# 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
For the summer stage, if the individuals are safe, the search strategy is similar to the jumping search method, but the gliding step is replaced by the \(Le^{\prime}vy\) flight. If the individuals are endangered, mutate one dimension of the individuals randomly to maintain the diversity of the population. Algorithm 3 shows the pseudo code of the progressive search method:

```
Input:\(pop\) Output:\(pop_{raw}\) if\(S_{t}^{t}\geq S_{min}\) if\(r\geq P_{dp}\) season = winter else  season = summer end ifseason == winter for\(p_{1}=1\) for\(p_{1}=1\) to \(popsize\) \(FS_{i}^{t+1}=FS_{i}^{t}+d_{g}\times G_{c}\times(F_{h}^{t}-FS_{i}^{t})\) end if\(r<P_{dp}\) select one dimension randomly and change it between \(FS_{L}\) and \(FS_{H}\)  end if end if ifseason == summer for\(p_{1}=1\) to \(popsize\) if\(r\geq P_{dp}\) \(FS_{i}^{t+1}=F_{h}^{t}+Le^{\prime}vy(x)\times(F_{h}^{t}-FS_{i}^{t})\) end if if\(r<P_{dp}\) select one dimension randomly and change it between \(U\) and \(L\)  end if end
```

**Algorithm 3**Pseudo Code of the Progressive Search Method

### Linear Regression Selection Strategy

According to the introduction in Sections 3.2 and 3.3, due to the fact that individuals generated in summer are more similar to \(F_{h}\), the jumping search method improves the convergence speed in an obvious manner. As the progressive search method retains the evolutionary information efficiently, the population diversity can be better maintained. Meanwhile, during the optimization process of the minimization problem, the best fitness value of the population is supposed to be a downward trend. If the population diversity is not abundant enough, the search space will not be developed sufficiently, meaning the algorithm will not convergence efficiently and the best fitness value will fluctuate or even become larger. Considering that different problems are suitable for different evolutionary strategies, this paper proposes a linear regression selection strategy to choose the appropriate updating method from the two methods mentioned above. The details are as follows:

Divide the whole evolutionary process into \(n\) substages evenly. The optimization starts with the jumping search method, calculating the linear regression equations of the best fitness value when asubstage is finished. There are three linear regression equations needed to be calculated: the best fitness value of the first half substage, the second half substage and the whole substage. If two or more equations' slopes are positive, the best fitness value may fluctuate or become larger, that is, the population diversity needs to be supplemented. The jumping search method is not suitable for the current problem. Therefore, the progressive search method is selected to finish the evolution. Otherwise, the population diversity is abundant enough, and will continue evolving through the jumping search method to converge faster. Figure 2 is the illustration of the linear regression selection strategy introduced above. Taking \(n=10\) and \(t=0.2\) T as an example, calculate the linear regression equations of the three regions: [0.1 T, 0.15 T], [0.15 T, 0.2 T] and [0.1 T, 0.2 T]. It can be seen from Figure 2 that the slope of ab is negative, and the slopes of bc and ac are positive, which means that the best fitness value will fluctuate or become larger, therefore, select the progressive search method to finish the evolution. Figure 3 is the procedure of ISSA. Figure 3: The procedure of improved squirrel search algorithm (ISSA). Figure 2: The illustration of the linear regression selection strategy. ## 4 Analysis of the Experimental Results

### Benchmark Functions

In order to test the performance of the proposed ISSA, a series of experiments are carried out. All the experiments work on CPU: Intel Core i5-7200 M, 4 G RAM, 2.70 GHZ, Windows 10 and Matlab R2016a.