# Modified Cuckoo Search Algorithm with Variational Parameters and Logistic Map
## 2 Preliminary
### Boundary Is Constrained by Logistic Map of Each Dimension
The VLCS algorithm uses logistic map of each dimension to handle global random \(\mathbf{r}\), which is used in initiation and boundary issue. Logistic map can guarantee that the location of each nest will only be calculated once. The VLCS algorithm uses \(\alpha_{new}\) and \(p_{new}\) to replace the fixed parameters of the CS algorithm and improve the performance of CS algorithm. The pseudo code of VLCS algorithm is shown in Algorithm 4. ```
Input: d represents the dimension;  Objective function f(x), x=(\(x_{1},...,x_{d}\))\({}^{T}\);  Global random number r, r=(\(r_{1},\ldots,r_{d}\))\({}^{T}\),r\({}_{d}\in(0,1)\); Output: Postprocess results and visualization
1 Initiation part uses Algorithm 2;
2 n is equal to the number of nests;
3 while\(cur\_iteration<total\_iteration\)do
4 Get a nest randomly by Levy flights with \(\alpha_{new}\);
5 The location of nest is bounded by Algorithm 3;
6 Then, evaluate its quality/fitness \(F_{i}\);
7 Choose a nest among n (say, j) with maximum fitness \(F_{j}\);
8 if\(F_{i}>F_{j}\)then
9 replace j by the new solution;
10 A fraction (\(p_{new}\)) of worse nests are abandoned;
11 New ones are built by Algorithm 3;
12 Keep the best solutions (or nests with quality solutions);
13 Rank the solutions and find the current best;
14 Postprocess results and visualization;
15 final ;
16 return Postprocess results and visualization;
```

**Algorithm 4**VLCS algorithm

## 4 Simulation Experiments

This paper selects 15 benchmark functions [34; 35; 36] to prove that the VLCS algorithm is better than the CS algorithm.The simulation environment is Matlab R2014a. The \(\alpha_{new}\) and \(p_{new}\) are shown in Equation (4). The benchmark functions and conditions are shown in Table 2. In Table 2, the dimensions of f02, f08 and f15 are two, and the others' dimensions are not less than ten. To compare with other algorithms, the _nest_ number of n is 15 [18; 24]. If the number of nest is changed, the relative convergence rates of the VLCS algorithm and the CS algorithm do not change. Of course, if n is increased, the VLCS algorithm and the CS algorithm will converge faster than other meta-heuristic algorithms. The CS [18] was proposed by [PERSON], who has already attested that it is better than other meta-heuristic algorithms. Section 3.1 illustrates the benefits of logistic map. Most of modified CS algorithms are mainly focused on choosing chaotic maps (e.g., [23; 24; 25]). These studies do not mention how to reduce repeated calculation in each dimension. This means the VLCS algorithm converges faster than most of other modified CS algorithms, because the VLCS algorithm uses logistic map of each dimension to initialize host nest location and update the location of host nest beyond the boundary. Therefore, this paper compares the CS algorithm and the VLCS algorithm, and shows the results in Figure 2. There is little difference between the VLCS algorithm and the CS algorithm in the function f04, f13 and f14 because the three functions are simpler than other functions in Figure 2 and do not need to carry out complex calculations. From the results, f07, f08, f09, f10 and f14 show two things clearly. Firstly, the convergence speed of the VLCS algorithm is faster than the CS algorithm, because, at the middle of the iteration, the CS algorithm easily falls into the local optimal solution and needs time to jumpout of the local optimal solution. Secondly, the convergence accuracy of the VLCS algorithm is better than the CS algorithm because the VLCS algorithm does not do redundant calculations in the same place, even in the same dimension. Actually, if the other pictures in Figure 2 are enlarged, it is easy to get the same conclusion. The VLCS algorithm solves the three disadvantages of the CS algorithm mentioned in Section 2.2. In addition, Table 2 contains high-dimensional and low-dimensional test functions. This means the VLCS algorithm applies not only to low-dimensional problems but also to high-dimensional problems.