# 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
The points in the figures refer to the individuals' positions, and the blue, green, black and red points refer to SSA, the jumping search method, the progressive search method and ISSA, respectively. The abscissa and the ordinate of each figure are the search range for the function; for F2 the search range is from \(-10\) to \(10\) and for F5 the search range is from \(-100\) to \(100\). The title of each figure is the name of F2 or F5. Table 4 shows the variance of the population's fitness values when the algorithms converge to about \(10^{-4}\) of benchmark functions. The functions contain the high dimensional unimodal functions F11 and F12 and the high dimensional multimodal functions F20 and F21. The number before '\(\pm\)' is the variance's mean and the number after '\(\pm\)' is the variance's standard deviation. '/' represents the algorithm failure to converge to \(10^{-4}\) after evaluating 30,000 times. Figure 4: Convergence curves of the proposed methods and SSA. It can be seen from Table 3 that SSA has the most dispersed individual distribution because of the individuals' re-initialization. For the three proposed strategies mentioned in this paper, individuals of the progressive search method distribute most dispersedly. ISSA takes second place, and the jumping search method has the most concentrated distribution. The progressive search method mutates individuals on a certain dimension, which will make more individuals distribute on lines \(x_{1}=0\) and \(x_{2}=0\). As a result, the benchmark functions have a great chance to converge to the optimal value. SSA re-initializes the whole individual, which makes the individuals distribute in the search space irregularly. From the analysis above, the progressive search method has the best population diversity while the jumping search method performs worst when the population evaluates to the fixed accuracy. Data in Table 4 show that the variances of SSA and the progressive search method are much larger than the variances of the jumping method and ISSA. Excessive population diversity slows down the convergence speed and some functions cannot converge to the fixed accuracy. Meanwhile, for the population diversity of the three proposed strategies in this paper, the progressive search method performs best, ISSA takes second place and the jumping search method performs worst. #### 4.3.3 Comparison of Comprehensive Performance on Four Methods

In order to compare the comprehensive performance of the four methods, Table 5 shows the mean and standard deviation of the optimal value obtained in 30 independent experiments on the 21 benchmark functions. For each method, the population size is 30, the total of evolution times is 30,000 and the \(N_{f_{3}}\) is 3. The number before '\(\pm\)' is the mean and the number after '\(\pm\)' is the deviation of the obtained best fitness value; '\(+\)', '\(-\)' and '\(=\)' represent that the means of the corresponding method are better than ISSA, worse than ISSA and equal to ISSA, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Name**} & \multicolumn{4}{c}{**Method**} \\ \cline{2-5}  & **SSA** & \begin{tabular}{c} **Jumping Search** \\ **Method** \\ \end{tabular} & **Progressive Search Method** & **ISSA** \\ \hline F11 & \begin{tabular}{c} 1.2837e+05 \(\pm\) \\ 7.3449e+04 \\ 5.7955e+04 \(\pm\) \\ 2.1202e+04 \\ \end{tabular} & \begin{tabular}{c} 2.5981e–04 \(\pm\) \\ 2.4867e–04 \\ 7.1891e–05 \(\pm\) \\ 4.6142e–05 \\ 6.0584e–05 \(\pm\) \\ 2.5350e–05 \\ 8.5775e–05 \(\pm\) \\ 8.4886e–05 \\ \end{tabular} & \
\begin{tabular}{c} 1.1115e+04 \(\pm\) 3.3201e+03 \\ 7.1891e–05 \(\pm\) \\ 1.3107e+03 \(\pm\) 543.9195 \\ 1.2767 \(\pm\) 0.2946 \\ 7.2621e+04 \(\pm\) 6.0527e+04 \\ 7.2621e+04 \(\pm\) 6.0527e+04 \\ \end{tabular} &
\begin{tabular}{c} 0.0017 \(\pm\) 0.0015 \\ 7.2366e–04 \(\pm\) 6.0848e–04 \\ 2.2921e–04 \(\pm\) 1.7366e–04 \\ 6.2753e–04 \(\pm\) 4.7156e–04 \\ 6.2753e–04 \(\pm\) 4.7156e–04 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The comparison of the population’s fitness value.