## 3 Proposed Methodology
### Phase 5: Application of the Opposition Nelder-Mead Algorithm
The effectiveness of the proposed methodology was assessed through rigorous testing on the CEC 2022 ([[https://github.com/P-N-Suganthan/2022-SO-BO](https://github.com/P-N-Suganthan/2022-SO-BO)]([https://github.com/P-N-Suganthan/2022-SO-BO](https://github.com/P-N-Suganthan/2022-SO-BO)) (accessed on 28 June 2023)) benchmark, which comprises a set of 12 hard and challenging test functions. Among these functions, one is unimodal in nature, whereas four are multimodal. Additionally, three functions are designed as hybrid, and the remaining four functions are composite. The use of the unimodal function aims to evaluate the methodology's exploitation capability, as it requires focusing on refining solutions within a simple search space. The inclusion of multimodal functions allows for assessing the methodology's exploration capability, as it necessitates exploring a complex search space with multiple optima. Furthermore, the hybrid and composite functions were employed to evaluate the methodology's ability to strike a balance between exploration and exploitation, as they combine different characteristics and complexities. The comprehensive testing on this diverse set of test functions provided valuable insights into the performance and robustness of the proposed methodology across various optimization scenarios. Table 2 depicts the features of the test problems suite. \begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{N^{\circ}}\) & **Functions** & \(\mathbf{F_{i}^{*}}\) \\ \hline Unimodal function & 1 & Shifted and full Rotated Zakharov Function & 300 \\ \hline \multirow{4}{*}{Basic functions} & 2 & Shifted and full Rotated Rosenbrock’s Function & 400 \\ \cline{2-4}  & 3 & Shifted and full Rotated Expanded Schaffer’s \(f_{6}\) Function & 600 \\ \cline{2-4}  & 4 & Shifted and full Rotated Non-Continuous Rastrigin’s Function & 800 \\ \cline{2-4}  & 5 & Shifted and full Rotated Lévy Function & 900 \\ \hline \multirow{4}{*}{Hybrid functions} & 6 & Hybrid Function 1 (N = 3) & 1800 \\ \cline{2-4}  & 7 & Hybrid Function 2 (N = 6) & 2000 \\ \cline{2-4}  & 8 & Hybrid Function 3 (N = 5) & 2200 \\ \hline \multirow{4}{*}{Composition functions} & 9 & Composition Function 1 (N = 5) & 2300 \\ \cline{2-4}  & 10 & Composition Function 2 (N = 4) & 2400 \\ \cline{1-1} \cline{2-4}  & 11 & Composition Function 3 (N = 5) & 2600 \\ \cline{1-1} \cline{2-4}  & 12 & Composition Function 4 (N = 6) & 2700 \\ \hline \end{tabular} \(D\): The dimensionality of the search space.

\end{table} 
Table 2: Information and features of the test problems suite. To thoroughly assess the effectiveness of the proposed methodology, it was subjected to a comparative analysis against 11 highly influential and powerful algorithms, specifically:

1. Co-PPSO: Performance of Composite PPSO on Single Objective Bound Constrained Numerical Optimization Problems of CEC 2022 [78]. 2. EA4 eigN100-10: Eigen Crossover in Cooperative Model of Evolutionary Algorithms applied to CEC 2022 Single Objective Numerical Optimization [79]. 3. IMPML-SHADE: Improvement of Multi-Population ML-SHADE [80]. 4. IUMOEAII: An improved IMODE algorithm based on Reinforcement Learning [81]. 5. jSObinexpEig: An adaptive variant of jSO with multiple crossover strategies employing Eigen transformation [82]. 6. MTT-SHADE: Multiple Topology SHADE with a tolerance-based composite framework for CEC 2022 Single Objective Bound Constrained Numerical Optimization [83]. 7. NL-SHADE-LBC: NL-SHADE-LBC algorithm with linear parameter adaptation bias change for CEC 2022 Numerical Optimization [84]. 8. NL-SHADE-RSP-MID: A version of the NL-SHADE-RSP algorithm with Midpoint for CEC 2022 Single Objective Bound Constrained Problems [85]. 9. OMCSOMA: Opposite Learning and Multi-Migrating Strategy-Based Self-Organizing Migrating algorithm with a convergence monitoring mechanism [86]. 10. S-LSHADE-DP: Dynamic Perturbation for Population Diversity Management in Differential Evolution [87]. 11. NLSOMACLP: NL-SOMA-CLP for Real Parameter Single Objective Bound Constrained Optimization [88].