A new imperialist competitive algorithm with spiral rising mechanism for solving path optimization problems

Introduction

It is difficult to find global optimal values by using exact algorithms when solving the global optimization problems: the time-complexity is too high or it is impossible to find the global optimal solution. In recent years, approximation algorithms have begun to play their part in solving such problems, but where traditional approximation algorithms fail to meet research needs, meta-heuristics stand out because of their good optimization ability. These meta-heuristics have reasonably dealt with such inapproximable optimization problems, so more people have been involved in related research since their first development (Salmeron et al., 2019; Zhang & Wang, 2016).

In Fausto et al. (2020), meta-heuristics are classified into four categories according to the different basic concepts invoked thus: evolution-based, swarm-based, physics-based, and human behavior-based algorithms. In each of the different nature-inspired meta-heuristic algorithms the purpose is to obtain a search accuracy of the highest order, the fastest search speed, and the most applicable range of intelligent optimization algorithms. After these intelligent optimization algorithms are proposed, they are often applied to engineering optimization problems either alone or in combination with other algorithms (Hu et al., 2021; Yu et al., 2022).

The imperialist competitive algorithm (ICA) proposed by Atashpaz-Gargari & Lucas (2007) is a meta-heuristic algorithm imitating human social behavior. It involves simulating the colonial aggression of imperialist countries and the competitive mechanism of the law of the jungle to seek a local optimal value until finding the overall optimal value. It has achieved quite good optimization results in dealing with practical problems and compared with the previous proposed genetic algorithms (GA) (Holland, 1992) and particle swarm optimization (PSO) (Kennedy & Eberhart, 1995), it has improved the optimization accuracy and convergence speed to a certain extent; because of these, some practical engineering problems related to parameter optimization or function optimization use ICA to search for global optimal values.

ICA has been widely used in engineering practice due to its excellent optimization capability (Yan, Liu & Huang, 2022; Yu et al., 2022; Kashikolaei et al., 2020). Osmani, Mohasef & Gharehchopogh (2022) enhanced the exploration ability of the ICA with artificial bee colony optimization so as to enhance the global optimization ability. Li, Su & Lei (2021) added the concept of national cooperation into the ICA and used the improved algorithm to solve the scheduling optimization problems. ICA is also used in structural damage detection (Gerist & Maheri, 2019). Subsequent improvements to the ICA have also continued, Li, Lei & Cai (2019) developed a two-level ICA (TICA) based on ICA and applied it to optimize hybrid-flow shop scheduling problems. Barkhoda & Sheikhi (2020) introduced immigration concepts to ICA (IICA), and used the improved IICA to optimize the layout of wireless sensor networks. It can be concluded from the above, when it comes to optimization problems, ICA opens up a new way of thinking for researchers in addition to the commonly used solutions such as PSO and GA.

The original ICA has good exploitation ability by virtue of the competition and elimination mechanism within and between empires, while the exploration ability of the algorithm may be relatively weak. Therefore, ICA has the risk of premature convergence and falling into local optimal values. Most of the above-mentioned algorithm improvements revolve around improving the exploration ability of the ICA and enhancing the diversity of solution sets, enhancing the global optimization capability of the algorithm. After improving the exploration ability of ICA, the exploration ability and exploitation ability of ICA will be more balanced, and it can try to avoid falling into the local optimal solution too early.

In order to improve the shortcomings of ICA in global optimization, this article also proposes a new and improved idea from the perspective of improving the exploration capability, pursuing to find the global optimal solution with higher search accuracy. The original ICA enhanced country diversity through use of an assimilation step, but it simply moved at a fixed step size and angle, which limited the randomness of the countries. The assimilation step in this paper is changed into one whereby colonial countries approach an imperialist country in a spiral rising way, so as to achieve better enhanced global search ability. As the movement mode close to the optimal value of colonial countries is optimized, the convergence speed of the optimization algorithm can also be improved to some extent.

The improved ICA is named the Spiral Rising Imperialist Competitive Algorithm (SR-ICA). Herein, we not only discuss the superiority of the improved algorithm theoretically, but also test its performance on many benchmark functions and evaluate it using a variety of methods.

