# Automated Generation of Hybrid Metaheuristics Using Learning-to-Rank
## 5 Experiment Evaluation
### Experimental Results and Discussion
Overall, the proposed LTR-MHA demonstrates the best final fitness values on most test functions, with its advantages being particularly evident on multimodal and composition functions (e.g., F6, F22, and F24), approaching the optimal region within just 100 iterations, whereas the other algorithms generally converge more slowly and are more prone to local optima. Although the Rand-MHA, with its random update strategy, surpasses traditional metaheuristic algorithms on certain functions (e.g., F3, F9, F14, and F24), its overall performance still lags behind the LTR-MHA and exhibits greater variability. Among the three conventional algorithms, the GA performs the best. The ISGA yields convergence behavior similar to the GA across most functions. The RLWOA converges rapidly on a majority of the tested functions, achieving final fitness values that are comparable to or only marginally lower than those of the LTR-MHA. HHO, despite occasionally achieving a rapid decline in the early iterations, suffers from premature convergence and stagnates in the later stages, resulting in noticeably poorer final solutions. These findings demonstrate that the multi-strategy fusion update mechanism learned by the LTR-MHA enables stable and efficient optimization across various types of complex search spaces. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c} \hline \hline
**FunesAlgorithm** & **Dim = 10** & \multicolumn{4}{c}{**Dim = 30**} & \multicolumn{4}{c}{**Dim = 50**} & \multicolumn{4}{c  
to 60 s due to its online reinforcement learning-based parameter adjustment. The LTR-MHA, requiring feature generation and online prediction via the model at each iteration, introduces additional computational complexity, resulting in the longest runtime among the compared algorithms. This difference arises from the algorithmic design of the LTR-MHA. In each iteration, for every individual in the population, the LTR-MHA constructs \(m=12\) candidate update strategies. Each candidate is represented by a feature vector composed of 10 components (as shown in Equation (19)). These \(m\) feature vectors are then passed into the trained predictive model to compute scores and select the most promising strategy. As a result, the per-iteration time complexity of the LTR-MHA is approximately \(O(n\cdot m\cdot(k+T_{pred}))\), where \(n\) is the population size, \(k=10\) is the feature dimension, \(T_{pred}\) is the model's prediction time, and \(D\) is the problem dimension. This is significantly higher than other metaheuristics that apply a single, fixed rule per individual per iteration. In conclusion, while the LTR-MHA clearly outperforms in convergence speed and solution precision, its increased computational cost requires a trade-off with time efficiency in practical applications. Notably, the development time of automated methods and manually designed approaches is generally difficult to compare directly, as such information is typically not  
Figure 3: Convergence curves for the functions with D = 30.  
disclosed in published studies. Furthermore, although the proposed algorithm incurs additional computational time in both the training and testing steps, the primary goal of this research is not to discover the optimal approach but rather to explore an approach capable of automatically generating highly generalizable and state-of-the-art search strategies. In the long term, this additional computational overhead is justified by the ability to efficiently solve a broad range of problem instances. The Wilcoxon rank-sum test was used at a significance level of 0.05 to determine whether the LTR-MHA's optimization results outperformed the compared algorithms significantly. Table 9 presents the statistical analysis results of the LTR-MHA against the other algorithms under the 30-dimension setting. In the table, the symbols '+', '=', and '-' demonstrate that the LTR-MHA performs significantly better than, similarly to, and significantly worse than the compared algorithm.