# Modified Cuckoo Search Algorithm with Variational Parameters and Logistic Map

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary

#### 2.1. Cuckoo Search Algorithm

Algorithm 1: Cuckoo Search via Lévy Flights |

#### 2.2. The Disadvantages of the Cuckoo Search Algorithm

- InitializationCuckoo search algorithm uses the random number to initiate these location of nests. Sometimes, the location of these nests will be the same, and sometimes the location of these nests are not properly dispersed in a defined area. Therefore, it causes repeated calculations and the easy chance to fall into local optimal solution [24].
- Parameters $\alpha $ and ${p}_{a}$In most cases, Yang and Deb used $\alpha =O(L/10)$ or $\alpha =O(L/100)$, where L is the characteristic scale of the problem of interest [29]. Yang and Deb also suggested ${p}_{a}$ = 0.25 [18]. In other words, $\alpha $ and ${p}_{a}$ are fixed number. The properties of the two parameters are the shortcomings of the algorithm, because ${p}_{a}$ and $\alpha $ should be changed with the progress of iterator, when CS algorithm search a local optimal solution and the global optimal solution.
- Boundary issueCS algorithm uses Lévy flights and random walk to find nest location [18,30]. The locations of some nests may be out of the boundary; when this happens CS algorithm uses the boundary value to replace these location. The bound dealing method will result in a lot of nests at the same location on the boundary, which is inefficient.

## 3. Modified Cuckoo Search Algorithm: VLCS

#### 3.1. Nest Location of Each Host Are Initialized by Logistic Map of Each Dimension

Algorithm 2: Nest location of each host is initialized by logistic map of each dimension |

#### 3.2. Step Size and ${p}_{a}$ Are Changed by Coefficient Function

#### 3.3. Boundary Is Constrained by Logistic Map of Each Dimension

**r**, which is changed by logistic map.The reason for using logistic map is the same as in Section 3.1. Algorithms 2 and 3 use the same

**r**. This

**r**can guarantee that the location of each nest is calculated only once.That means the

**r**reduces repeated calculations and accelerates the convergence process. The pseudo code of boundary processing is shown in Algorithm 3.

Algorithm 3: Boundary is constrained by logistic map of each dimension |

#### 3.4. Proposed VLCS algorithm

Algorithm 4: VLCS algorithm |

## 4. Simulation Experiments

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Test Function | Dimension | Range | Optimum |
---|---|---|---|

${f}_{01}={\sum}_{i=1}^{d}{x}_{i}^{2}$ | 15 | ${x}_{i}\in [-5.12,5.12]$ | 0 |

${f}_{02}=-cos(x)cos(y)exp[-{(x-\pi )}^{2}-{(y-\pi )}^{2}]$ | 2 | $x,y\in [-100,100]$ | −1 |

${f}_{03}={\sum}_{i=1}^{n-1}[100{({x}_{i+1}-{x}_{i}^{2})}^{2}+{({x}_{i}-1)}^{2}]$ | 15 | $x\in [-5,5]$ | 0 |

${f}_{03}=-{\sum}_{i=1}^{5}cos[(i+1)x+1]{\sum}_{i=1}^{5}cos[(i+1)y+1]$ | 2 | $x,y\in [-10,10]$ | −186.7309 |

${f}_{04}=\frac{1}{4000}{\sum}_{i=1}^{d}{x}^{2}-{\prod}_{i=0}^{d}cos(\frac{{x}_{i}}{\sqrt{i}})+1$ | 15 | $x\in [-600,600]$ | 0 |

${f}_{05}=\begin{array}{c}-20exp[-20\sqrt{\frac{1}{d}{\sum}_{i=1}^{d}{x}_{i}^{2}}]\hfill \\ -exp[\frac{1}{d}{\sum}_{i=1}^{d}cos(2\pi {x}_{i})]+(20+e)\hfill \end{array}$ | 15 | $x\in [-32.768,32.768]$ | 0 |

${f}_{06}={\sum}_{i=1}^{d-1}[{(1-{x}_{i})}^{2}+100{({x}_{i+1}-{x}_{i}^{2})}^{2}]$ | 16 | $x\in [-10,10]$ | 0 |

${f}_{07}={\sum}_{i=1}^{d}[-{x}_{i}sin(\sqrt{|{x}_{i}|})]$ | 10 | $x\in [-500,500]$ | −4189.829 |

${f}_{08}=10d+{\sum}_{i=1}^{d}[{x}_{i}^{2}-10cos(2\pi {x}_{i})]$ | 10 | $x\in [-5.12,5.12]$ | 0 |

${f}_{09}=-{\sum}_{i=1}^{d}sin({x}_{i}){[sin(\frac{i{x}_{i}^{2}}{\pi})]}^{2m}$ | 5 | $m=10,x\in [0,\pi ]$ | −4.6877 |

${f}_{10}=10\ast d+{\sum}_{i=1}^{d}({x}_{i}^{2}-10cos(2\pi {x}_{i}))$ | 20 | $x\in [-10,10]$ | −654.6 |

${f}_{11}={\sum}_{i=1}^{d}{({\sum}_{j=1}^{i}{x}_{j})}^{2}$ | 30 | $x\in [-100,100]$ | 0 |

${f}_{12}={\sum}_{i=1}^{d}{({x}_{i}+0.5)}^{2}$ | 30 | $x\in [-100,100]$ | 0 |

${f}_{13}={\sum}_{i=1}^{d}i{x}_{i}^{4}+random[0,1)$ | 30 | $x\in [-1.28,1.28]$ | 0 |

${f}_{14}={\sum}_{i=1}^{d}[{x}_{i}^{2}-10cos(2\pi {x}_{i})+10]$ | 30 | $x\in [-5.12,5.12]$ | 0 |

${f}_{15}={({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}-6)}^{2}+10(1-\frac{1}{8\pi})cos{x}_{1}+10$ | 2 | ${x}_{1},{x}_{2}\in [-5,15]$ | 0.398 |

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Liu, L.; Liu, X.; Wang, N.; Zou, P. Modified Cuckoo Search Algorithm with Variational Parameters and Logistic Map. *Algorithms* **2018**, *11*, 30.
https://doi.org/10.3390/a11030030

**AMA Style**

Liu L, Liu X, Wang N, Zou P. Modified Cuckoo Search Algorithm with Variational Parameters and Logistic Map. *Algorithms*. 2018; 11(3):30.
https://doi.org/10.3390/a11030030

**Chicago/Turabian Style**

Liu, Liping, Xiaobo Liu, Ning Wang, and Peijun Zou. 2018. "Modified Cuckoo Search Algorithm with Variational Parameters and Logistic Map" *Algorithms* 11, no. 3: 30.
https://doi.org/10.3390/a11030030