f _min
$f_{8} (x) = - \sum_{i = 1}^{D} (x_{i} sin (\sqrt{\| x_{i} \|}))$	30,50	[-500, 500]	-418.9829D
$f_{9} (x) = \sum_{i = 1}^{D} (x_{i}^{2} - 10 cos (2 π x_{i}) + 10)$	30,50	[-5 . 12, 5 . 12]	0
$f_{10} (x) = - 20 exp (- 0.2 \sqrt{\frac{1}{D} \sum_{i = 1}^{D} x_{i}^{2}}) - exp (\frac{1}{D} \sum_{i = 1}^{D} cos 2 π x_{i}) + 20 + e$	30,50	[-32, 32]	0
$f_{11} (x) = \sum_{i = 1}^{D} x_{i}^{2} / 4000 - \prod_{i = 1}^{D} cos (x_{i} / \sqrt{i}) + 1$	30,50	[-600, 600]	0
$f_{12} (x) = \frac{π}{D} {10 {sin}^{2} (π y_{1}) + \sum_{i = 1}^{D - 1} (y_{i} - 1)^{2} [1 + 10 {sin}^{2} (π y_{i + 1})] + (y_{D} - 1)^{2}}$	30,50	[-50, 50]	0
$+ \sum_{i = 1}^{D} u (x_{i}, 10, 100, 4)$
$y_{i} = 1 + \frac{1}{4} (x_{i} + 1)$
$u (x_{i}, a, k, m) = {\begin{matrix} k (x_{i} - a)^{m}, x_{i} > a \\ 0, - 1 \leq x_{i} \leq a \\ k (- x_{i} - a)^{m}, x_{i} < - a \end{matrix}$
$f_{13} (x) = 0.1 {{sin}^{2} (3 π x_{1}) + \sum_{i = 1}^{D - 1} (x_{i} - 1)^{2} [1 + {sin}^{2} (3 π x_{i + 1})] + (x_{D} - 1)^{2} [1 + {sin}^{2} (2 π x_{D}]}$	30,50	[-50, 50]	0
$+ \sum_{i = 1}^{D} u (x_{i}, 5, 100, 4)$

Table 3

Fixed-dimenstion multimodal benchmark functions

Function	D	Range	f _min
$f_{14} (x) = [\frac{1}{500} + \sum_{j = 1}^{25} \frac{1}{j + \sum_{i = 1}^{2} (x_{i} - a_{ij})^{6}}]^{- 1}$	2	[-65 . 536, 65 . 536]	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
$f_{16} (x) = 4 x_{1}^{2} - 2.1 x_{1}^{4} + \frac{x_{1}^{6}}{3} + x_{1} x_{2} - 4 x_{2}^{2} + 4 x_{2}^{4}$	2	[-5, 5]	-1.0316
$f_{17} (x) = (x_{2} - \frac{5.1}{4 π^{2}} x_{1}^{2} + \frac{5}{π} x_{1} - 6)^{2} + 10 (1 - \frac{1}{8 π}) cos x_{1} + 10$	2	[-5, 5]	0.398
$f_{18} (x) = [1 + (x_{1} + x_{2} + 1)^{2} (19 - 14 x_{1} + 3 x_{1}^{2} - 14 x_{2} + 6 x_{1} x_{2} + 3 x_{2}^{2})]$	2	[-2, 2]	3
$[30 + (2 x_{1} - 3 x_{2})^{2} (18 - 32 x_{1} + 12 x_{1}^{2} + 48 x_{2} - 36 x_{1} x_{2} + 27 x_{2}^{2})]$
$f_{19} (x) = - \sum_{i = 1}^{4} c_{i} exp [- \sum_{j = 1}^{n} a_{ij} (x_{j} - p_{ij})^{2}]$	3	[0, 1]	-3.86
$f_{20} (x) = - \sum_{i = 1}^{6} c_{i} exp [- \sum_{j = 1}^{n} a_{ij} (x_{j} - p_{ij})^{2}]$	6	[0, 1]	-3.32
$f_{21} (x) = - \sum_{i = 1}^{5} [(x - a_{i}) (x - a_{i})^{T}]^{- 1}$	4	[0, 10]	-10.1532
$f_{22} (x) = - \sum_{i = 1}^{7} [(x - a_{i}) (x - a_{i})^{T}]^{- 1}$	4	[0, 10]	-10.4028
$f_{23} (x) = - \sum_{i = 1}^{10} [(x - a_{i}) (x - a_{i})^{T}]^{- 1}$	4	[0, 10]	-10.5363