The rest of the article is organized as follows: Section 2 reviews the original ICA and other knowledge related to the improved algorithm. Section 3 introduces the improved algorithm and describes the implementation thereof. Sections 4 and 5 verify the performance of the improved ICA through several experiments and apply the improved SR-ICA optimization algorithm to a path planning problem to observe its optimization ability in robot path planning application. Finally, Section 6 concludes.

Related Work

The Proposed ICA with an Added Concept of Spiral Rising

This section describes the basic steps of the SR-ICA. Figure 1 illustrates the optimization process of the original ICA, SR-ICA is improved based on this algorithm.

As mentioned above, the imperialist countries achieved better rule over the colonies by means of cultural, political, economic and folk customs assimilation. The process of assimilation is simulated in ICA by using colonies to move in a fixed step size towards imperialist countries. However, in the theory of “spiral rise”, the general direction and trend of the development of things is a progressive movement from low to high, simple to complex, and its direction and trend always appear as “spiral rise” movement. The basic characteristics of this movement behavior are forward, circuitous and periodicity. In this paper, the assimilation step of ICA is improved by referring to the “spiral rise” theory, and the original “straight approach” movement mode of colonies is improved to “spiral rising” movement.

The whale optimization algorithm (WOA) (Mirjalili & Lewis, 2016) proposed in 2016 imitates the movement mode of humpback whales in the hunting process and approaches the global optimal value by spiral contraction, WOA has good optimization ability. The improved algorithm completely changed the assimilation mechanism in ICA by referring to the movement mode of whale hunting in WOA, and the colonies moved closer to the imperialist countries in a “spiral rising” movement mode.

Figure 2 Forming the initial empires.

After the initial empires are formed, the colonial countries of each empire will move closer to the imperialist country of the empire at the center.

The whole process can be divided into two parts: (1) shrinking approach and (2) spiral circuitous approach.

1. Shrinking approach: in a single empire, the imperialist country will be the local optimal solution, all colonies will shrink and move closer to it, and the positions of the colonies will be updated during the shrinking approach. These behaviors are represented by the following equations: (9)${\displaystyle \overrightarrow{D}=\left|\overrightarrow{C}\cdot emp\left(k\right).imp.pos\left(t\right)-emp\left(k\right).col\left(i\right).pos\left(t\right)\right|}$ (10)${\displaystyle emp\left(k\right).col\left(i\right).pos\left(t+1\right)=emp\left(k\right).imp.pos\left(t\right)-\overrightarrow{A}\cdot \overrightarrow{D}}$

   where t is the tth iteration, $\overrightarrow{A}$ and $\overrightarrow{C}$ are coefficient vectors, $emp\left(k\right).imp.pos$ refers to the position vector of the imperialist country of the kth empire, $emp\left(k\right).col\left(i\right).pos$ is the position vector of the ith colonial country of the kth empire.

   Coefficient vectors $\overrightarrow{A}$ and $\overrightarrow{C}$ are given by: (11)${\displaystyle \overrightarrow{A}=2\overrightarrow{a}\cdot \overrightarrow{r}-\overrightarrow{a}}$ (12)${\displaystyle \overrightarrow{C}=2\cdot \overrightarrow{r}}$

   where, $\overrightarrow{a}$ linearly decreases from 2 to 0 during the iterative process, and $\overrightarrow{r}$ is a random vector on [0,1].

   The shrink approach is achieved through the reduction of $\overrightarrow{a}$, because the change in $\overrightarrow{A}$ is related to $\overrightarrow{a}$. In this process, $\overrightarrow{A}$ is a random value on [-a, a], so the change in $\overrightarrow{A}$ is affected by $\overrightarrow{a}$. When $\overrightarrow{A}$ takes a random value on [−1,1], it means that the movement strategy of the colonial countries is anywhere between the current positions and the position of their imperial power. Figure 3 shows some of the positions to which a colony might move in a 2-d space, as achieved by changing $\overrightarrow{A}$.

