# An Iterative Optimization Algorithm for Planning Spacecraft Pathways Through Asteroids
## 1 Introduction and Related Work  
Costly black-box optimization poses significant challenges across various fields, including engineering and finance, as it involves problems where the objective function does not have an explicit formal representation and demands considerable computational resources in terms of time, memory, or monetary cost. In these cases, algorithms can only learn knowledge about the problem by performing multiple evaluations of the objective function. However, because typically these evaluations are expensive, the algorithms often face strict computational budget limitations, thus restricting the number of allowable evaluations. For continuous variables, Bayesian optimization methods are frequently employed, as discussed in [1, 2]. These techniques build a surrogate model of the objective function, usually using a Gaussian process or a Kriging model, in an iterative manner. By performing a large number of computationally cheap evaluations of the surrogate function, they identify potential candidate solutions, which are then tested against the actual objective function. Therefore, the true evaluations have a twofold purpose. On one hand, they allow the discovery of solutions of the problem at hand, and on the other hand they are used to improve the surrogate model for the next iteration. This strategy minimizes the need for expensive evaluations and enables the use of traditional optimization techniques that rely on the surrogate model for guidance. In this study, we deal with a category of combinatorial optimization problems known as permutation problems, where solutions consist of permutations of elements from a specified set [3, 4]. Specifically, we examine a recently introduced black-box permutation optimization problem, the Asteroid Routing Problem (ARP), presented by [PERSON] et al. in [5]. The ARP is inspired by the _11 th Global Trajectory Optimisation Competition_ (for more details, see the official competition webpage at [[https://sophia.estec.esa.int/gtoc_portal/?page_id=782](https://sophia.estec.esa.int/gtoc_portal/?page_id=782)]([https://sophia.estec.esa.int/gtoc_portal/?page_id=782](https://sophia.estec.esa.int/gtoc_portal/?page_id=782)), accessed on 10 November 2024) and involves a spacecraft that, after being launched from Earth, has to visit a given set of asteroids while minimizing both energy consumption and the total time required. Although this study does not focus on the astrophysical details of the problem, it is important to highlight that the ARP's primary motivation is linked to the growing demand for technological devices, such as mobile phones and computers, which is causing a rapid depletion of Earth's mineral resources like silicon, boronite, and quartz. As a result, asteroid or near-Earth object mining is emerging as a potential solution to address the scarcity of such mineral resources. Interestingly, the ARP is just one of many computational optimization problems related to space engineering. Closely related problems are active debris removal, which has been described in [6], and the design of rendezvous tours through the Sun-Jupiter asteroid, depicted in [7]. Other variants of the well-known Traveling Salesman Problem have been proposed for minimizing the energy and time required for on-orbit servicing [8], and for optimizing debris selection and trajectories in debris collecting missions [9]. From a computational point-of-view, the ARP is a permutation optimization problem where evaluating the objective function necessitates executing a computationally expensive procedure. This makes the ARP an interesting benchmark for costly black-box permutation optimization. Other related benchmark problems have been proposed in [10] and as competition challenges by the European Space Agency (ESA), as reported, e.g., in [11; 12; 13; 14]. Moving out from the field of space engineering, other related problems are fourth-party logistics [15] and the schedule risk control problem of IT outsourcing projects [16].