The PGCSOS is validated by comparing it with SOS [9] and variant of SOS namely pbest_SOS [11], as well as recently developed well-known meta-heuristics search algorithms, i.e., GWO [7], whale optimization algorithm (WOA) [36], multi-verse optimizer (MVO) [37], salp swarm algorithm (SSA) [38], QPSO [39], and TLBO_GC [31]. For the sake of fairness, the most common parameter settings were used that exist in the original literature except for the maximum iterations and population size for the nine algorithms used in validation, a total of 30 individuals are allowed to determine the best solution over 1000 iterations in each run for each algorithm, and in order to eliminate stochastic discrepancy, all algorithms are executed in 30 independent runs. For unimodal functions and multimodal functions, two different dimension sizes, i.e. D = 30, 50 are tested.

Tables 4 –8 include the comparison results of test functions, the best results among the different methods are written in boldface, according to the statistical results (average and standard deviation) of Tables 4-8, PGCSOS is able to provide very competitive results. The improved algorithm outperforms other algorithms with regard to the quality of the solutions for all the majority of the functions. From Table 4 and Table 6, firstly, it can be seen that the PGCSOS shows superior results on 5 out of 7 unimodal test functions with D = 30 expect for the f₅ and f₇, PGCSOS gets mean optimal result for f₅ with D = 30, for the f₇ with D = 30, solution accuracy obtained by SOS is slightly the better than the PGCSOS. secondly, the performance of the SOS algorithm is improved on functions f₁₂ and f₁₃ with D = 30, whereas results are nearly identical to functions f₉ and f₁₁ with D = 30 by the PGCSOS algorithm. The pbest_SOS algorithm outperforms other opponents for function f₈ and the PGCSOS as the second best after the pbest_SOS obtains mean result for f₁₀. While comparing the mean results for f₁₃, QPSO outperforms well, the PGCSOS as the second best after the QPSO. It can be seen from Table 5 and Table 7 that the PGCSOS provides better or similar results on 8 fixed-dimension multimodal functions expect for the f₂₁ and f₂₂. As seen from Table 4 and Table 8, for functions with D = 50, the PGCSOS has better results compared to other compared techniques for functions f₁-f₆, f₁₂ and f₁₃, and the PGCSOS has similar results on functions f₉, f₁₁, compared to the SOS and pbest_SOS. PGCSOS obtains the mean optimal result on f₁₀ with D = 50, WOA on f₈ with D = 50. SOS outperforms well for f₇ with D = 50, the PGCSOS as the second best after the SOS. Meanwhile, the proposed algorithm also can find the global optimum on the seven test functions (f₉, f₁₁, f₁₄, f₁₇). Thus in exploring the promising regions of the search space PGCSOS is better than others in most of the problems. It can be concluded that PGCSOS exhibits the good convergence performance on almost all the benchmark functions. It is evident that the PGCSOS has a better searching abilities compared to all the considered various meta-heuristic algorithms. Tables 4 –8 show that PGCSOS provides an efficient strategy for finding the optimal solution of an optimization problem with a fast convergence rate.