2. Spiral circuitous approach: in a single empire, the second strategy for colonial countries to approach the imperial power is termed spiral circuitous, as given by: (13)${\displaystyle \overrightarrow{\dot{D}}=\left|emp\left(k\right).imp.pos\left(t\right)-emp\left(k\right).col\left(i\right).pos\left(t\right)\right|}$ (14)${\displaystyle emp\left(k\right).col\left(i\right).pos\left(t+1\right)=\overrightarrow{\dot{D}}\cdot e^{bl}\cdot \mathit{cos}\left(2\pi l\right)+emp\left(k\right).imp.pos\left(t\right)}$ Where b is a constant for defining the shape of the logarithmic spiral; l is a random number on [−1,1]. Figure 4 2-d diagram of SR-ICA’s spiral circuitous.

   The addition of spiral rising mechanism can increase the spatial diversity of colonies, so in the whole process of assimilation, the positional changes are governed (among colonial countries) by the spiral rising mechanism. To synchronize the two movements, the position update factor p is introduced and the probability is set to 0.5. The mathematical model of this synchronization behavior is as follows: (15)${\displaystyle emp\left(k\right).col\left(i\right).pos\left(t+1\right)=\left\{\begin{array}{cc}emp\left(k\right).imp.pos\left(t\right)-\overrightarrow{A}\cdot \overrightarrow{D},\hfill & p<0.5\hfill \\ \overrightarrow{\dot{D}}\cdot e^{bl}\cdot \mathit{cos}\left(2\pi l\right)+emp\left(k\right).imp.pos\left(t\right),\hfill & p\ge 0.5\hfill \end{array}\right.}$

   The value of p is a random value on [0,1], $\overrightarrow{D}$ and $\overrightarrow{\dot{D}}$ are the distances between countries with two different modes of motion.

   The following Fig. 5 shows the comparisonof assimilation step before and after improvement.

   The spiral rising path is a zig-zag and round-about way, which approaches the global optimal solution in a special cyclic way. In the process of assimilation, the approach of colonial countries to imperialist countries in this way is conducive to increasing the optimal search space; on the premise of ensuring the enhancement of exploration ability, the exploitation ability will not be weakened, and the rate convergence will not decrease. The algorithm structure of the improved algorithm SR-ICA is presented in Table 1.

Experimental Verification and Comparisons

Various experiments are designed to verify the optimized performance of the proposed SR-ICA. First, SR-ICA is compared with its homologous optimization algorithms, the solution accuracy and rate of convergence of SR-ICA are evaluated by benchmark functions. Then the statistical significance tests and the complexity analysis of calculation time are carried out. In addition, the improved algorithm is compared with some other commonly used advanced optimization algorithms, and the performance of the algorithms is analyzed according to the experimental results. We also apply SR-ICA to solve high-dimensional problems, observing whether it can maintain its optimization ability in the face of high-dimensional problems, finally this paper will use SR-ICA to solve the path planning problem, proposing a new solution to the path planning problems.

Benchmark functions

Benchmark functions are generally used as a benchmark when evaluating the performance of intelligent optimization algorithms (Zhang et al., 2020). Different benchmark functions can be used to investigate the ability of intelligent optimization algorithms to deal with problems in different optimization situations. To measure the optimization ability of SR-ICA, 19 benchmark functions are selected (Table 2, Fig. 6).

Table 2:

Benchmark functions.

