An efficient DP algorithm on a tree-structure for finite horizon optimal control problems

TL;DR

This paper introduces a novel tree-structure dynamic programming algorithm for finite horizon optimal control problems that avoids the curse of dimensionality associated with grid-based methods, enabling high-dimensional problem solving.

Contribution

The paper presents a new tree-structure DP algorithm that eliminates the need for fixed space grids, improving scalability for high-dimensional optimal control problems.

Findings

Effective in high-dimensional settings

Reduces computational complexity compared to grid-based methods

Numerical tests demonstrate improved performance

Abstract

The classical Dynamic Programming (DP) approach to optimal control problems is based on the characterization of the value function as the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. The DP scheme for the numerical approximation of viscosity solutions of Bellman equations is typically based on a time discretization which is projected on a fixed state-space grid. The time discretization can be done by a one-step scheme for the dynamics and the projection on the grid typically uses a local interpolation. Clearly the use of a grid is a limitation with respect to possible applications in high-dimensional problems due to the curse of dimensionality. Here, we present a new approach for finite horizon optimal control problems where the value function is computed using a DP algorithm on a tree structure algorithm (TSA) constructed by the time discrete dynamics. In this way there is no need to build a fixed space triangulation and to project on it. The tree will guarantee a perfect matching with the discrete dynamics and drop off the cost of the space interpolation allowing for the solution of very high-dimensional problems. Numerical tests will show the effectiveness of the proposed method.