## 3 Single-Objective Parameter Tuning
### Empirical Procedure  
This section presents the empirical procedure followed during the investigation of single-objective tuning algorithms. The purpose of this experiment is to demonstrate that tuning with respect to different performance objectives results in different optimal parameter configurations for the same meta-heuristic algorithm. An iterated F-race implementation, as presented in Section 3.1, was used to tune a PSO using a global-best neighbourhood topology [19] and an inertia weight component [20]. Four numeric control parameters were tuned:  
* \(w\), the inertia weight;
* \(c_{1}\), the cognitive component;
* \(c_{2}\), the social component;
* \(N\), the number of particles in the swarm.  
Parameters were sampled in the ranges \(w\in[-1,1]\), \(c_{1}\in[0,2]\), \(c_{2}\in[0,2]\), and \(N\in[10,30]\cap\mathbb{Z}\). Sampling was also performed such that the Poli stability condition  
\[c_{1}+c_{2}<\frac{24(1-w^{2})}{7-5w} \tag{10}\]was satisfied [21]. This ensures that the PSO algorithm converges to a stable state and computational cost is not wasted by evaluating parameter configurations that are known to prevent a PSO from converging to a stable solution.  
To start off the F-race algorithm, 1000 candidate configurations were sampled uniformly over the parameter space. At each iteration of F-race, \(n_{min}=10\) was used as the threshold for surviving candidates and new candidates were sampled such that \(n=20\) candidates are analysed by F-race in the next iteration. The Friedman test used a significance level of \(\alpha=0.05\).  
Four performance objectives were used during the tuning process, with each evaluation metric optimised by F-race separately:  
* Accuracy: The difference between the found objective function value and the known optimal value.
* Efficiency: The number of iterations until the found solution is within some level of accuracy to the known optimal solution.
* Stability: The deviation of the found solutions from the mean solution taken over some number of independent iterations.
* Success rate: The percentage of a number of independent solutions that reach a certain level of accuracy.  
The performance of the algorithm for the stability and success rate objectives was analysed using 10 independent runs.  
Tuning was performed on the following functions:  
* [PERSON] [22]: \[f(\mathbf{y})=10D+\sum_{i=0}^{D-1}\left(y_{i}^{2}-10\cos(2\pi y_{i})\right)\]
* [PERSON] [23]: \[f(\mathbf{y}) =-a\exp\left(-b\sqrt{\frac{1}{D}\sum_{i=0}^{D-1}y_{i}^{2}}\right)\] \[-\exp\left(\frac{1}{D}\sum_{i=1}^{D}\cos(cy_{i})\right)+a+e\]
* [PERSON] and [PERSON]: \[f(\mathbf{y})=\frac{1}{2}\sum_{i=0}^{D-1}\left(y_{i}^{4}-16y_{i}^{2}+y_{i}\right)\]  
where \(D\) is the dimensionality of the function in each case. For this investigation, values of \(D=20\), \(a=20\), \(b=0.2\), and \(c=2\pi\) were used.  
The accuracy levels used for efficiency and success rate metrics are presented in Table 1. The efficiency threshold defines the objective function value at which convergence is considered efficient, while the success rate threshold defines the value at which a trial is considered successful. The ranges of the search spaces for each function are given in Table 2.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Function** & **Optimal Value** & **Efficiency Threshold** & **Success Rate Threshold** \\ \hline Rastrigin & 0 & 25 & 15 \\ Ackley & 0 & 5 & 2 \\ Styblinski–Tang & \(-\)783.3 & –600 & –780 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Performance thresholds for efficiency and success rate metrics.