Group	Test function	Dim	Search range	fmin	Name
Multimodal	$f_1\left(x\right)=-20exp\left(-0.2\sqrt{\frac{1}{d}{\sum }_{i=1}^d{x}_i^2}\right)-exp\left(\frac{1}{d}{\sum }_{i=1}^{d}cos\left(2\pi x_i\right)\right)+20+exp\left(1\right)$	30	[−32,32]	0	Ackley
	$f_{\mathrm{2}}\left(x\right)=10d+{\sum }_{i=1}^d\left[x_i^2-10cos\left(2\pi x_i\right)\right]$	30	[−10,10]	0	Rastrigin
	$f_{\mathrm{3}}\left(x\right)={\sum }_{i=1}^d\left|x_{i}sin\left(x_i\right)+0.1x_i\right|$	30	[−10,10]	0	Alpine
	$f_{\mathrm{4}}\left(x\right)={sin}^2\left(\pi w_1\right)+{\sum }_{i=1}^{d-1}{\left(w_i-1\right)}^2\left[1+10{sin}^2\left(\pi w_i+1\right)\right]+{\left(w_d-1\right)}^2\left[1+{sin}^2\left(2\pi w_d\right)\right]$$w_i=1+\frac{x_i-1}{4}$	30	[−10,10]	0	Levy
	$f_{\mathrm{5}}\left(x\right)={\sum }_{i=1}^d\frac{x_i^2}{4000}-{\Pi }_{i=1}^{d}cos\left(\frac{x_i}{\sqrt{i}}\right)+1$	30	[−600,600]	0	Griewank
	$f_{\mathrm{6}}\left(x\right)={\sum }_{i=1}^d\left|x_i\right|+{\prod }_{i=1}^d\left|x_i\right|$	30	[−10,10]	0	Schwefe2.l22
	$\begin{array}{c}{f}_{\mathrm{7}}\left(x\right)=0.1\cdot {sin}^2\left(3\pi x_1\right)+{\sum }_{i=1}^d{\left(x_i-1\right)}^2\left[1+{sin}^2\left(3\pi x_{i+1}\right)\right]\hfill \\ +{\left(x_d-1\right)}^2\cdot \left[1+{sin}^2\left(2\pi x_d\right)\right]\hfill \end{array}$	30	[−5,5]	0	levy_montalo
Unimodal	$f_{\mathrm{8}}\left(x\right)={\sum }_{i=1}^d{x}_i^2$	30	[−100,100]	0	Sphere
	$f_{\mathrm{9}}\left(x\right)={\sum }_{i=1}^d{x}_i^2+{\left({\sum }_{i=1}^{d}0.5i{x}_i\right)}^2+{\left({\sum }_{i=1}^{d}0.5i{x}_i\right)}^4$	30	[−5,10]	0	Zakharove
	$f_{\mathrm{10}}\left(x\right)={\sum }_{i=1}^d\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right]$	30	[−5,10]	0	Rosenbrock
	$f_{\mathrm{11}}\left(x\right)={\left(x_1-1\right)}^2+{\sum }_{i=2}^{d}i{\left(2x_i^2-x_{i-1}\right)}^2$	30	[−5,10]	0	Dixonprice
	$f_{\mathrm{12}}\left(x\right)={\sum }_{i=1}^{d}i{x}_i^2$	30	[−10,10]	0	Sumsquare
	$f_{\mathrm{13}}\left(x\right)={\sum }_{i=1}^{d}i{x}_i^4+random\left[\right.0,1\left.\right)$	30	[−1.28,1.28]	0	Quartic
	$f_{\mathrm{14}}\left(x\right)={\sum }_{i=1}^d{\left(10^6\right)}^{\frac{i-1}{d-1}}{x}_i^2$	30	[−100,100]	0	Elliptic
Fixed low dimensional	$f_{\mathrm{15}}\left(x\right)=-\frac{1+cos\left(12\sqrt{x_1^2+x_2^2}\right)}{0.5\left(x_1^2+x_2^2\right)+2}$	2	[−5.12,5.12]	−1	Drop_wave
	$f_{\mathrm{16}}\left(x\right)=0.5+\frac{{sin}^2\left(x_1^2-x_2^2\right)-0.5}{{\left[1+0.001\left(x_1^2+x_2^2\right)\right]}^2}$	2	[−100,100]	0	Schaffer
	$f_{\mathrm{17}}\left(x\right)=-0.0001{\left(\left|sin\left(x_1\right)sin\left(x_2\right)exp\left(\left|100-\frac{\sqrt{x_1^2+x_2^2}}{\pi }\right|\right)\right|+1\right)}^{0.1}$	2	[−10,10]	−2.06	Cross_in_tray
	$f_{18}\left(x\right)=0.26\left(x_1^2+x_2^2\right)-0.48x_1{x}_2$	2	[−10,10]	0	Matyas
	f19(x) =  − cos(x1)cos(x2)exp(−(x1 − π)2 − (x2 − π)2)	2	[−100,100]	−1	Easom

