This paper introduces a new formulation that finds the optimum for the Moving-Target Traveling Salesman Problem (MT-TSP), which seeks to find a shortest path for an agent, that starts at a depot, visits a set of moving targets exactly once within their assigned time-windows, and returns to the depot. The formulation relies on the key idea that when the targets move along lines, their trajectories become convex sets within the space-time coordinate system. The problem then reduces to finding the shortest path within a graph of convex sets, subject to some speed constraints. We compare our formulation with the current state-of-the-art Mixed Integer Conic Program (MICP) solver for the MT-TSP. The experimental results show that our formulation outperforms the MICP for instances with up to 20 targets, with up to two orders of magnitude reduction in runtime, and up to a 60\% tighter optimality gap. We also show that the solution cost from the convex relaxation of our formulation provides significantly tighter lower bounds for the MT-TSP than the ones from the MICP.

翻译：本文提出了一种新的数学规划形式，用于求解移动目标旅行商问题（Moving-Target Traveling Salesman Problem, MT-TSP）的最优解。该问题旨在为智能体找到一条最短路径：从起点出发，在指定时间窗内精确访问每个移动目标一次，并返回起点。该公式的核心思想是：当目标沿直线运动时，其轨迹在时空坐标系中成为凸集。此时，问题可简化为在凸集构成的图中寻找满足速度约束的最短路径。我们将所提公式与当前最先进的移动目标旅行商问题混合整数锥规划（MICP）求解器进行了对比。实验结果表明：对于包含多达20个目标的实例，我们的公式在运行时间上实现了两个数量级的缩减，最优性间隙缩小了60%。此外，我们公式的凸松弛解所提供的最优下界显著优于MICP方法。