# A Quantum-Inspired Predator-Prey Algorithm for Real-Parameter Optimization
## 4 Empirical Evaluation  
### Experimental Settings  
We assess the efficacy of QPPA for selected real-parameter optimization benchmark problems and compare it with related algorithms from the literature. We first outline the experimental configuration employed to evaluate the performance of the selected algorithms. We select a set of fourteen benchmark objective functions, as shown in Table 1, which provides the function name, formulation, and search space. We compare QPPA with established optimization methods from the literature using the hyperparameters given in their original papers, as shown in shown in Table 2. We make an exception for the case of PSO for the F7 objection function (HGBat) given in Table 1, where we fine-tune the hyperparameters in trial experiments. In our algorithmic analysis, we investigate two different settings. In Setting 1, we focus on evaluating the algorithm's efficiency and convergence behavior in a lower-dimensional solution space. Hence, in Setting 1, we use a population size (\(N_{p}=30\)), maximum iterations (\(T=1000\)), and dimension (\(D=10\)). In Setting 2, we maintain a consistent population size (\(N_{p}=30\) and iteration count (\(T=1000\)) and consider a much higher-dimensional space (\(D=100\)). This choice allows us to showcase the algorithm's scalability and robustness when solving more complex, high-dimensional optimization problems. By systematically varying these settings, we aim to comprehensively understand the algorithm's strengths and limitations across a wide range of optimization scenarios. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Function** & **Name** & **Formulation** & **Search Space** \\ \hline F1 & [PERSON] & \(\Sigma_{i=1}^{D}x_{i}^{2}+(\Sigma_{i=1}^{D}0.5 ix_{i})^{2}+(\Sigma_{i=1}^{D}0.5 ix_{ i})^{4}\) & \([-100,100]\) \\ F2 & [PERSON] & \(\Sigma_{i=1}^{D-1}100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}]\) & \([-10,10]\) \\ F3 & Exp. Schaffer & \(g(x,y)=0.5+\frac{(\sin^{2}(\sqrt{x^{2}+y^{2}})-0.5)}{(1+0.4001(x^{2}+y^{2}))^{ 2}}\) & \([-100,100]\) \\  & \(f(x)=g(x_{1},x_{2})+g(x_{2},x_{3})+\ldots+g(x_{D-2},x_{D-1})+g(x_{D-1},x_{D})\) & \\ F4 & Rastrigin & \(10D+\Sigma_{i=1}^{D}[x_{i}^{2}-10\,cos(2\pi x_{i})]\) & \([-100,100]\) \\ F5 & Levy & \(\sin^{2}(\pi w_{1})+\Sigma_{i=1}^{D-1}(x_{i}-1)^{2}\big{[}1+10\sin^{2}(\pi w_{ i}-1)\big{]}+(w_{D}-1)^{2}\big{[}1+\sin^{2}(2\pi w_{D})\big{]}\) & \([-30,30]\) \\  & \(w_{i}=1+\frac{x_{i}-1}{2},\forall i=1,\ldots,D\) & \([-100,100]\) \\ F6 & Bent Cigar & \(x_{i}^{2}+10^{\xi\Sigma_{i=2}^{D}x_{i}^{2}}\) & \([-100,100]\) \\ F7 & HGBat & \(\left|\left(\sum_{i=1}^{D}x_{i}^{2}\right)^{2}-\left(\sum_{i=1}^{D}x_{i}^{2} \right)\right|^{0.5}+\frac{0.5\sum_{i=1}^{D}x_{i}^{2}+\Sigma_{i=1}^{D}x_{i}}{D}+0.5\) & \([-1.28,1.28]\) \\ F8 & High Conditioned Elliptic & \(\sum_{i=1}^{D}(10^{6})^{\frac{i-1}{D-1}x_{i}^{2}}\) & \([-500,500]\) \\ F9 & Happycat & \(\left|\sum_{i=1}^{D}x_{i}^{2}-D\right|^{1/4}+\frac{(0.5\sum_{i=1}^{D}x_{i}^{2} +\Sigma_{i=1}^{D}x_{i})}{D}+0.5\) & \([-32,32]\) \\ F10 & Modified Schwefel & \(418.9829\times D-\sum_{i=1}^{D}g(z_{i})\) & \([-50,50]\) \\  & & \(z_{i}=x_{i}+4.209687462275036E+002\) \\ F11 & Ackley & \(-20\exp\left(-0.2\sqrt{\frac{1}{D}\sum_{i=1}^{D}x_{i}^{2}}\right)-\exp\left( \frac{1}{D}\sum_{i=1}^{D}\cos(2\pi x_{i})\right)+20+e\) & \([-50,50]\) \\ F12 & Discus & \(10^{6}x_{1}^{2}+\sum_{i=2}^{D}x_{i}^{2}\) & \([-65.536,65.536]\) \\ F13 & Griewank & \(\Sigma_{i=1}^{D}\frac{x_{i}^{2}}{0.000}-\Pi_{i=1}^{D}cos(\frac{x_{i}}{\sqrt{i} })+1\) & \([-5,5]\) \\ F14 & Schaffer & \(\left[\frac{1}{D-1}\sum_{i=1}^{D-1}\left(\sqrt{s_{i}}\cdot\left(\sin(50.0s_{i}^{ 0.2})+1)\right)\right]^{2}\), & \([-5,5]\) \\  & & \(s_{i}=\sqrt{x_{i}^{2}+x_{i+1}^{2}}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Benchmark objective functions from Congress on Evolutionary Computation, 2022.