DOI: 10.7717/peerjcs.1075/table-2

The 19 benchmark functions can be divided into three types. The first type is multi-modal functions, which includes seven benchmark functions (F1 to F7). The benchmark functions of this type have multiple local minimum values, which can verify the exploration ability of the algorithms: because they have multiple local minima, these types of functions can be used to evaluate the ability of the optimization algorithms to escape from local optimal values. The second type are unimodal functions: of the 19 benchmark functions, seven benchmark functions (F8-F14) are of this type. They have a unique global optimal value, so there are no local optima issues arising therewith. Unimodal functions can evaluate the optimization precision and convergence speed of the optimization algorithms, and the exploitation ability of the optimization algorithms can be seen from the experimental results. The third type fixed low-dimensional functions (F15-F19) have fixed dimensions and make it easier to find the optimal value than the first two benchmark functions. The dimension settings, search ranges, and global minima of these baseline functions are detailed in Table 2.

Statistical significance tests of experimental results

To compare the performance of the four optimization algorithms mentioned above, a Friedman test (Pelusi et al., 2020) was used to analyze their significance in this section. A Friedman test is performed on ICA, WOA, DE-ICA, and SR-ICA at the significance level of 5%, and the test results are displayed in Table 7.

Table 7:

Friedman test results.

Average Rank
F1-F7		F8-F14
Algorithm	Rank	Algorithm	Rank
ICA	3.88	ICA	3.57
WOA	2.25	WOA	2.79
DE-ICA	2.88	DE-ICA	2.57
SR-ICA	1.00	SR-ICA	1.07
p-value:	0.000	p-value:	0.003

DOI: 10.7717/peerjcs.1075/table-7

On the left-hand side of Table 7, seven multi-modal functions are given to solve the ranking results obtained from Friedman test with four different optimization algorithms. The lower the rank value, the better the optimization performance of the algorithm. The performance of SR-ICA was the best: the p-value for this set of comparisons is also given below (a p-value of less than 0.05 indicates that these algorithms are not correlated). On the right-hand side, we show the Friedman test of four optimization algorithms for seven unimodal functions, with the p-value of 0.003. This result shows that SR-ICA was significantly superior to all other optimization algorithms considered for comparison in terms of solution accuracy.

The F1-F7 benchmark functions ICA and WOA were subjected to a Friedman test with SR-ICA respectively, and the p-values were both 0.005. F8-F14 benchmark functions ICA and WOA are tested by Friedman with SR-ICA, and the p-values obtained are 0.008 and 0.014, respectively. These p-values are less than 0.05, which verifies that SR-ICA is significantly different from ICA and WOA, and there is no correlation between the algorithms. It can also be seen from the optimization ranking results of the algorithms in Table 7 that compared with the other three algorithms, the proposed SR-ICA can achieve the best comprehensive performance.

Comparison with other advanced optimization algorithms

To verify the superiority of SR-ICA in solving optimization problems, in this part, we choose some commonly used advanced intelligent optimization algorithms for comparison, use these algorithms to process the above multi-modal and unimodal benchmark functions, observing the experimental results to analyze the optimization performance of SR-ICA.

In this part, we chose six types of intelligent optimization algorithms for comparative experiments. These six optimization algorithms are relatively representative among the four categories of nature-inspired meta-heuristic algorithms mentioned in the introduction (they are also intelligent optimization algorithms that are often used to solve practical optimization problems Askari, Younas & Saeed, 2020) and they are: differential evolution (DE) (Lampinen & Storn, 2004) of evolution-based optimization algorithms, particle swarm optimization (PSO), the grey wolf optimizer (GWO) (Mirjalili, Mirjalili & Lewis, 2014) of swarm-based optimization algorithms, teaching-learning based optimization (TLBO) (Rao, Savsani & Vakharia, 2011) and brain storm optimization (BSO) (El-Abd, 2017) of human behavior-based optimization algorithms, and the sine-cosine algorithm (SCA) (Mirjalili, 2016) from among the physics-based optimization algorithms.

