## 5 CGE-SSA Multi-Objective Optimization
### Improved Sparrow Algorithm
#### 5.2.2 Improved Gold Sine Strategy Update Discovery  
The traditional SSA finder position updating process is prone to premature convergence into local optimization; to solve this problem, a sinusoidal function was used for the optimization search, thus expanding the search space of the algorithm and improving its global optimization ability. The golden section coefficient was also introduced to enable the algorithm to search more fully in the local region, in order to avoid falling into local optimality [33]. In order to make it easier for the algorithm to move beyond the local optimal solution and explore a wider solution space, the optimal solution perturbation strategy was added to the golden sine search. The improved finder update formula is as follows:  
\[X_{i,j}^{t+1}=\left\{\begin{array}{ll}X_{i,j}^{t}\cdot\left|\sin(r_{1}) \right|+r_{2}\cdot\sin(r_{1})\cdot\left|x_{1}\cdot X_{best1}^{t}-x_{2}\cdot X _{i,j}^{t}\right|&\text{if }R<ST\\ X_{i,j}^{t}+Q\cdot L&\text{if }R\geq ST\end{array}\right. \tag{7}\]  
\[X_{best1}^{t}=X_{best}^{t}+\left|X_{i,j}^{t}-X_{best}^{t}\right|\cdot A^{+}\cdot L \tag{8}\]  
where \(X_{i,j}\) is the position of the sparrow; \(X_{best}\) is the global optimal position; \(X_{best1}\) is the global optimal position after the perturbation; \(Q\) is a random number obeying a normal distribution; \(r_{1}\in[0,2\pi]\) and \(r_{2}\in[0,\pi]\) are random numbers determining the distance and direction of motion of the individual for the next iteration, respectively; \(x_{1}=-\pi+2\pi(1-\pi)\), \(x_{2}=-\pi+2\pi\tau\) are the coefficients obtained based on the golden section coefficient \(\tau\); \(\tau=\left(\sqrt{5}-1\right)/2\); \(L\) is a \(1\times\) dim matrix whose elements are all \(1\); and \(A^{+}=A^{\text{T}}\) (\(AA^{\text{T}}\))\({}^{-1}\), where \(A\) is a \(1\times\) dim matrix with elements \(-1\) or \(1\).  
Figure 8: Population distribution map: (**a**) random population distribution; (**b**) circular sinusoidal chaos map population distribution; (**c**) histogram of randomized population distribution; and (**d**) histogram of circular sinusoidal chaotic mapping population distribution.  
In order to verify that the sinusoidal function can expand the search space, and thus improve global optimization ability, the benchmark test function \(f_{10}\) was used to compare the optimization ability of the algorithm with and without the sinusoidal function [34]. The number of populations was set to 100, the maximum number of iterations to 100, with the cycle recurring 50 times, and the function f iteration curve is shown in Figure 9, where it can be seen that, when there is a sinusoidal function, the iteration curve converges to \(8.88\times 10^{-16}\); meanwhile, when there is no sinusoidal function, the iteration curve does not converge, and the optimal value of the search is \(1.01\times 10^{-5}\). It can thereby be concluded that the position updating formula can effectively improve the algorithm's convergence speed and accuracy when there is a sinusoidal function.  
When \(R\geq ST\), some sparrows have found the danger, at which point the discoverer moves randomly to the neighborhood of the current position, according to a normal distribution, and conducts a local search. When \(R<ST\), the global optimal position is not a fixed value, and adding random perturbation to increase the search range allows the finder to further expand the search space under the effect of the golden sine strategy in order to conduct a global search.