# A Gradient-Based Cuckoo Search Algorithm for a Reservoir-Generation Scheduling Problem
## 5 Case Study
### Benchmark Function Tests  
To evaluate the performance of the proposed ICS, we chose several famous typical benchmark functions, shown in Table 2. Results obtained by proposed ICS algorithms are compared with that simulated by improved harmony search (IHS) [35], the Kbest gravitational search algorithm (KGSA) [36] and basic CS. Of the functions in Table 2, the Ackley function, Griewank function, Rastrigin function, Rosenbrock function and Happy Cat function are multimodal functions. The Sphere function, Bent Cigar function, Discus function and Schwefel 2.22 function are unimodal functions. These functions will be tested under sophisticated features such as shifted and rotated features.  
Dimensions of the benchmark functions are set to 10. The maximum evaluation count is set to 100,000. The default parameters of ICS are \(ns=30\), \(p_{a}^{s}=0.3\), \(p_{a}^{e}=0.1\), \(sl=0.01\), \(u=0\), \(c=1.5\). The parameters of IHS and GSA are set to the default values in [35;36]. All of the experiments are performed using a computer with 2.7 GHz Intel i7-4800 MQ with 8 GB RAM. The source code is compiled with Java SE8.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Formula** & **Search Domain** & **Optimum** \\ \hline Ackley & \(f_{1}=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}x_{i}^{2}}\right)\) & \([-32.768,32.768]\) & \(f(0,\cdots,0)=0\) \\  & \(-\exp\left(\frac{1}{n}\sum\limits_{i=1}^{n}\cos(2\pi x_{i})\right)+e+20\) & & \\ [PERSON] & \(f_{2}=1+\frac{1}{4000}\sum\limits_{i=1}^{n}x_{i}^{2}-\frac{1}{i=1}\cos\left( \frac{x_{i}}{\sqrt{i}}\right)\) & \([-600,600]\) & \(f(0,\cdots,0)=0\) \\ Rastrigin & \(f_{3}=An+\sum\limits_{i=1}^{n}\left[x_{i}^{2}-A\cos(2\pi x_{i})\right]\) & \([-5.12,5.12]\) & \(f(0,\cdots,0)=0\) \\ Rosenbrock & \(f_{4}=\sum\limits_{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+\left( x_{i}-1\right)^{2}\right]\) & \([-5,10]\) & \(f(1,\cdots,1)=0\) \\ Sphere & \(f_{5}=\sum\limits_{i=1}^{n}x_{i}^{2}\) & \([-100,100]\) & \(f(0,\cdots,0)=0\) \\ Bent Cigar & \(f_{6}=x_{1}^{2}+10^{6}\sum\limits_{i=2}^{D}x_{i}^{2}\) & \([-100,100]\) & \(f(0,\cdots,0)=0\) \\ Discus & \(f_{7}=10^{6}x_{1}^{2}+\sum\limits_{i=2}^{D}x_{i}^{2}\) & \([-100,100]\) & \(f(0,\cdots,0)=0\) \\ Happy Cat & \(f_{8}=\left|\sum\limits_{i=1}^{D}x_{i}^{2}-D\right|^{1/4}+\left(0.5\sum \limits_{i=1}^{D}x_{i}^{2}+\sum\limits_{i=1}^{D}x_{i}\right)/D+0.5\) & \([-100,100]\) & \(f(-1,\cdots,-1)=0\) \\ Schwefel 2.22 & \(f_{9}=\sum\limits_{i=1}^{n}\left[x_{i}\right]+\prod\limits_{i=1}^{n}\left|x_{i}\right|\) & \([-10,10]\) & \(f(0,\cdots,0)=0\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Test functions.  
Figure 5: Topology of reservoirs in the Jinsha River.  
Table 3 shows the results of the algorithms including means and standard deviation based on 100 independent tests for each benchmark function. As demonstrated in Table 3, ICS performs superior to IHS, KGSA and CS in most functions except the shifted Rosenbrock function, shifted Happy Cat function, and rotated and shifted Ackley function. Especially on the Sphere, Bent Cigar and Discus, the means and standard deviations of ICS are extremely close to the optimal solutions, much better than other algorithms. While on Rosenbrock function, shifted Happy Cat function, rotated and shifted Ackley function, KGSA performs slightly better than ICS. Overall, ICS can find a satisfactory solution, meaning that the improvement is effective.