# Robust Estimators in Geodetic Networks Based on a New Metaheuristic: Independent Vortices Search
## 3 Independent Vortices Search
### Vortex Search Algorithm  
The Vortex Search algorithm (VS) is a single solution MH for numerical function optimization. It generates new candidates around the current best result, moving along the search space when achieving a better solution. The radius defines the limit for generating new candidate solutions and decreases over cycles [32]. VS and MVS define the initial radius (radii) of the center (centers) by using Equation (19), covering the whole search space. The algorithms obtain their first solution by applying Equation (20), causing the search to begin at the center of the space:  
\[r_{0j}=\frac{x_{min,j}-x_{max,j}}{2}, \tag{19}\]  
\[\mu_{0j}=\frac{x_{min,j}+x_{max,j}}{2}, \tag{20}\]  
where \(r_{0j}\) is the initial radius of the \(j\)th parameter, \(x_{min,j}\) and \(x_{max,j}\) are the respective minimum and maximum limits of \(j\)th parameter, and \(\mu_{0j}\) is the center of the vortex and the initial solution for the parameter \(j\). In each cycle, the radius decreases to limit the generation space for new candidate solutions. In the later cycles, the VS produces a fine adjustment as the current candidates get closer to the best solution. The radius decrease is obtained by Equation (21), and provides satisfactory control of the exploration and investigation [32]:  
\[r_{c}=r_{0}\cdot\left(1/R\right)\cdot\gamma\left(R,a_{c}\right), \tag{21}\]  
where \(r_{c}\) is the radius size in cycle \(c\), \(R\) is a constant (set at 0.1) that controls the resolution of the algorithm's search, and \(\gamma\left(R,a_{c}\right)\) is the incomplete inverse gamma function, given by Equation (22):  
\[\gamma\left(R,a_{c}\right)=\int_{0}^{R}e^{-t}t^{a_{c}-1}dt\quad a_{c}>0, \tag{22}\]  
where \(a_{c}\) is a parameter that defines the shape for each cycle, as defined by Equation (23):  
\[a_{c}=1-\frac{c}{C_{max}}. \tag{23}\]  
The equation of the shape parameter uses the cycle count \(c\) and the maximum number of iterations \(C_{max}\). The vector of candidate solutions \(s\) is randomly generated at the center \(\mu\) using the Multivariate Normal Distribution (MND) with the standard deviation vector of \(r\). Figure 1 shows an example of the radius convergence by applying Equation (21) and the candidate generation limits by the MND. The algorithm checks that the candidate values are within the search space by applying Equation (24), where \(\epsilon\) is a random real value between [0, 1]:  
\[x_{i,j}= \begin{cases}x_{i,j},&\text{if }x_{min,j}\leq x_{i,j}\leq x_{max,j},\\ \epsilon\cdot\left(x_{max,j}-x_{min,j}\right)+x_{min,j},&\text{otherwise}.\end{cases}  \tag{24}\]  
For each candidate solution, a fitness function tests the result: If a candidate produces a better outcome than the current one, VS moves the center (\(\mathbf{\mu}\)) to the new solution and discards the remaining candidates. Otherwise, the algorithm keeps the best obtained result and produces new solutions. Both VS and MVS use all the above equations. However, by using several vortices, MVS adjusts the positions of its centers to each cycle, based on the vortex, with a better result. This separates positioning the center \(\mu_{l}\) for the generation of new solutions, and the best solution \(s_{l}\) of a vortex \(l\). For this, MVS applies Equation (25):  
\[\mu_{l}=s_{l}+\epsilon\cdot\left(s_{l}+s_{best}\right), \tag{25}\]  
where \(\mu_{l}\) is the new position of the center \(l\), \(s_{l}\) is the best solution of the center \(l\), and \(s_{best}\) is the result of the vortex with the best solution. At each cycle, all centers (except for \(s_{best}\)) gain new positions using the above equation [32]. As has been shown, VS has no crossover nor mutations between candidate solutions, unlike Genetic Algorithms (GAs) and the Artificial Bee Colony (ABC) method. At each cycle, it saves the best solution--either the current best or a new one--for the next cycle, discarding the remaining ones. This way, VS has no need to use individual selection strategies, a mutation rate, or other strategies present in other MHs.