$A^*$ for Graphs of Convex Sets

TL;DR

The paper addresses the Shortest Path Problem in the Graph of Convex Sets (SPP-GCS) by fusing heuristic search with convex-relaxation techniques. It introduces A^*-GCS, which uses a cut-set growth strategy anchored by relaxations and admissible heuristics to obtain lower bounds and, when possible, feasible solutions, with a reversal of the traditional relaxation flow that is particularly effective for Euclidean costs. Theoretical guarantees ensure finite termination and valid bounds, while extensive numerical experiments on 2D mazes, axis-aligned bars, and large 3D maps demonstrate significant reductions in relaxation sizes and substantial speedups over full-relaxation baselines. The results underscore the practicality of computing bounds and near-optimal solutions for geometric planning problems, and point to scalable applicability through careful initialization and heuristic design.

Abstract

We present a novel algorithm that fuses the existing convex-programming based approach with heuristic information to find optimality guarantees and near-optimal paths for the Shortest Path Problem in the Graph of Convex Sets (SPP-GCS). Our method, inspired by $A^*$, initiates a best-first-like procedure from a designated subset of vertices and iteratively expands it until further growth is neither possible nor beneficial. Traditionally, obtaining solutions with bounds for an optimization problem involves solving a relaxation, modifying the relaxed solution to a feasible one, and then comparing the two solutions to establish bounds. However, for SPP-GCS, we demonstrate that reversing this process can be more advantageous, especially with Euclidean travel costs. In other words, we initially employ $A^*$ to find a feasible solution for SPP-GCS, then solve a convex relaxation restricted to the vertices explored by $A^*$ to obtain a relaxed solution, and finally, compare the solutions to derive bounds. We present numerical results to highlight the advantages of our algorithm over the existing approach in terms of the sizes of the convex programs solved and computation time.

Figures (8)

• Figure 1: Illustration of the SPP-GCS. The graph has six vertices, with the origin and the destination corresponding to singleton sets. The dotted lines represent the edges in the graph. Each vertex $t$ in the graph is associated with a heuristic value $h(t)$. A feasible path is shown with solid (red) line segments.
• Figure 2: Setup in the SPP*-GCS showing the cut-set $S$, and subsets $N_S$ and $S'$. $N_S$ contains all the vertices in $V\setminus S$ that are adjacent to $S$, and $S'$ is any nonempty subset of $N_S$. A feasible path for SPP*-GCS is shown with solid (red) line segments.
• Figure 3: 2D maps and their GCS. In the GCS corresponding to the Maze, the red segments show the convex sets and the black lines show the edges. Similarly, in the GCS corresponding to the axis-aligned bars, the shaded squares show the convex sets and the black lines show the edges.
• Figure 4: The feasible path (thick red lines) and the cut-set vertices (shown in color blue) produced by the two-step $A^*$ based heuristic for different weights. There are 415 vertices in this GCS corresponding to map no. 2. $A^*$-$GCS$ in this case terminated in 1 iteration for all the weights and produced the same bounds. By first finding the cut-set ($S_{A^*}$) and then applying $A^*$-$GCS$ with $S_{init}:=S_{A^*}$, we reduced the size of the relaxation with $|V|-1=414$ vertices into a relaxation with $|S_{A^*}|$ vertices.
• Figure 5:

Theorems & Definitions (16)

• Lemma 1
• proof
• Lemma 2
• proof
• Remark 1
• Remark 2
• Lemma 3
• proof
• Lemma 4
• proof