## 4 Methodology
**Function tolerance:** After finding the current best position \(\vec{x}_{*}(t)\) in an iterative procedure, its fitness is compared against the previous one \(\vec{x}_{*}(t-1)\), with the objective function \(f(\vec{x})\), employing the following criterion,

\[c_{f,ft}(t,f,X)\triangleq H\bigg{(}\varepsilon_{ft}-\frac{|f(\vec{x}_{*}(t)-f( \vec{x}_{*}(t-1))|}{|\sup\{1,f(\vec{x}_{*}(t)\}|}\bigg{)}. \tag{17}\]

The function tolerance criterion \(c_{f,ft}(f)\) depends of a threshold value \(\varepsilon_{ft}\) established to limit the closeness of these two function values. **Domain tolerance:** Similar to the **Function tolerance** finalisation, but this one only compares the positions using a distance metric, e.g., the \(\ell^{2}\)-norm, such as

\[c_{f,dt}(t,f,X)\triangleq H(\varepsilon_{dt}-||\vec{x}_{*}(t)-\vec{x}_{*}(t-1 )||_{2}), \tag{18}\]

since \(\varepsilon_{dt}\) is the threshold value to evaluate the closeness of these positions. In this work, we dismiss the finalisers found from the analysed metaheuristics because we delegate this function to a superior technique. Further information is provided in Section 3, where we describe the proposed model as well as its main motivation. Notwithstanding, we implemented only the **Iteration number** finaliser in the basic metaheuristics used for comparison purposes (cf. Section 4). ## References

* [PERSON] and [PERSON] (2018) [PERSON]; [PERSON]; [PERSON] A history of metaheuristics. _Handb. Heuristics_**2018**, \(2\), 791-808. [CrossRef]
* [PERSON] (1988) [PERSON]; [PERSON] Genetic algorithms and machine learning. _Mach. Learn._**1988**, \(3\), 95-99. [CrossRef]
* [PERSON] et al. (1983) [PERSON]; [PERSON]; [PERSON] Optimization by Simulated Annealing Optimization by Simulated Annealing. _Science_**1983**, _220_, 671-680. [CrossRef] [PubMed]
* [PERSON] et al. (2021) [PERSON]; [PERSON]; [PERSON]; [PERSON] Solving microelectronic thermal management problems using a generalized spiral optimization algorithm. _Appl. Intell._**2021**. [CrossRef]
* [PERSON] et al. (2021) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Archimedes optimization algorithm: A new metaheuristic algorithm for solving optimization problems. _Appl. Intell._**2021**, _51_, 1531-1551. [CrossRef]
* [PERSON] (2015) [PERSON] Metaheuristics--The metaphor exposed. _Int. Trans. Oper. Res._**2015**, _22_, 3-18. [CrossRef][PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON] Towards a Generalised Metaheuristic Model for Continuous Optimisation Problems. _Mathematics_**2020**, \(8\), 2046. [CrossRef]
* [PERSON] et al. (2019) [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON] No Free Lunch Theorem: A Review. In _Approximation and Optimization_; [PERSON], [PERSON], Eds. ; Springer: Cham, Swizerland, 2019; pp. 57-82. [CrossRef]
* [PERSON] and [PERSON] (2004) [PERSON]; [PERSON] UPSO: A unified particle swarm optimization scheme. _Lect. Ser. Comput. Comput. Sci._**2004**, \(1\), 868-873. * [PERSON] et al. (2017) [PERSON]; [PERSON]; [PERSON] A bibliometric analysis of Genetic Algorithms throughout the history. _Comput. Ind. Eng._**2017**, _110_, 395-403. [CrossRef]
* [PERSON] and [PERSON] (2020) [PERSON]; [PERSON]; [PERSON] A Survey of Automatic Parameter Tuning Methods for Metaheuristics. _IEEE Trans. Evol. Comput._**2020**, _24_, 201-216. [CrossRef]
* [PERSON] and [PERSON] (2021) [PERSON]; [PERSON] Adaptive binary artificial bee colony algorithm. _Appl. Soft Comput._**2021**, _101_, 107054. [CrossRef]
* [PERSON] et al. (2021) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Multi-objective adaptive differential evolution for SVM/SVR hyperparameters selection. _Pattern Recognit._**2021**, _110_, 107649. [CrossRef]
* [PERSON] et al. (2018) [PERSON]; [PERSON]; [PERSON] Adaptive and Multilevel Metaheuristics. In _Handbook of Heuristics_; Springer International Publishing: Cham, Swizerland, 2018; pp. 3-21. [CrossRef]
* [PERSON] et al. (2017) [PERSON]; [PERSON]; [PERSON] H3 AD: A hybrid hyper-heuristic for algorithm design. _Inf. Sci._**2017**, _414_, 340-354.