A Novel Particle Swarm Optimization Algorithm (NIPSO) Integrating and Improved Sine Chaos Map-with Code

A Novel Particle Swarm Optimization Algorithm (NIPSO) Integrating Improved Sine Chaos Mapping

Abstract: In order to deal with the problems of the traditional particle swarm optimization (PSO), such as uneven initial position, easy to achieve local optimum, and low search accuracy, a new PSO algorithm based on improved Sine chaotic map is proposed. An improved Sine chaotic mapping technology is used to replace the traditional pseudo-random number method to generate the initial particle population to enrich the diversity of the population. On the basis of the original basic position update formula, two new position update mechanisms are added, and a Gaussian mutation operator is introduced respectively to achieve the dynamic balance between the algorithm exploration performance and development performance, and to make the particles effectively jump out of the local optimum during the iterative process

1. Particle swarm optimization algorithm

The specific principle of the basic particle swarm optimization algorithm reference blog

2. Improved particle swarm optimization algorithm

2.1 Improved Sine Chaos Mapping

In view of the shortcomings of ordinary chaotic maps, the improved chaotic maps are often selected as the initialization of the swarm intelligence optimization algorithm. The two most improved ones are Logistic map and Sine map. In view of the improved Sine chaotic map has good space ergodicity, rich population diversity, uniform distribution in the phase plane, simple structure and high efficiency, this paper chooses the improved Sine chaotic map As an initialization method for the improved particle swarm optimization algorithm. The improved Sine chaos formula is as follows:
$$\{\begin{array}{l}{a}_{i+1}=\sin \left(cf.a_i\right)\\ b_{i+1}=\sin \left(cf.b_i\right)\\ y_{i+1}=\left(a_{i+1}+b_{i+1}\right)\%1\end{array}\text{\hspace{1em}( 5 )}$$
In the formula:$a_i$Japanese $b_i$The initial value ranges from $(0,1)$ ; control parameter$\kappa$ =$1200;y_{i+1}$It is the iterative chaotic sequence value; $\%$ is the remainder operation.

2.2 Particle swarm improvement

In order to dynamically coordinate the global search and local search at different stages of the optimization process, the following new particle position update equation is proposed to improve the speed and accuracy of finding the optimal solution without weakening its global search ability.
$$\begin{array}{rl}& {\mathbf{X}}_i(t+1)=\\ & \left\{\begin{array}{lr}{\mathbf{X}}_i(t)+{\mathbf{V}}_i(t+1),& r_3>0.5\sin \left(\frac{t}{2t{\_}_{\max }}\right)\\ X_i\left(t\right)+u\left({\mathbf{X}}_{\mathrm{gb}}\left(t\right)-X_i(t)\right),l_{i\max }\left(t\right)>a& \\ (1+v){\mathbf{X}}_i(t),& l_{i\text{ max }}\left(t\right)\le \end{array}\right\}{r}_3\le 0.5\sin \left(\frac{t}{2t{\_}_{\max }}\right)\end{array}\text{\hspace{1em}( 6 )}$$
where:$t_{\max }$is the total number of iterations; $r_3$for $[0,1]$ random number between;$\alpha$ is the variation factor;$v$ sum$u$ is a random number that obeys Gaussian distribution;$l_{i\max }\left(t\right)$ is X i ( t ) \boldsymbol{X}_i(t)in all dimensions${\mathbf{X}}_i\left(t\right)$ and$X_{\mathrm{gb}}(t)$ 之间的最大差值 的权重, 可以描述为
$$\{\begin{array}{l}{l}_{\text{imax }}(t)=\max \left\{l_i(t),l_{i2}\left(t\right),\dots ,l_{}\left(t\right),\dots ,l_{in}(t)\right\}\\ l_{}(t)=\frac{|x_{}(t)-x_{}(t)|}{|x_{}(t)|}\end{array}\text{\hspace{1em}( 7 )}$$
where:$x_{}\left(t\right)$ is thettthiiat iteration$t$The kkth$of\; the\; i$ particle position$k-$ dimensional value,$x_{gb}\left(t\right)$ is thettthThe kkthglobal best position at iteration$t$$k-$ dimensional value, and$l_{}(t)$ 则是 $x_{}\left(t\right)$ and$x_{gb}$The difference between$($$t$$)$$x_{gb}\left(t\right)$ ratio. It should be noted that the random number$uwa$ vv$Let\; v$ be (8) a smooth function
$$\{\begin{array}{l}u\sim \mathrm{N}\left(1,p_u^2\right),p_u=\sqrt{m\cos \left(\frac{(t-1)}{2t{\_}_{\max }}\right)}\\ v\sim N\left(r_4,p_v^2\right),p_v=r_5\end{array}\text{\hspace{1em}( 8 )}$$
where:$r_4$is $[-1.5,1.5]$ , a random number between$r_5$for $[0,1]$ random number between,$\mu$ is an adaptive coefficient determined by equation (9).