Table 4

Experimental results obtained by SOS, pbest_SOS and PGCSOS for f₁ - f₁₃

Function (D = 30)	Indexes	SOS	pbest_SOS	PGCSOS	Function (D = 50)	Indexes	SOS	pbest_SOS	PGCSOS
f ₁	Mean	1.07E-270	5.96E-253	1.47E-301	f ₁	Mean	1.07E-259	1.35E-240	1.06E-278
	Std	0.00E+00	0.00E+00	0.00E+00		Std	0.00E+00	0.00E+000	0.00E+00
f ₂	Mean	1.21E-136	8.67E-127	1.20E-153	f ₂	Mean	3.52E-131	2.05E-121	5.03E-143
	Std	2.44E-136	1.33E-126	2.18E-153		Std	5.41E-131	2.56E-121	9.52E-143
f ₃	Mean	3.22E-94	6.42E-78	4.16E-97	f ₃	Mean	1.08E-75	1.57E-60	4.23E-77
	Std	1.29E-93	3.50E-77	1.62E-96		Std	2.72E-75	8.21E-60	1.51E-76
f ₄	Mean	7.54E-111	6.11E-102	1.70E-117	f ₄	Mean	3.18E-104	2.92E-095	2.39E-107
	Std	9.64E-111	1.17E-101	4.46E-117		Std	5.09E-104	7.77E-095	5.01E-107
f ₅	Mean	2.38E+01	2.48E+01	1.89E+01	f ₅	Mean	5.10E-01	4.52E+01	4.10E+01
	Std	5.10E-01	2.55E-01	4.18E-01		Std	2.38E+01	6.27E-01	4.40E-01
f ₆	Mean	9.51E-15	1.72E-13	2.42E-32	f ₆	Mean	1.30E-03	7.18E-03	1.11E-27
	Std	5.02E-14	7.55E-13	3.17E-32		Std	3.20E-03	2.38E-02	1.95E-27
f ₇	Mean	6.64E-05	3.82E-04	2.48E-04	f ₇	Mean	5.87E-05	2.76E-04	4.14E-04
	Std	6.22E-05	1.94E-04	1.97E-04		Std	5.76E-05	1.96E-04	2.11E-04
f ₈	Mean	-11467.21	-11574.91	-9628.932	f ₈	Mean	-16876.30	-16724.79	-9783.364
	Std	6.09E+02	4.39E+02	6.56E+02		Std	1.05E+03	1.09E+03	8.14E+02
f ₉	Mean	0.00E+00	0.00E+00	0.00E+00	f ₉	Mean	0.00E+00	0.00E+00	0.00E+00
	Std	0.00E+00	0.00E+00	0.00E+00		Std	0.00E+00	0.00E+00	0.00E+00
f ₁₀	Mean	3.85E-15	3.26E-15	3.38E-15	f ₁₀	Mean	4.32E-15	4.20E-15	4.09E-15
	Std	1.35E-15	1.70E-15	1.66E-15		Std	6.49E-16	9.01E-16	1.08E-15
f ₁₁	Mean	0.00E+00	0.00E+00	0.00E+00	f ₁₁	Mean	0.00E+00	0.00E+00	0.00E+00
	Std	0.00E+00	0.00E+00	0.00E+00		Std	0.00E+00	0.00E+00	0.00E+00
f ₁₂	Mean	2.77E-17	1.92E-13	2.12E-32	f ₁₂	Mean	6.62E-06	1.93E-06	1.86E-28
	Std	1.42E-16	9.55E-13	2.59E-32		Std	1.42E-05	3.27E-06	6.67E-28
f ₁₃	Mean	1.38E-02	1.97E-02	9.77E-03	f ₁₃	Mean	1.24E-01	7.32E-02	1.88E-02
	Std	3.42E-02	3.22E-02	2.12E-02		Std	1.29E-01	6.14E-02	3.21E-02

Table 5

Experimental results obtained by SOS, pbest_SOS and PGCSOS for fixed-dimension multimodal functions

