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 generalization of this classical problem in which the position of each vertex in the graph is a continuous decision variable, constrained to lie in a corresponding convex set. The length of an edge is then defined as a convex function of the positions of the vertices it connects. Problems of this form arise naturally in motion planning of autonomous vehicles, robot navigation, and even optimal control of hybrid dynamical 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 mixed-integer convex formulation based on perspective functions. This formulation has a very tight convex relaxation and makes it possible to efficiently find globally-optimal paths in large graphs and in high-dimensional spaces.

翻译：以图表为参照, 最短路径问题需要找到一系列边缘, 其最小累积长度将源顶点与目标顶点连接起来。 我们考虑对这个古老问题的概括化, 即图形中每个顶点的位置是一个连续的决定变量, 并被限制在相应的锥形中。 然后, 边缘的长度被定义为它连接的顶点位置的矩形函数。 这种形式的问题自然出现在自主飞行器的移动规划中, 机器人导航, 甚至混合动力系统的最佳控制中。 如此广泛应用的代价是这一问题的复杂性, 这个问题很容易被看成是硬的。 我们的主要贡献是基于视角功能的强大的混合整数矩形配方。 这种配方具有非常紧紧的锥形松动, 并有可能在大图表和高维空间中有效地找到全球最佳路径 。