An Efficient Policy Iteration Algorithm for Dynamic Programming Equations

TL;DR

This paper introduces a hybrid algorithm that accelerates solving Hamilton-Jacobi-Bellman equations by combining value iteration and policy iteration, improving efficiency and convergence in optimal control problems.

Contribution

The paper proposes a novel coupling of value and policy iteration methods with an adaptive switching strategy to enhance computational efficiency for static HJB equations.

Findings

Effective in dimensions two, three, and four

Reduces computation time compared to traditional methods

Demonstrates superlinear convergence in relevant cases

Abstract

We present an accelerated algorithm for the solution of static Hamilton-Jacobi-Bellman equations related to optimal control problems. Our scheme is based on a classic policy iteration procedure, which is known to have superlinear convergence in many relevant cases provided the initial guess is sufficiently close to the solution. In many cases, this limitation degenerates into a behavior similar to a value iteration method, with an increased computation time. The new scheme circumvents this problem by combining the advantages of both algorithms with an efficient coupling. The method starts with a value iteration phase and then switches to a policy iteration procedure when a certain error threshold is reached. A delicate point is to determine this threshold in order to avoid cumbersome computation with the value iteration and, at the same time, to be reasonably sure that the policy iteration method will finally converge to the optimal solution. We analyze the methods and efficient coupling in a number of examples in dimension two, three and four illustrating its properties.