The Sea Exploration Problem Revisited

Sea exploration is important for countries with large areas in the ocean under their control, since in the future it may be possible to exploit some of the resources in the seafloor. The sea exploration problem was presented by Pedroso et al. [13] (unpublished); we maintain most of the paper's structure, to provide the needed theoretical background and context. In the sea exploration problem, the aim is to schedule the expedition of a ship for collecting information about the resources on the seafloor. The goal is to collect data by probing on a set of carefully chosen locations, so that the information available is optimally enriched. This problem has similarities with the orienteering problem, where the aim is to plan a time-limited trip for visiting a set of vertices, collecting a prize at each of them, in such a way that the total value collected is maximum. In our problem, the score at each vertex is associated with an estimation of the level of the resource on the given surface, which is done by regression using Gaussian processes. Hence, there is a correlation among scores on the selected vertices; this is the first difference with respect to the standard orienteering problem. The second difference is the location of each vertex, which in our problem is a freely chosen point on a given surface. Results on a benchmark test set are presented and analyzed, confirming the merit of the approach proposed. In this paper, additional methods are presented, along with a small topological result and subsequent proof of the convergence of these same methods to the optimal solution, when we have instant access to the ground truth and the underlying function is piecewise continuous.