## 4 Methodology
random variables with \(\mathcal{U}(-0.5,0.5)\) or another distribution, and \(I_{n}=f(\vec{x}_{n})\), \(\forall\,n\in\{1,\ldots,N\}\), is the light intensity of agent \(n\). **Spiral Dynamic (SD):** The candidate position \(\vec{y}_{n}(t)\) is determined from the logarithmic spiral trajectory given from the current position \(\vec{x}_{n}(t)\) and the the rotation centre \(\vec{x}_{*}(t)\), such as,

\[\vec{y}_{n}=\vec{x}_{n}(t)+\alpha\vec{r}\odot(\vec{x}_{n}(t)-\vec{x}_{*}(t)), \tag{10}\]0 (A10)

where \(\theta\in[0^{\circ},360^{\circ}]\) is the rotation angle, and \(\vec{r}\) is a vector of i.i.d. random variables with \(\mathcal{U}(r_{0}-\sigma_{r},r_{0}+\sigma_{r})\), since \(r_{0}\in[0,1]\) is the convergence rate or central radius and \(\sigma_{r}\in[0,\min\{r_{0},1-r_{0}\}]\). Plus, \(\mathbf{R}_{D}(\theta)\in\mathbb{R}^{D\times D}\) is the rotation matrix determined as the product of all the \(D(D-1)/2\) combinations of two-dimensional rotation matrices in \(\mathbb{R}^{D}\). Consider that this rotation centre is given by the best current position found by the population. **Central Force (CF):** This perturbator determines the candidate position \(\vec{y}_{n}(t)\) by using a Newtonian gravitational attraction approach given by

\[\vec{y}_{n}=\vec{x}_{n}(t)+\alpha\vec{r}\odot(\vec{x}_{n}(t)-\vec{x}_{*}(t)), \tag{10}\]1 (A11a) where \[\vec{y}_{n}=\vec{x}_{n}(t)+\alpha\vec{r}\odot(\vec{x}_{n}(t)-\vec{x}_{*}(t)), \tag{10}\]2 is the acceleration vector for the \[\vec{y}_{n}=\vec{x}_{n}(t)+\alpha\vec{r}\odot(\vec{x}_{n}(t)-\vec{x}_{*}(t)), \tag{10}\]3 -th agent and \[\vec{y}_{n}=\vec{x}_{n}(t)+\alpha\vec{r}\odot(\vec{x}_{n}(t)-\vec{x}_{*}(t)), \tag{10}\]4 is the time interval between steps, i.e., \[\vec{y}_{n}=\vec{x}_{n}(t)+\alpha\vec{r}\odot(\vec{x}_{n}(t)-\vec{x}_{*}(t)), \tag{10}\]5. This acceleration component is obtained such as, \[\vec{a}_{n}(t)=\alpha_{0}\sum_{ \begin{subarray}{c}k=1\\ k\
eq n\end{subarray} }^{N}H(M_{k}(t)-M_{n}(t))(M_{k}(t)-M_{n}(t))^{\alpha_{1}} \frac{\vec{x}_{k}(t)-\vec{x}_{n}(t)}{||\vec{x}_{k}(t)-\vec{x}_{n}(t)||_{2}^{ \alpha_{2}}},\] (A11b) where \[M_{n}=f(\vec{x}_{n}(t))\], \[\forall\,n\in\{1,\ldots,N\}\], is the mass of agent \[n\], \[\alpha_{0}\in[0,0.01]\] is the gravitational constant, and \[\alpha_{1}\in[0,0.01]\], \[\alpha_{2}\in[1,2]\] are two positive factors. **Gravitational Search (GS):** This operator is founded in kinematic and gravitational interactions. So, the candidate position \[\vec{y}_{n}(t)\] is achieved as given \[\vec{y}_{n}=\vec{x}_{n}(t)+\vec{v}_{0}\odot\vec{v}_{n}(t),\] (A12a)since \(\vec{v}_{n}(t)\) is the velocity of the agent \(n\). This parameter is attained as

\[\vec{v}_{n}(t)=\vec{r}\odot\vec{v}_{n}(t-1)+\vec{a}_{n}(t), \tag{11b}\]

where \(\vec{r}_{0}\) is a vector of i.i.d. random variables with \(\mathcal{U}(0,1)\). Plus, \(\vec{a}_{n}(t)\) is the acceleration component calculated by

\[\vec{a}_{n}(t)=G(t)\sum_{ \begin{subarray}{c}k=1\\ k\
eq n\end{subarray} }^{N}M_{k}(t)\vec{r}_{k}\odot\frac{\vec{x}_{k}(t)-\vec{x }_{n}(t)}{||\vec{x}_{k}(t)-\vec{x}_{n}(t)||_{2}+\varepsilon_{g}}, \tag{11c}\]

since \(G(t)\in\mathbb{R}_{+}\) is the gravitational factor that can be time dependent, \(M_{n}(t)\) is the mass of agent \(n\), \(\vec{r}_{k}\) is a vector of i.i.d. random variables with \(\mathcal{U}(0,1)\), and \(\varepsilon_{g}\in\mathbb{R}_{+}\) is a small constant. In particular, \(G(t)\) and \(M_{n}(t)\) are determined with

\[G(t)=\alpha_{0}e^{-\alpha_{1}t}\quad\text{and}\quad M_{n}(t)=\frac{f(\vec{x}_ {0}(t))-f(\vec{x}_{n}(t))}{Nf(\vec{x}_{0}(t))-\sum_{k=1}^{N}f(\vec{x}_{k}(t))}, \tag{11d}\]

where \(\alpha_{0}\in[0,1]\) is the initial gravitational value, \(\alpha_{1}\in[0,0.1]\) is the damping ratio, and \(f(\vec{x}_{0}(t))\) is the worst fitness value at \(\vec{x}_{0}(t)\), such that \(\vec{x}_{0}(t)=\text{argsup}\bigcup_{n=1}^{N}\{f(\vec{x}_{n}(t))\}\). ### Selectors

From the prior mentioned metaheuristics, we chose the four most representative selection operators, i.e., **Direct, Greedy, Probabilistic**, and **Metropolis selectors**.