Function	Indexes	SOS	pbest_SOS	PGCSOS	Function	Indexes	SOS	pbest_SOS	PGCSOS
f ₁₄	Mean	0.99800	0.99800	0.99800	f ₁₉	Mean	-3.86278	-3.86278	-3.86278
	Std	8.23E-17	0.00E+00	0.00E+00		Std	2.71E-15	2.71E-15	2.71E-15
f ₁₅	Mean	3.07E-04	3.07E-04	3.07E-04	f ₂₀	Mean	-3.26651	-3.26651	-3.28446
	Std	8.77E-19	1.36E-19	1.72E-19		Std	6.03E-02	6.03E-02	5.51E-02
f ₁₆	Mean	-1.03163	-1.03163	-1.03163	f ₂₁	Mean	-9.47347	-9.98327	-9.81333
	Std	6.71E-16	6.78E-16	6.78E-16		Std	1.76E+00	9.31E-01	1.29E+00
f ₁₇	Mean	0.39789	0.39789	0.39789	f ₂₂	Mean	-9.87141	-10.2258	-10.0486
	Std	0.00E+00	0.00E+00	0.00E+00		Std	1.62E+00	9.70E-01	1.35E+00
f ₁₈	Mean	3.00000	3.00000	3.00000	f ₂₃	Mean	-10.3561	-10.1759	-10.5364
	Std	1.33E-15	1.23E-15	1.32E-15		Std	9.87E-01	1.37E+00	1.78E-15

Table 6

Comparison results of PGCSOS and other state-of-the-art algorithms for f₁ - f₁₃ with D = 30

Function	Indexes	GWO	WOA	MVO	SSA	QPSO	TLBO_GC	PGCSOS
f ₁	Mean	2.09E-58	1.28E-152	3.30E-01	1.29E-08	0.00E+00	2.48E-141	1.47E-301
	Std	4.54E-58	3.82E-152	8.70E-02	2.86E-09	0.00E+00	7.86E-141	0.00E+00
f ₂	Mean	1.86E-34	5.92E-105	4.04E-001	1.27E+00	2.84E-18	2.98E-77	1.20E-153
	Std	1.55E-34	2.10E-104	1.12E-001	1.25E+00	1.55E-17	5.69E-77	2.18E-153
f ₃	Mean	5.02E-11	2.13E+04	4.63E+01	3.34E+02	8.54E-16	7.48E-18	4.16E-97
	Std	2.74E-10	1.15E+04	1.66E+01	2.52E+02	2.36E-15	1.92E-17	1.62E-96
f ₄	Mean	2.73E-13	2.53E+01	9.40E-01	8.05E+00	8.15E-06	3.59E-50	1.70E-117
	Std	7.64E-13	2.44E+01	4.45E-01	2.54E+00	4.46E-05	5.29E-50	4.46E-117
f ₅	Mean	2.73E+01	2.72E+01	1.94E+02	1.38E+02	2.61E+01	2.58E+01	1.89E+01
	Std	6.29E-01	5.21E-01	3.64E+02	2.69E+02	1.90E-01	5.24E-01	4.18E-01
f ₆	Mean	8.37E-01	9.46E-02	3.27E-01	1.26E-08	5.29E-05	1.38E-02	2.42E-32
	Std	3.28E-01	1.14E-01	9.59E-02	3.47E-09	1.77E-04	5.20E-02	3.17E-32
f ₇	Mean	2.57E-02	1.88E-03	1.67E-02	9.40E-02	5.58E-03	1.19E-04	2.48E-04
	Std	2.16E-02	1.93E-03	7.09E-03	4.40e-02	3.97E-03	7.73E-05	1.97E-04
f ₈	Mean	-5780.690	-10921.20	-7895.149	-7484.350	-9023.100	-9202.033	-9628.932
	Std	6.25E+02	1.73E+03	7.10E+02	6.36E+02	6.45E+02	6.10E+02	6.56E+02
f ₉	Mean	1.17E+00	5.68E-15	1.12E+02	5.89E+01	0.00E+00	4.17E+01	0.00E+00
	Std	4.49E+00	2.29E-14	2.95E+01	2.24E+01	0.00E+00	1.29E+01	0.00E+00
f ₁₀	Mean	2.40E-14	3.97E-15	1.13E+00	2.09E+00	5.30E-07	3.61E-15	3.38E-15
	Std	5.26E-15	2.76E-15	7.73E-01	7.09E-01	2.90E-06	1.53E-15	1.66E-15
f ₁₁	Mean	2.65E-03	1.00E-02	5.85E-01	1.07E-02	3.858E-16	1.31E-03	0.00E+00
	Std	8.56E-03	2.95E-02	1.03E-01	1.10E-02	1.35E-15	7.19E-03	0.00E+00
f ₁₂	Mean	6.02E-02	5.77E-03	1.67E+00	5.59E+00	6.03E-07	3.61E-03	2.12E-32
	Std	2.48E-02	4.36E-03	1.08E+00	3.01E+00	7.92E-07	1.89E-02	2.59E-32
f ₁₃	Mean	8.20E-01	2.30E-01	9.19E-02	5.59E+00	3.33E-03	3.38E-02	9.77E-03
	Std	2.54E-01	1.66E-01	6.34E-02	3.01E+00	5.13E-03	4.94E-02	2.12E-02

