Efficiency of tree-search like heuristics to solve complex mixed-integer programming problems applied to the design of optimal space trajectories

In the past, space trajectory optimization was limited to optimal design of transfers to single destinations, where optimality refers to minimum propellant consumption or transfer time. New technologies, and a more daring approach to space, are today making the space community consider missions that target multiple destinations. In the present paper, we focus on missions that aim to visit multiple asteroids within a single launch. The trajectory design of these missions is complicated by the fact that the asteroid sequences are not known a priori but are the objective of the optimization itself. Usually, these problems are formulated as global optimization (GO) problems, under the formulation of mixed-integer non-linear programming (MINLP), on which the decision variables assume both continuous and discrete values. However, beyond the aim of finding the global optimum, mission designers are usually interested in providing a wide range of mission design options reflecting the multi-modality of the problems at hand. In this sense, a Constraint Satisfaction Problem (CSP) formulation is also relevant. In this manuscript, we focus on these two needs (i.e. tackling both the GO and the CSP) for the asteroid tour problem. First, a tree-search algorithm based upon the Bellman’s principle of optimality is described using dynamic programming approach to address the feasibility of solving the GO problem. This results in an efficient and scalable procedure to obtain global optimum solutions within large datasets of asteroids. Secondly, tree-search strategies like Beam Search and Ant Colony Optimization with back-tracking are tested over the CSP formulations. Results reveal that BS handles better the multi-modality of the search space when compared to ACO, as this latter solver has a bias towards elite solutions, which eventually hinders the diversity needed to efficiently cope with CSP over graphs.