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：本文提出了一种求解移动目标旅行商问题（MT-TSP）的新公式，该问题旨在为智能体寻找最短路径：智能体从仓库出发，在指定时间窗内精确访问一组移动目标一次，然后返回仓库。该公式基于一个关键思想：当目标沿直线运动时，其轨迹在时空坐标系中形成凸集。于是问题简化为在凸集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求解器进行了比较。实验结果表明，对于最多包含20个目标的实例，该公式在运行时间上比混合整数锥规划方法降低了两个数量级，最优性间隙提升了60%。我们还表明，与混合整数锥规划方法相比，该公式的凸松弛解为移动目标旅行商问题提供的下界显著更严格。