Table 7

Comparison results of PGCSOS and other state-of-the-art algorithms for fixed-dimension multimodal functions

Function	Indexes	GWO	WOA	MVO	SSA	QPSO	TLBO_GC	PGCSOS
f ₁₄	Mean	6.60526	2.37652	0.99800	0.99800	0.99800	2.76568	0.99800
	Std	4.43E+00	2.47E+00	2.52E-11	2.46E-16	0.00E+00	3.09E+00	0.00E+00
f ₁₅	Mean	8.68E-03	5.88E-04	1.04E-02	8.18E-04	1.31E-03	4.44E-03	3.07E-04
	Std	9.81E-03	2.87E-04	1.55E-02	2.77E-04	1.97E-03	8.10E-03	1.72E-19
f ₁₆	Mean	-1.03163	-1.03163	-1.03163	-1.03163	-1.03163	-1.03163	-1.03163
	Std	2.91E-08	4.17E-10	1.86E-07	1.11E-14	6.71E-16	5.13E-16	6.78E-16
f ₁₇	Mean	0.39789	0.39789	0.39789	0.39789	0.39789	0.39789	0.39789
	Std	2.33E-07	7.53E-07	6.18E-08	2.41E-15	0.00E+00	0.00E+00	0.00E+00
f ₁₈	Mean	3.00001	3.00001	3.00000	3.00000	3.00000	3.00000	3.00000
	Std	8.65E-06	3.86E-05	1.63E-06	7.68E-014	1.08E-15	2.66E-15	1.32E-15
f ₁₉	Mean	-3.86054	-3.85952	-3.86278	-3.86278	-3.86173	-3.86278	-3.86278
	Std	3.03E-03	3.59E-03	3.98E-07	4.05E014	2.73E-03	2.42E-15	2.71E-15
f ₂₀	Mean	-3.26394	-3.23771	-3.27024	-3.22105	-3.21994	-3.24441	-3.28446
	Std	8.91E-02	1.22E-01	6.02E-02	4.60E-02	1.11E-01	6.13E-02	5.51E-02
f ₂₁	Mean	-9.56346	-9.41209	-8.21049	-8.55908	-7.36582	-6.79571	-9.81333
	Std	1.84E+00	2.29E+00	2.63E+00	2.76E+00	2.69E+00	2.65E+00	1.29E+00
f ₂₂	Mean	-10.4023	-8.67094	-9.36479	-8.85542	-8.65402	-7.00379	-10.0486
	Std	4.22E-04	2.98E+00	2.41E+00	2.91E+00	2.81E+00	2.87E+00	1.35E+00
f ₂₃	Mean	-10.2653	-7.93607	-9.48320	-8.40949	-9.69643	-7.04670	-10.5364
	Std	1.48E+00	3.34E+00	2.44E+00	3.14E+00	2.21E+00	2.71E+00	1.78E-15

Table 8

Comparison results of PGCSOS and other state-of-the-art algorithms for f₁ - f₁₃ with D = 50

