Given a graph, the shortest-path problem requires finding a sequence of edges with minimum cumulative length that connects a source vertex to a target vertex. We consider a variant of this classical problem in which the position of each vertex in the graph is a continuous decision variable constrained in a convex set, and the length of an edge is a convex function of the position of its endpoints. Problems of this form arise naturally in many areas, from motion planning of autonomous vehicles to optimal control of hybrid systems. The price for such a wide applicability is the complexity of this problem, which is easily seen to be NP-hard. Our main contribution is a strong and lightweight mixed-integer convex formulation based on perspective operators, that makes it possible to efficiently find globally optimal paths in large graphs and in high-dimensional spaces.

翻译：给定一个图，最短路径问题要求找到连接源顶点与目标顶点且累积长度最小的边序列。我们研究该经典问题的一个变体：图中每个顶点的位置是受限于凸集的连续决策变量，而边的长度是其端点位置的凸函数。此类问题自然出现在众多领域，从自动驾驶车辆的路径规划到混合系统的最优控制。广泛适用性的代价在于该问题的复杂性——不难看出其为NP难题。我们的主要贡献是基于透视算子构建一种精简而高效的混合整数凸规划模型，使得在大规模图和高维空间中高效寻找全局最优路径成为可能。