## 4 Methodology
Besides, consider that \(||\cdot||_{2}\) refers to the \(\ell^{2}\)-norm. Recall that \(\vec{x}_{n}(t)\in\mathfrak{X}(t)\) stands as the current position of the \(n\)-th agent in a population \(X(t)\) of size \(N\), and \(\vec{x}_{*}(t)\) corresponds to the best solution found at the current time \(t\). Furthermore, we organised the perturbators according to their sophistication level in terms of mathematical formulation, which is not necessarily related to the numerical complexity. The first four heuristics comprise the simplest ones implemented in the literature. They use basic ideas such as samples and walks guided by probability distributions. These operators are widespread with multiple variants and schemes [54]. Many of them are embedded in more sophisticated algorithms. Subsequently, the next four perturbators obey those inspired on evolutionary processes, which conform two well-known MHs such as GA [2] and DE [55]. The ninth and tenth simple heuristics correspond to those inspired on the idealised social behaviour of many living creatures [56]. We extracted them from two well-known metaheuristics as PSO [39] and fa [40]. Lastly, the final three perturbators concern those from MHs that model trajectories and dynamics common in the classical mechanics, such as SOA [41], CFO [53], and GSA [43]. **Random Sample (RX):** It neglects the \(n\)-th current position \(\vec{x}_{n}(t)\) and replaces it with a candidate solution \(\vec{y}_{n}\) obtained via a random sample operation such as \(\vec{y}_{n}=\vec{r}\), where \(\vec{r}\in\mathfrak{S}\) is a vector of i.i.d. random numbers with \(\vec{r}\
i r_{i}\sim\mathcal{U}(x_{i},x_{n})\). Likewise the **Restart** action but more general, this perturbation is an implementation of the **RX** initialiser as a perturbator. **Random Search (RS):** It determines the candidate solution \(\vec{y}_{n}\) with a random search operation over the current position \(\vec{x}_{n}(t)\), such as \(\vec{y}_{n}=\vec{x}_{n}(t)+\alpha\vec{r}\), where \(\alpha\in\left]0,1\right]\) is thestep size factor and \(\vec{r}\in\mathfrak{S}\) is a vector of i.i.d. random numbers with \(\mathcal{U}(x_{l},x_{u})\) or other probability distribution. **Random Flight (RF):**: It calculates the candidate solution \(\vec{y}_{n}\) by employing a random flight operation with respect to the current position \(\vec{x}_{n}\) as given

\[\vec{y}_{n}=\vec{x}_{n}(t)+\alpha\vec{r}\odot(\vec{x}_{n}(t)-\vec{x}_{*}(t)), \tag{10}\]

where \(\alpha\in\mathbb{R}_{+}\) is the spatial step size, \(\vec{r}\in\mathbb{R}_{+}^{D}\) is a vector of i.i.d. random numbers with either uniform \(\mathcal{U}(0,1)\), normal standard \(\mathcal{N}(0,1)\), and symmetric Levy stable \(\mathcal{L}(1.5)\) distributions. **Local Random Walk (RW):**: It achieves the candidate solution \(\vec{y}_{n}\) by implementing a local random walk on the current position \(\vec{x}_{n}\), such as

\[\vec{y}_{n}=\vec{x}_{n}+\alpha\vec{r}\odot H(p-\vec{q})\odot(\vec{x}_{z_{1}}( t)-\vec{x}_{z_{2}}), \tag{11}\]

where \(\alpha,p\in\left]0,1\right]\) are the step size factor and the probability of change. \(\vec{r}\) and \(\vec{q}\) are both vectors of i.i.d. random variables with \(\mathcal{U}(0,1)\) or \(\mathcal{N}(0,1)\). \(\vec{x}_{z_{1}}\) and \(\vec{x}_{z_{2}}\), \(z_{1}\
eq z_{2}\), are individuals selected from the population by following a given scheme. It is common to randomly choose them from the population, i.e., \(z_{1},z_{2}\sim\mathcal{U}\{1,N\}\) and \(z_{1}\
eq z_{2}\). **Genetic Crossover (GC):**: It obtains the candidate solution \(\vec{y}_{n}\) with a somehow complex procedure over the current population \(X(t)\). This procedure first determines a ranked version of the population \(\hat{\mathcal{X}}(t)\) with respect to the cost function value i.e., \(\hat{\mathcal{X}}(t)=\{\hat{\vec{x}}_{1},\ldots,\hat{\vec{x}}_{N}\}\) with \(f(\hat{\vec{x}}_{1})<\cdots<f(\hat{\vec{x}}_{N})\). Such preliminary adjustment is known by several authors as natural selection [49].