2026/04/26 by Khaled Surur, Melkior Ornik
Computer Science · Engineering · #eess.SY #cs.SY
This paper investigates the problem of computing the shortest path between two states under resource constraints in environments with resource-replenishment regions. Namely, the length of the path is limited by a budget that can be restored within polytopic replenishment regions. We show that the optimal path in this problem exhibits a distinct geometric structure: it consists of straight-line segments, changes direction at the edges of replenishment regions, and enters a sequence of replenishment regions in a feasible-connected space. We propose an approach to solve the continuous problem in two steps: using a graph-based approach, followed by convex programming. First, we define a graph whose nodes are possible waypoints of feasible paths, and the edges are the Euclidean distances between these nodes. To obtain a discrete set of nodes that ensure a feasible and near-optimal solution, we utilize a wavefront algorithm. With a sufficiently small spacing between wavefronts, the solution of the shortest path problem on this graph yields the optimal sequence of polytopes to visit. Next, we use convex optimization on this sequence of polytopes to refine the solution to optimality. A numerical experiment is presented to demonstrate the effectiveness of the approach.