# An Iterative Optimization Algorithm for Planning Spacecraft Pathways Through Asteroids
## 4 The Algorithm Fat-RLS
The value of \(d\) decreases monotonically according to a \"skewed S-shaped\" function influenced by \(\beta\). This approach encourages a gradual transition from explorative behavior to exploitative behavior throughout the iterations, which is a widely recognized best practice in the design of meta-heuristic algorithms [38]. Figure 1: Illustrativecexamples for forward and backward insertions. For \(\beta\geq 1\), the \"skewed S-shaped\" \(s:[0,1]\rightarrow[0,1]\) is defined as  
\[s_{\beta}(p)=1-\frac{1}{1+\left(\frac{1-p}{p}\right)^{\beta}}. \tag{6}\]  
Consequently, the perturbation strength \(d\) for each iteration is calculated by rounding to the nearest integer the product of \(d_{\text{ini}}\) and the value of the \(s\) function when applied to the evolution percentage \(p\) (i.e., the ratio of the current iteration number to the total iteration budget). Figure 2 illustrates the behavior of the function \(s_{\beta}(p)\) for various values of \(\beta\). Specifically, when \(\beta=1\), the function simplifies to \(s_{1}(p)=1-p\), resulting in a linear decline in perturbation strength. As \(\beta\) increases, the curve becomes steeper in the middle section, which prolongs the initial exploration phase and facilitates a more abrupt and rapid transition to the final exploitative phase. Furthermore, since FAT-RLS is designed for low-budget optimization scenarios, we incorporate a tabu-based mechanism. This mechanism serves not only to prevent the trajectory of an FAT-RLS execution from revisiting previously explored solutions but also to ensure that the same elements of the current permutation are not shifted too frequently by the perturbation operation. Hence, FAT-RLS maintains a tabu queue, denoted as \(TQ\), with a maximum size of \(k\), which is a hyperparameter of the algorithm. During each iteration, an insertion \((i,j)\) cannot be selected if the element \(\pi(i)\) designated for the shift is marked as tabu (i.e., if \(\pi(i)\in TQ\)). When a non-tabu insertion \((i,j)\) is selected and executed on the current permutation, the shifted element is added to the tabu queue \(TQ\). If the queue reaches its maximum capacity, the oldest item is removed to accommodate the new entry. In summary, FAT-RLS utilizes a straightforward randomized local search framework that conducts a single objective function evaluation per iteration. To enhance its effectiveness in low-budget contexts, the algorithm implements both an adaptive perturbation strength strategy and a tabu-based mechanism. The adaptive perturbation strength is implemented by considering insertion moves with a \"shift length\" that decreases during the iterations, while the tabu-based mechanism is realized by maintaining a queue data structure containing the items shifted during the last iterations. Therefore, these compo  
Figure 2: The “skewed S”-shaped function for \(\beta=1,2,3\). nents serve to limit the number of applicable perturbations in each iteration, facilitating a more rapid convergence toward potentially promising solutions. For completeness, we provide the pseudocode for FAT-RLS in Algorithm 1. This algorithm is designed to minimize (without loss of generality) an objective function of the form \(f:\Sigma_{n}\rightarrow\mathbb{R}\). It accepts three hyperparameters as input: \(d_{\text{ini}}\in[1,n)\), \(\beta\geq 1\), and \(k\leq n\), as previously described. ```
1:procedureFAT-RLS(\(d_{\text{ini}},\ \beta,\ k\))
2:/*Variables Initialization */
3:\(\sigma\leftarrow\) random permutation from \(\Sigma_{n}\)
4:Evaluate\(f(\sigma)\)
5:nfev\(\leftarrow\) 1
6:TQ\(\leftarrow\) \(\mathcal{O}\)
7:/*Main loop */
8:whilenfev\(<\) budgetdo
9:/*Compute perturbation strength */
10:\(p\leftarrow\)nfev/budget
11:\(d\leftarrow\)round\(\big{(}1+s_{\beta}(p)\cdot(d_{\text{ini}}-1)\ \big{)}\)
12:/*Perturbation */
13:\(\pi\leftarrow\sigma\)
14:Sample \(i,j\) s.t.