## 3 Methodology
### Function Problems  
The base problem functions used as benchmarks are: Elliptical (scale: 500), Ackley (scale: 32), Rosenbrock (scale: 7.5; offset: 1.5), Rastrigin (scale: 5.12), Drop-wave (scale: 5.12), Zero-sum (scale: 10), and Salomon (scale: 100) [22,23]. For 2 to 3 dimensions, most of these functions would be considered somewhat challenging, as they have many local minima. To add complexity to each problem, random offsets (up to 80% off center) and ill-conditioning (up to 10 times the scale), rotations, and partial-separability (30% overlap) were applied in steps as shown in Table 1[24]. The dimensions shown in the table are for the 14D problem; however, these steps were similarly carried out for 7D and 21D problems. Ill-conditioning, rotation, and overlap effects were applied to all 7, 14, and 21 dimensions, respectively, while the 4 copies of each function had independently allocated dimensions. For the cases of overlap and rotation, some of these functions are partially dependent on the same inputs, further complicating the search process.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Abbrex.** & **Name** & **Augments** & **Dimensions** \\ \hline \(f_{0}\) & In-Sample & (one of each base function) offset and ill-conditioning & (2 each = 14) \\ \(f_{1}\) & In-Sample & (one of each base function) offset, ill-conditioning, and rotation & (2 each = 14) \\ \(f_{2}\) & In-Sample & (one of each base function) offset, ill-conditioning, rotation, and overlap & (2 each = 14) \\ \hline \(f_{3}\) & Elliptical & (4 copies) offset and ill-conditioning & (-14/4 each = 14) \\ \(f_{4}\) & Elliptical & (4 copies) offset, ill-conditioning, and rotation & (-14/4 each = 14) \\ \(f_{5}\) & Elliptical & (4 copies) offset, ill-conditioning, rotation, and overlap & (-14/4 each = 14) \\ \hline \(f_{6}\) & Ackley & (4 copies) offset all ill-conditioning & (-14/4 each = 14) \\ \(f_{7}\) & Ackley & (4 copies) offset, ill-conditioning, and rotation & (-14/4 each = 14) \\ \(f_{8}\) & Ackley & (4 copies) offset, ill-conditioning, rotation, and overlap & (-14/4 each = 14) \\ \hline \(f_{9}\) & Rosenbrock & (4 copies) offset all all-conditioning & (-14/4 each = 14) \\ \(f_{10}\) & Rosenbrock & (4 copies) offset, ill-conditioning, and rotation & (-14/4 each = 14) \\ \(f_{11}\) & Rosenbrock & (4 copies) offset, ill-conditioning, rotation, and overlap & (-14/4 each = 14) \\ \hline \(f_{12}\) & Rastrigin & (4 copies) offset all all-conditioning & (-14/4 each = 14) \\ \(f_{13}\) & Rastrigin & (4 copies) offset, ill-conditioning, and rotation & (-14/4 each = 14) \\ \(f_{14}\) & Rastrigin & (4 copies) offset, ill-conditioning, rotation, and overlap & (-14/4 each = 14) \\ \hline \(f_{15}\) & Drop-Wave & (4 copies) offset all ill-conditioning & (-14/4 each = 14) \\ \(f_{16}\) & Drop-Wave & (4 copies) offset, ill-conditioning, and rotation & (-14/4 each = 14) \\ \(f_{17}\) & Drop-Wave & (4 copies) offset, ill-conditioning, rotation, and overlap & (-14/4 each = 14) \\ \hline \(f_{18}\) & Zero-Sum & (4 copies) offset all all-conditioning & (-14/4 each = 14) \\ \(f_{19}\) & Zero-Sum & (4 copies) offset, ill-conditioning, and rotation & (-14/4 each = 14) \\ \(f_{20}\) & Zero-Sum & (4 copies) offset, ill-conditioning, rotation, and overlap & (-14/4 each = 14) \\ \hline \(f_{21}\) & Salomon & (4 copies) offset all-conditioning & (-14/4 each = 14) \\ \(f_{22}\) & Salomon & (4 copies) offset, ill-conditioning, and rotation & (-14/4 each = 14) \\ \(f_{23}\) & Salomon & (4 copies) offset, ill-conditioning, rotation, and overlap & (-14/4 each = 14) \\ \hline \end{tabular}
\end{table}
Table 1: The list of functions evaluated on for each algorithm.