A Scalable Method for Optimal Path Planning on Manifolds via a Hopf-Lax Type Formula

TL;DR

This paper introduces a scalable, grid-free algorithm for optimal path planning on manifolds, leveraging a generalized Hopf-Lax formula, suitable for high-dimensional problems in motion planning and data analysis.

Contribution

The work presents a novel primal dual hybrid gradient algorithm based on a Hopf-Lax type formula for efficient, scalable path planning on manifolds without grid-based PDE methods.

Findings

Algorithm demonstrates high efficiency in examples.

Scales well for high-dimensional problems.

Outperforms traditional grid-based methods.

Abstract

We consider the problem of optimal path planning on a manifold which is the image of a smooth function. Optimal path-planning is of crucial importance for motion planning, image processing, and statistical data analysis. In this work, we consider a particle lying on the graph of a smooth function that seeks to navigate from some initial point to another point on the manifold in minimal time. We model the problem using optimal control theory, the dynamic programming principle, and a Hamilton-Jacobi-Bellman equation. We then design a novel primal dual hybrid gradient inspired algorithm that resolves the solution efficiently based on a generalized Hopf-Lax type formula. We present examples which demonstrate the effectiveness and efficiency of the algorithm. Finally, we demonstrate that, because the algorithm does not rely on grid-based numerical methods for partial differential equations, it scales well for high-dimensional problems.