Function	Indexes	GWO	WOA	MVO	SSA	QPSO	TLBO_GC	PGCSOS
f ₁	Mean	1.54E-43	8.64E-146	2.23E+00	8.72E-08	4.20E-14	3.31E-120	1.06E-278
	Std	2.78E-43	4.73E-145	5.39E-01	2.30E-08	5.38E-14	1.18E-119	0.00E+00
f ₂	Mean	4.37E-26	1.12E-101	5.56E+01	3.58E+00	4.49E-22	1.36E-64	5.03E-143
	Std	4.01E-26	4.35E-101	7.33E+01	2.01E+00	2.35E-21	1.84E-64	9.52E-143
f ₃	Mean	6.18E-06	1.38E+05	1.99E+03	4.87E+03	5.83E-15	1.01E-07	4.23E-77
	Std	2.21E-05	3.09E+04	6.11E+02	3.11E+03	5.77E-15	1.93E-07	1.51E-76
f ₄	Mean	2.21E-09	6.53E+01	6.65E+00	1.69E+01	4.28E-04	1.08E-43	2.39E-107
	Std	3.67E-09	2.40E+01	2.11E+00	3.94E+00	1.63E-04	2.48E-43	5.01E-107
f ₅	Mean	4.71E+01	4.76E+01	5.61E+02	3.59E+02	4.67E+01	4.60E+01	4.10E+01
	Std	5.39E-01	5.90E-01	7.34E+02	5.86E+02	2.49E-01	8.77E-01	4.40E-01
f ₆	Mean	2.32E+00	4.44E-01	2.33E+00	9.33E-08	2.84E-02	1.92E-01	1.11E-27
	Std	5.75E-01	2.76E-01	6.21E-01	2.84E-08	2.01E-02	2.10E-01	1.95E-27
f ₇	Mean	1.10E-03	2.16E-03	6.22E-02	2.97E-01	5.60E-03	1.60E-04	4.14E-04
	Std	7.56E-04	2.47E-03	1.57E-02	7.58E-02	4.53E-03	1.05E-04	2.11E-04
f ₈	Mean	-8953.507	-19200.28	-12562.80	-12362.23	-13340.20	-14772.77	-9783.364
	Std	1.81E+03	2.39E+03	7.59E+02	1.01E+03	1.32E+03	7.81E+02	8.14E+02
f ₉	Mean	5.89E-01	1.89E-15	2.36E+02	9.29E+01	6.63E-06	7.64E+01	0.00E+00
	Std	1.65E+00	1.04E-14	3.33E+01	2.76E+01	7.45E-06	1.80E+01	0.00E+00
f ₁₀	Mean	3.24E-14	3.73E-15	2.38E+00	3.03E+00	5.21E-04	4.44E-15	4.09E-15
	Std	3.20E-15	2.36E-15	4.99E-01	6.44E-01	3.04E-04	0.00E+00	1.08E-15
f ₁₁	Mean	2.70E-03	3.20E-03	9.26E-01	9.31E-03	2.71E-13	0.00E+00	0.00E+00
	Std	5.67E-03	1.75E-02	5.28E-02	8.45E-03	2.45E-13	0.00E+00	0.00E+00
f ₁₂	Mean	8.49E-02	9.05E-03	3.95E+00	9.80E+00	5.56E-04	4.70E-03	1.86E-28
	Std	2.74E-02	5.37E-03	2.05E+00	3.13E+00	4.83E-04	1.26E-02	6.67E-28
f ₁₃	Mean	8.20E-01	6.63E-01	4.04E-01	5.53E+01	5.56E-01	2.67E-01	1.88E-02
	Std	2.54E-01	3.12E-01	1.90E-01	1.89E+01	3.58E-01	1.83E-01	3.21E-02

Due to the space limit, we present only the convergence curves for 4 unimodal functions (f₂, f₄-f₆) and 8 multimodal functions (f₈, f₁₀, f₁₂-f₁₃,f₁₇, f₁₈, f₂₁, f₂₃) in Fig. 1, Fig. 2 and Fig. 3, respectively, similar other graphics are not included in the study. As shown in Fig. 1, PGCSOS has a higher convergence accuracy and faster convergence speed on most functions with D = 30 except for function f₈. As shown in Fig. 2, for most of fixed-dimension multimodal function, PGCSOS not only has the higher convergence accuracy, but also converges faster. The convergence curve of PGCSOS for D = 50 unimodal functions and multimodal functions is shown in Fig. 3, consequently, it could be concluded that the proposed PGCSOS apparently improves the performance (efficiency) of the SOS and indeed outperforms well in terms of the convergence speed with an accurate solution. Therefore, PGCSOS is generally effective for both unimodal functions and multimodal functions with regards to the accuracy and convergence speed.

