## 4 Methodology
Figure 7 presents the performance values for the resulting uMHs from the two experimental setups, when solving the problem domains with different characteristics and dimensionalities. In this figure, rows correspond to the combination of problem categories (differentiability and unimodality), and columns are the number of dimensions. It is pretty easy to notice that the conditions of Experiment 2 promote better performing uMHs than those of Experiment 1. Nevertheless, this is not a general rule. For example, when dealing with multimodal problem domains (say, DU = 10 and 00), the performance distribution for both experimental setups is virtually the same in the context of boxplots. One can explain such a fact with the dimensionality of these problems, making them 'easy' to solve. Naturally, if we have to choose a procedure to find an adequate solver, we would prefer to implement Experiment 1 because it requires fewer resources. In the other cases, it is evident that those additional requirements render notably better performing metaheuristics. Another point to emphasise is the precision of these performance values, with Experiment 2 allowing more concentrated values than the other. For our subsequent discussion, we compare the achieved unfolded metaheuristics against the collection of basic metaheuristics, which are also provided by the CUSTOMHyS framework [31], using the same performance metric. These basic methods are those used for extracting the simple heuristics but implemented with different default hyper-parameters. This gives us a total of 66 MHs. The first comparison is also the most straightforward: performance-to-performance. Thus, we calculated the Success Rate (SR) for an achieved unfolded metaheuristic (uMH) for a given problem domain \((\bar{x},f)\) compared against (\(M=66\)) basic MHs implemented in the same domain, such as,

\[\text{SR}(\text{uMH})\triangleq\frac{1}{M}\sum_{i=1}^{M}H(Q(\text{uMH}|\bar{x },f)-Q(\text{MH}_{i}|\bar{x},f)), \tag{6}\]

where \(H:\mathbb{R}^{D}\rightarrow\mathbb{Z}_{2}^{D}\) is the Heaviside step function and \(Q(\bar{h}|\bar{x},f)\) is the performance metric from (5). SR means a weighted counting of how many basic MHs were beaten (outperformed) by the designed uMH per problem and dimensionality. Although analysing the individual SR for each problem domain cannot be insightful, we aggregated the SR values from problems sharing the same category and dimensionality employing the average operation. Thus, we obtained an Average Success Rate (ASR) measurement that Figure 8 shows. With that in mind, we must notice that our approach, whichever experimental configuration, surmounts the performance of at least one basic metaheuristic in almost all problems (i.e., \(\text{ASR}>0\)). This is crucial because even for an excellent solver for a given problem, we must know its limitations and advantages. However, employing the approach we are proposing in this manuscript, we as practitioners are able to find automatically a heuristic-based solver for any problem. It may not be the best one, but it is a good starting point for further enhancements. These words are supported with the non-zero values of ASR that Figure 8 depicts; it means that the uMH achieved by SAHH has beaten at least one basic MH. Moreover, we must emphasise that ASRs from

Figure 7: Performance values achieved by the unfolded metaheuristics when solving optimisation problems, with different characteristics and dimensionalities, via the proposed approach and using two experimental setups. Experiment 1 were, in general, improved when we implemented the second setup. The most notorious scenarios where our approach surmounts the basic metaheuristics are the multimodal (DU = 10 and 00) problems. On the contrary, unimodal problem domains seem to be difficult rims, paradoxically, due to their 'easiness', for tackling these basic metaheuristics. In other words, the relatively low success rates do not imply that our approach has trouble solving these problems. It just means that many basic MHs can achieve good performance values.