These six representative optimization algorithms are compared with the SR-ICA proposed herein. The experimental results are shown in Tables 8 and 9. The parameters of the other six optimization algorithms are summarized in Table 10. Similarly, for the six intelligent optimization algorithms, this paper refers to the relevant literatures to set their parameters to the optimal situation.

Table 8:

Experimental results: multi-modal benchmark functions.

F	DE	PSO	GWO	BSO	TLBO	SCA	SR-ICA
F1	Ave: 4.75E+00 Std: 6.97E−01	Ave: 4.37E−03 Std: 4.30E−03	Ave: 3.32E−14 Std: 4.79E−15	Ave: 1.40E−01 Std: 3.41E−01	Ave: 5.51E−15 Std: 1.66E−15	Ave: 1.54E+00 Std: 6.92E−01	Ave: 8.88E−16 Std: 8.88E−16
F2	Ave: 2.22E+02 Std: 1.46E+01	Ave: 4.25E+01 Std: 1.30E+01	Ave: 6.39E−01 Std: 2.60E+00	Ave: 4.68E+01 Std; 1.08E+01	Ave: 3.83E+00 Std: 4.11E+00	Ave: 6.36E+01 Std: 3.69E+01	Ave: 0.00E+00 Std: 0.00E+00
F3	Ave: 1.13E+01 Std: 1.57E+00	Ave: 5.77E−03 Std: 3.52E−03	Ave: 3.86E−04 Std: 5.10E−04	Ave: 5.08E−01 Std: 4.37E−01	Ave: 6.64E−09 Std: 3.58E−08	Ave: 6.94E−01 Std: 1.15E+00	Ave: 8.78E−13 Std: 4.81E−12
F4	Ave: 1.61E+00 Std: 5.93E−01	Ave: 3.50E−03 Std: 1.63E−02	Ave: 6.92E−01 Std: 1.59E−01	Ave: 6.04E+00 Std: 3.82E+00	Ave: 1.64E−01 Std: 1.45E−01	Ave: 2.83E+00 Std: 1.07E+00	Ave: 5.16E−14 Std: 1.64E−13
F5	Ave: 2.64E+00 Std: 7.25E−01	Ave: 8.54E−03 Std: 9.60E−03	Ave: 3.71E−04 Std: 2.03E−03	Ave: 1.27E+01 Std: 4.82E+00	Ave: 0.00E+00 Std: 0.00E+00	Ave: 1.01E+00 Std: 2.35E−01	Ave: 0.00E+00 Std: 0.00E+00
F6	Ave: 1.63E+01 Std: 5.36E+00	Ave: 6.76E−03 Std: 4.09E−03	Ave: 1.91E−19 Std: 1.68E−19	Ave: 5.51E−02 Std: 7.19E−02	Ave: 5.82E−25 Std: 1.69E−25	Ave: 6.10E−02 Std: 7.01E−02	Ave: 3.3E−182 Std: 0.00E+00
F7	Ave: 1.20E+00 Std: 9.31E−01	Ave: 3.77E−05 Std: 3.44E−05	Ave: 2.70E−01 Std: 5.10E−01	Ave: 3.33E−03 Std: 1.82E−02	Ave: 1.66E−02 Std: 3.77E−02	Ave: 1.98E+01 Std: 2.42E+00	Ave: 6.69E−14 Std: 2.74E−13

DOI: 10.7717/peerjcs.1075/table-8

Table 9:

Experimental results: unimodal benchmark functions.

