Global Convergence of Successive Approximations for Non-convex Stochastic Optimal Control Problems

TL;DR

This paper establishes the global convergence of a modified successive approximation method for non-convex stochastic optimal control problems, providing convergence rates under certain conditions without requiring convex control domains.

Contribution

It introduces a novel error estimate involving a higher order backward adjoint equation, enabling global convergence analysis without convex control domain assumptions.

Findings

Convergence to the global minimum under certain convexity conditions.

A convergence rate is derived for generalized linear-quadratic systems.

The method handles control-dependent diffusions effectively.

Abstract

This paper focuses on finding approximate solutions to stochastic optimal control problems with control domains being not necessarily convex, where the state trajectory is subject to controlled stochastic differential equations. The control-dependent diffusions make the traditional method of successive approximations (MSA) insufficient to reduce the value of cost functional in each iteration. Without adding extra terms over which to perform the Hamiltonian minimization, the MSA becomes sufficient by our novel error estimate involving a higher order backward adjoint equation. Under certain convexity assumptions on the coefficients (no convexity assumptions on the control domains), the value of the cost functional descends to the global minimum as the number of iterations tends to infinity. In particular, a convergence rate is available for a class of generalized linear-quadratic systems.