# An Improved Squirrel Search Algorithm for Global Function Optimization
## 3 The improved Squirrel Search Algorithm
### The Jumping Search Method
#### 3.2.3 Characters of the Jumping Search Method
A Holm test is carried out and the results are shown in Table 13:

It can be seen from Table 13 that \(P_{1}<\alpha(k-1)\), \(P_{2}<\alpha(k-2)\), \(P_{3}<\alpha(k-3)\), \(P_{4}<\alpha(k-4)\), \(P_{5}>\alpha(k-5)\), and the original hypothesis is rejected at the 5% significance level. Therefore, compared with AND-RSN-PSO, distABC, IGSA/PSO and MDE, ISSA has a significantly better performance. ISSA has a smaller average rank, though it does not outperform PSO-GWO. In summary, the ISSA proposed in this paper has obvious advantages on convergence precision and stability in terms of the high dimensional functions. Figure 5a-u shows the convergence curves of the algorithms mentioned above on F1-F21 in order to compare the convergence performances intuitively. The yellow, cyan, purple, green, black, blue and red curves refer to MDE, distABC, IGSA/PSO, AND-RSN-PSO, PSO-GWO, SSA and ISSA, respectively. The abscissa of each figure is the iteration and the ordinate of each figure and is the best value found so far. The title of each figure is given the name of F1-F21. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(i\) & **Algorithm** & \(z=(R_{i}-R_{6})/\sqrt{\frac{k(k+1)}{\alpha}=(R_{i}-R_{6})/\sqrt{\frac{\kappa(k+ 4)}{6\alpha 11}}}=\) & \(P_{i}\) & \(\alpha/(k-i)\) \\ \hline
1 & AND-RSN-PSO & \((5-1)/0.7977=5.0144\) & 0 & 0.01 \\
2 & distABC & \((5-1)/0.7977=5.0144\) & 0 & 0.01 \\
3 & IGSA/PSO & \((4.1818-1)/0.7977=3.9887\) & 6e\(-05\) & 0.0167 \\
4 & MDE & \((3.3636-1)/0.7977=2.9630\) & 0.0030 & 0.025 \\
5 & PSO-GWO & \((2.4545-1)/0.7977=1.8234\) & 0.068 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: The results of the Holm test. Figure 5: _Cont._

## 5 Conclusions

This paper proposes an improved squirrel search algorithm. In terms of SSA, the winter searching method cannot develop the search space sufficiently, and the summer searching method is too random to guarantee the convergence speed. ISSA introduces the jumping search method and the progressive search method. For the jumping search method, the 'escape' operation in winter supplements the population diversity and fully exploits the search space, while the 'death' operation in summer explores the search space more sufficiently and improves the convergence speed. Aside from this, the mutation in the progressive search method retains the evolutionary information more effectively and maintains the population diversity. ISSA selects the suitable method by the linear regression selection strategy according to the variation tendency of the best fitness value, which improves the robustness of the algorithm. Compared with SSA, ISSA pays more attention to developing the search space in winter, and pays more attention to exploring around the elite individual in summer, which keeps a good balance between development and exploration improves the convergence speed and the convergence accuracy. Moreover, ISSA selects a proper search strategy along with the optimization processing, so ISSA has greater possibilities in finding the optimal solution. The experimental results on 21 benchmark

Figure 5: The convergence curves of ISSA and other algorithms. functions and the statistical tests show that the proposed algorithm can promote the convergence speed, improve the convergence accuracy and maintain the population diversity at the same time. Furthermore, ISSA has obvious advantages in convergence performances compared with other five intelligent evolutionary algorithms. [PERSON] designed the experiments and revised the paper; [PERSON] shared the ideas and mainly wrote the paper. This research was funded by the National Natural Science Foundation of China under grants NO.61501107 and NO.61603073, and the Project of Scientific and Technological Innovation Development of Jilin NO.201750227 and NO.201750219. The authors declare no conflict of interest. ## References

* [PERSON] and [PERSON] (2016) [PERSON]; [PERSON] Average Convergence Rate of Evolutionary Algorithms. _IEEE Trans. Evol. Comput._**2016**, _20_, 316-321. [CrossRef]
* [PERSON] et al.