Fig. 1

Comparison of performances of algorithms for f₂, f₅, f₈, f₁₀ with D = 30.

Fig. 2

Comparison of performances of algorithms for f₁₇, f₁₈, f₂₁, f₂₃.

Fig. 3

Comparison of performances of algorithms for f₄, f₆, f₁₂, f₁₃ with D = 50.

Moreover, to make the experimental results more convincing, Friedman test is used [40] on the mean values found by the algorithms given in Tables 4 –7 (D = 30 and fixed-dimension) and the ranking values and p-value are presented in Table 9, boldface in the Table 9 indicates the best results by all algorithms. Friedman test ranks the algorithm from the best to the worst as PGCSOS, SOS, pbest_SOS, QPSO, TLBO_GC, WOA, GWO, MVO and SSA. The p-value computed through the statistics of Friedman test strongly indicated the existence of significant differences among nine algorithms. Here it is observed that the PGCSOS algorithm is the best one among these algorithms.

Table 9

Friedman test results for f₁ - f₁₃ with D = 30 and f₁₄ - f₂₃

Algorithm	GWO	WOA	MVO	SSA	QPSO	TLBO_GC	SOS	pbest_SOS	PGCSOS	p-value
Friedman rank	6.22	5.87	6.74	6.98	5.13	5.61	2.93	3.11	2.41	3.91E-16
Final rank	7	6	8	9	4	5	2	3	1

The Friedman test on the results (D = 50) given in Table 4 and Table 8 is reported and the ranking values are presented in Table 10, boldface in the Table 10 indicates the best results by the different algorithms. Friedman test ranks the SOS as the second best after the PGCSOS, as can be seen from Table 10 and the other algorithms were ranked from the third best to the worst as pbest_SOS, TLBO_GC, WOA, QPSO, GWO, SSA and MVO. Here it can be concluded again that the PGCSOS is the best one among these algorithms.

Table 10

Friedman test results for f₁ - f₁₃ with D = 50

Algorithm	GWO	WOA	MVO	SSA	QPSO	TLBO_GC	SOS	pbest_SOS	PGCSOS	p-value
Friedman rank	6.46	5.54	7.92	7.77	5.62	4.50	2.27	2.81	2.12	5.63E-12
Final rank	7	5	9	8	6	4	2	3	1

4.2 Comparison on real-world problems

In order to further testify the effectiveness of PGCSOS on real-world applications, two problems [41] from real life: gear train design problem and parameter estimation for frequency modulated (FM) sound waves were chosen.

The first problem is to minimize the gear ratio for a compound gear train that contains three gears. The mathematical model of this problem can be described as follows: $f (x) = (\frac{1}{6.931} - \frac{x_{1} x_{2}}{x_{3} x_{4}})^{2}$ (8) where x_i ∈ [12, 60] , i = 1, 2, 3, 4 .

The second problem is to estimate the parameters of a FM synthesizer. The parameter vector has six components: x = (a₁, ω₁, a₂, ω₂, a₃, ω₃), and the formula of the estimated sound wave is given as: $\begin{matrix} y (t) = a_{1} \cdot sin (ω_{1} \cdot t \cdot θ + a_{2} \cdot \\ sin (ω_{2} \cdot t \cdot θ + a_{3} \cdot sin (ω_{3} \cdot t \cdot θ))) \end{matrix}$ (9) and the equation of the target sound waves is given by: $\begin{matrix} y_{0} (t) = sin (5 \cdot t \cdot θ - 1.5 \cdot sin (4 . 8_{2} \cdot \\ t \cdot θ + 2 \cdot sin (4.9 \cdot t \cdot θ))) \end{matrix}$ (10) where $θ = \frac{2 π}{100}$ and the parameters are defined in the range [-6.4, 6.35]. The goal function is defined as follows: $f (x) = \sum_{t = 0}^{100} (y (t) - y_{0} (t))^{2}$ (11) The comparison on two real-world problems over 30 runs are shown in Table 11. In Table11 the best results are highlighted with bold letters. A set of well-known algorithms have been used for comparison and they are came from Li et al. [41]. As seen from Table 11, PGCSOS solved the first problem easily, and obtained the best values in all aspects of min, max, mean and standard deviation, respectively, the proposed algorithm is very competitive than other algorithms. For the second problem, it can be analyzed that the PGCSOS find the best solution at least once and obtained the second best standard deviation, which means PGCSOS has obvious advantages in finding the best solution and maintaining stability when facing real-word problems. These results demonstrate the efficiency of the PGCSOS method to solve the real-world problems.