F	DE	PSO	GWO	BSO	TLBO	SCA	SR-ICA
F8	Ave: 1.66E+02 Std: 8.52E+01	Ave: 2.82E−05 Std: 3.27E−05	Ave: 5.41E−33 Std: 7.56E−33	Ave: 8.80E−06 Std: 2.79E−06	Ave: 1.25E−49 Std: 6.40E−50	Ave: 7.28E+00 Std: 1.06E+01	Ave: 0.00E+00 Std: 0.00E+00
F9	Ave: 8.49E+00 Std: 6.29E+00	Ave: 3.59E+01 Std: 7.01E+00	Ave: 7.03E−16 Std: 1.14E−15	Ave: 8.07E+00 Std: 7.60E+00	Ave: 2.68E−03 Std: 1.81E−03	Ave: 1.27E+01 Std: 7.29E+00	Ave: 7.71E−87 Std: 4.22E−86
F10	Ave: 2.57E+02 Std: 1.04E+02	Ave: 6.04E+01 Std: 4.32E+01	Ave: 2.62E+01 Std: 6.40E−01	Ave: 5.25E+01 Std: 3.81E+01	Ave: 2.00E+01 Std: 6.30E−01	Ave: 6.76E+01 Std: 7.70E+01	Ave: 9.84E−13 Std: 2.63E−12
F11	Ave: 4.76E+01 Std: 2.59E+01	Ave: 1.32E+00 Std: 9.43E−01	Ave: 6.67E−01 Std: 1.08E−05	Ave: 1.92E+00 Std: 2.13E+00	Ave: 6.67E−01 Std: 4.79E−13	Ave: 8.59E+00 Std: 1.44E+01	Ave: 2.57E−01 Std: 8.51E−02
F12	Ave: 2.12E+01 Std: 1.00E+01	Ave: 3.57E−04 Std: 5.68E−04	Ave: 5.90E−34 Std: 8.56E−34	Ave: 1.32E−01 Std: 2.13E−01	Ave: 1.82E−50 Std: 8.84E−51	Ave: 6.32E−01 Std: 7.23E−01	Ave: 0.00E+00 Std: 0.00E+00
F13	Ave: 1.39E−01 Std: 4.07E−02	Ave: 8.70E−02 Std: 2.92E−02	Ave: 4.61E−04 Std: 2.70E−04	Ave: 4.67E−02 Std: 1.94E−02	Ave: 1.15E−03 Std: 3.75E−04	Ave: 5.47E−02 Std: 4.15E−02	Ave: 3.03E−05 Std: 2.06E−05
F14	Ave: 3.20E+05 Std: 1.27E+05	Ave: 4.26E−01 Std: 4.67E−01	Ave: 1.28E−29 Std: 2.08E−29	Ave:1.94E+06 Std: 1.59E+06	Ave: 1.84E−45 Std: 9.27E−46	Ave: 1.13E+03 Std: 1.51E+03	Ave: 0.00E+00 Std: 0.00E+00

DOI: 10.7717/peerjcs.1075/table-9

Table 10:

Parameter settings for optimization algorithms.

Algorithm	Parameter settings
DE	N = 200; MaxIt=300 F = 0.5; CR=0.8; mutationStrategy=1; crossStrategy=2
PSO	N = 200; MaxIt=300 Vmax=6; wMax=0.9; wMin=0.2; c1=2; c2=2
GWO	N = 200; MaxIt=300; e = 2.717
BSO	N = 200; MaxIt=300; n_c = 2; prob_one_cluster = 0.8
TLBO	N = 200; MaxIt=300
SCA	N = 200;MaxIt=300; a = 2
SR-ICA	nPop=200; nEmp=3; alpha=1; zeta=0.1; MaxIt=300 pRevolution=0.05; mu=0.1; e = 2.717; b = 5; P = 0.5

DOI: 10.7717/peerjcs.1075/table-10

Compared with the six optimization algorithms mentioned above, the minimum values found by SR-ICA in the optimization problems of the 14 benchmark functions are the best. The five benchmark functions F2, F5, F8, F12, and F14 can all find their minimum values using SR-ICA, among which only the F5 benchmark function TLBO can also find its minimum value.

