Shortest Paths in Graphs of Convex Sets

TL;DR

This work tackles shortest-path problems where vertex positions are continuous within convex sets and edge lengths are convex functions of endpoint positions, framing the problem as NP-hard. It develops a strong, scalable mixed-integer convex program (MICP) based on perspective operators to globally optimize paths in graphs of convex sets, complemented by a set-based convex relaxation that is typically tight in practice. The paper also connects the framework to control applications, including minimum-time control and hybrid systems, and provides extensive numerical results showing favorable performance against alternative relaxations. The proposed approach offers a practical, principled route to globally optimal motion-planning and hybrid-control problems, with broad applicability and accessible implementations.

Abstract

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.

Figures (7)

• Figure 1: Example of an SPP in GCS. The source set is on the left and the target set is on the right. The graph edges are arrows, and the shortest path is shown in dashed green. The dotted red lines connect the optimal positions of the vertices along the shortest path.
• Figure 1: Graphs for the formulation of the optimal-control problems in Section \ref{['sec:optimal_control']} as SPPs in GCS.
• Figure 1: Two-dimensional SPP in GCS from Section \ref{['sec:example_2d']}. The tightness of the convex relaxation of our MICP \ref{['eq:micp']} is analyzed for two edge lengths (the Euclidean distance \ref{['eq:2norm']} and the Euclidean distance squared \ref{['eq:2norm_squared']}) and different sizes of the sets $\mathcal{X}_v$ (parameterized by the scalar $\sigma$). As a baseline, we also report the optimal value of the relaxation of the McCormick formulation \ref{['eq:mc_relaxation']}.
• Figure 2: Projection onto two dimensions of a random instance of the SPP in GCS from Section \ref{['sec:example_large_scale']}. The problem parameters have nominal value.
• Figure 3: Relaxation gap versus MICP solution time for the 500 random instances described in Section \ref{['sec:example_large_scale']}. Two edge lengths are analyzed: the Euclidean distance \ref{['eq:2norm']} and the Euclidean distance squared \ref{['eq:2norm_squared']}. For each edge length, 100 nominal instances are generated with the nominal problem parameters, and four other batches of 100 instances each are obtained by increasing a different subset of the parameters. Our relaxation is almost always exact with the Euclidean length. While, with the Euclidean length squared, it is more sensitive to the dimension $n$ of the space and the density of the graph $G$. (Note the different horizontal scales of the two plots.)

Theorems & Definitions (36)

• Theorem 3.1
• Proof 1
• Theorem 3.2
• Definition 4.1
• Remark 4.2
• Example 4.3
• Definition 4.4
• Remark 4.5
• Example 4.6
• Example 4.7