Where: $r_4$is $[-1.5,1.5]$ , a random number between$r_5$for $[0,1]$ random number between,$The\; \mu$ equation ( 9 ) is the standard deviation parameter
$$m=\{\begin{array}{ll}\frac{20}{3}{l}_{i\text{ max }}^2(t)-7{min}_{i\text{ max }}\left(t\right)+\frac{7}{3},& a<l_{i\max }\left(t\right)\le 0.5\\ 0.3,& l_{i\text{ max }}\left(t\right)>0.5\end{array}\text{\hspace{1em}( 9 )}$$
According to formulas (8) and (9), it can be seen that the random number$The\; concentration\; of\; u$ is not only affected by the number of iterations, but also by the parameter$l_{\text{max }}\left(t\right)$ influence.

Considering the influence of inertia weight and acceleration constant coefficient on finding the global optimum, the influence of individual and global optimum fitness is added, so that the inertia weight can be adjusted adaptively, and the acceleration coefficient is directly expressed in the literature. Therefore, the inertia coefficient $w_i$and acceleration coefficients $c_1、c_2$can be expressed as:
$$w_i=\{\begin{array}{cc}\left|\frac{w_{\max }f\left(X_{\mathrm{gb}}(t)\right)}{f\left({\mathbf{X}}_{\mathrm{pbi}}(t)\right)}-\frac{\left(w_{\max }-w_{\min }\right)t}{t_{\max }}\right|,& f\left({\mathbf{X}}_{pbi}\left(t\right)\right)>0\\ w_{\max }-\frac{\left(w_{\max }-w_{\min }\right)t}{t_{\max }},& f\left(X_{pbi}\left(t\right)\right)=0\\ \left|\frac{w_{\max }f\left(X_{pbi}\left(t\right)\right)}{f\left(X_{gb}\left(t\right)\right)}-\frac{\left(w_{\max }-w_{\min }\right)t}{t_{\max }}\right|,& f\left(X_{pbi}\left(t\right)\right)<0\end{array}\text{\hspace{1em}(10)}$$

$$\{\begin{array}{l}{c}_1=2.5-\frac{2}{t_{\max }}\\ c_2=0.5+\frac{2}{t_{\max }}\end{array}\text{\hspace{1em}( 11 )}$$
in formula:$w_{\text{min }}$is the minimum value of inertia weight, $w_{\max }$is the maximum value of the inertia weight, $f_{\mathrm{pb}i}$is the individual optimal fitness of the particle, $f_{\mathrm{gb}}$is the global optimal fitness of the particle.

3. Experimental results

4. References

[1] Liu Lei, Jiang Bowen, Zhou Hengyang, etc. A New Particle Swarm Optimization Algorithm Fused with Improved Sine Chaos Mapping [J/OL]. Journal of Xi'an Jiaotong University: 1-11 [2023-05-25]. http://kns. cnki.net/kcms/detail/61.1069.T.20230512.1619.002.

5. Matlab code

6. Python code