In view of the experimental data from the seven optimization algorithms and the convergence of each optimization process (Fig. 8), SR-ICA has high convergence precision and a rapid rate of convergence. The optimization performance of the improved algorithm is assessed from different angles by unimodal and multi-modal functions. After adding the spiral-rise mechanism, the population diversity of the original ICA is improved, and its exploration ability is greatly enhanced. Compared with other commonly used advanced intelligent optimization algorithms, the ability of SR-ICA to solve optimization problems is also very good.

Optimization Analysis of Path Planning Problems

In this section, the SR-ICA was applied to the mobile robot path planning problem (MRPPP), and the path optimization comparison with other optimization algorithms is made to analyze the ability of SR-ICA method to find the optimal path.

In this experiment, MRPPP refers to finding a path with the shortest distance from a given starting point to a given end point that can avoid all obstacles in the area. This is an important technique used in design of mobile robots (Deng et al., 2021). The practical application of the optimization algorithms is to choose the path planning problems, use SR-ICA and other optimization algorithms to find the optimal path of robot movement.

This simulation experiment was an off-line path planning task, and the planning environment was fixed, which is known as static environment planning. Obstacles are set in advance in the known global static 30 × 30 raster matrix environment, and the starting coordinates of the robot are (1,1), and the end coordinates are (30,30). The distribution of obstacles is shown in Fig. 9.

When finding the optimal path, our requirement is to find the path with the shortest distance without touching obstacles, and to apply the optimization algorithms to finding the shortest path. In this way, the objective function of the optimization algorithms is as shown in Eq. (17): (17)${\displaystyle L\left(Path\right)=\sum _{i=0}^{n-1}\sqrt{{\left(x{l}_{i+1}-x{l}_i\right)}^2+{\left(y{l}_{i+1}-y{l}_i\right)}^2}.}$

In the practical application, we select the optimization algorithm SR-ICA for optimal path optimization, and also choose ICA, WOA, and PSO for purposes of comparison. When using these four optimization algorithms to find the optimal paths, if the path found fails to avoid all of the obstacles, we will regard it as having failed to find the optimal path. SR-ICA optimization of the path problem is made, the results of solving the above path planning problem are shown in Fig. 10.

The results of the four optimization algorithms used to solve the above path planning problem are listed in Table 12, and the optimization performance comparison results are shown in Fig. 11. The parameter settings of these four optimization algorithms when solving the problem are summarized in Table 13. The population was set to 100 when using these four optimization algorithms and the number of iterations was set to 150.

According to the path distance results obtained by the different methods used to solve the path planning problem of the set off-line single robot in Table 12, it can be seen that the optimized path result distance of SR-ICA is the shortest, indicating that this optimization method can find a more optimized effective path than three other optimization algorithms when solving this path planning problem.

SR-ICA achieved relatively good optimization results in both the optimization applications of benchmark functions and the application in large-scale high-dimensional problems and robot path planning problems. All of these indicate that the optimization ability of the algorithm is improved by the addition of the spiral rising strategy, which also provides a new idea for the improvement of the optimization algorithms in the future.

Conclusions

This article proposes a new improved intelligent optimization algorithm SR-ICA based on ICA. SR-ICA adopts a spiral rising mode in the position updating process, which increased the search space of the solution group through such particle motions, and improved the precision of the optimization results. SR-ICA had a stronger ability to avoid local optimal solutions, which improved the global search ability of the algorithm.

From the experimental results of 19 benchmark functions, it can be seen that SR-ICA had higher search accuracy of optimal solutions, faster search speed and higher stability of search results. In addition, SR-ICA was more suitable for handling high-dimensional and large-scale problems because of its own search mechanism, and the search accuracy did not decrease significantly with the increase of dimensionality, which provides an improved new idea for solving this kind of optimization problems.

The SR-ICA proposed in this paper achieves better optimization results in both mathematical models and engineering optimization problem. In future studies, we will focus on reducing the temporal complexity of SR-ICA. When solving optimization problems, we desire faster solution speed, therefore, we will conduct research aimed at reducing the time-complexity of SR-ICA, and hope that the optimization performance of SR-ICA can be further improved.

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