Table 11

Comparison on two real-world problems (The results of last ten algorithms are as per Li et al. [41])

Algorithms	Gear ratio				Estimation error
	Min	Max	Mean	Std	Min	Max	Mean	Std
PGCSOS	0	3.33E-15	1.11E-16	6.09E-16	0	18.50	6.78	6.78
GWO	1.34E-17	1.66E-11	1.83E-12	3.68E-12	4.04	25.28	18.48	5.70
SLPSO	2.70E-12	6.19E-09	2.22E-09	9.83E-09	0	13.79	4.18	26.99
APSO	2.70E-12	1.31E-08	1.59E-09	1.44E-08	0	34.22	11.33	41.13
CLPSO	2.70E-12	1.36E-09	1.99E-10	2.22E-09	0.07	14.08	3.82	23.53
CPSOH	1.54E-10	2.02E-06	2.80E-07	2.33E-06	3.45	42.52	27.08	60.61
FIPS	8.88E-10	8.90E-07	3.27E-08	8.77E-07	0	15.11	5.93	25.75
SPSO	2.70E-12	2.56E-07	1.39E-08	2.57E-07	0	18.27	9.88	33.85
JADE	2.70E-12	1.36E-09	2.10E-10	2.26E-09	0	13.92	7.55	26.18
HRCGA	2.70E-12	1.18E-09	1.53E-10	1.88E-09	0	17.59	8.41	32.54
FPSO	2.06E-09	1.70E-04	1.57E-05	1.80E-04	0	15.82	5.22	28.31
G-CMA-ES	2.70E-12	7.32E-01	2.44E-02	1.31E-01	3.326	55.09	38.75	16.77

From these observations, all simulation results assert that the proposed improved variant is very helpful in improving the efficiency of the SOS in the terms of result quality. it is clear that PGCSOS are superior to other compared methods on most of the test problems, the search capability of PGCSOS is better than that other algorithms on the most of functions and real-life problems. Overall, PGCSOS can exhibit highly competitive search performance among all compared algorithms, it can be concluded that these findings strengthen the effectiveness of PGCSOS algorithm in solving unconstrained global optimization problems both empirically and statistically.

5 Conclusion

In this paper, SOS has been extended to PGCSOS, and the parasitism phase is modified. In the parasitism phase of PGCSOS, mutual information and perturbed global crossover operator are combined to generate a new artificial parasite vector to strengthen the diversity of solutions in the search process and balance the exploration and exploitation abilities effectively. In the experiments, 23 classical test functions and two classical real-life problems are used to evaluate performance of PGCSOS. Moreover, PGCSOS is compared with several recently developed algorithms, The experimental results indicate that PGCSOS can obtain competitive results on the majority of the test functions and two classical real-life problems than other comparative algorithms.

For future studies, it would be interesting to hybridize SOS algorithm with other algorithms like β-hill climbing to improve its performance. In addition, the comparison with other SOS algorithms reported in the literature will be further studied to examine the efficiencies of the proposed algorithm, and employing PGCSOS algorithm for solving other real-life engineering problems such as antenna array design problems, filter design problems, and prediction of nonlinear time series will be further research work.

Footnotes

Acknowledgments

This research work was supported by (1) National Natural Science Foundation of China (No. 61877046); (2) Shangluo University Key Disciplines Project (Discipline name: Mathematics); (3) Science and Technology Innovation Team Construction Project of Shangluo University (No. 18SCX002). The authors would like to thank the Editor-in-Chief, the Associate Editor and the anonymous reviewers for their insightful comments and constructive suggestions for improving the quality of the paper. The codes of part algorithms is taken from .

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