$A^*$ for Graphs of Convex Sets

TL;DR

This paper introduces an $A^*$-inspired algorithm for the Shortest Path Problem in the Graph of Convex Sets, combining convex programming and heuristic search to efficiently find near-optimal solutions with guarantees.

Contribution

It proposes a novel $A^*$-based method that reverses traditional relaxation approaches, improving efficiency and solution bounds for SPP-GCS with Euclidean costs.

Findings

Reduces convex program sizes compared to existing methods.

Achieves faster computation times.

Provides optimality guarantees for solutions.

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.