Domain Decomposition for Stochastic Optimal Control

TL;DR

This paper introduces a domain decomposition approach combined with sum of squares and semidefinite programming to efficiently solve large-scale linear stochastic optimal control problems by localizing polynomial approximations.

Contribution

It presents a novel domain decomposition scheme with ADMM optimization to improve scalability and accuracy in stochastic optimal control solutions.

Findings

Enables solving larger SOC problems with lower polynomial degrees.

Improves convergence and conditioning of the optimization process.

Captures both local and global properties of the value function.

Abstract

This work proposes a method for solving linear stochastic optimal control (SOC) problems using sum of squares and semidefinite programming. Previous work had used polynomial optimization to approximate the value function, requiring a high polynomial degree to capture local phenomena. To improve the scalability of the method to problems of interest, a domain decomposition scheme is presented. By using local approximations, lower degree polynomials become sufficient, and both local and global properties of the value function are captured. The domain of the problem is split into a non-overlapping partition, with added constraints ensuring $C^1$ continuity. The Alternating Direction Method of Multipliers (ADMM) is used to optimize over each domain in parallel and ensure convergence on the boundaries of the partitions. This results in improved conditioning of the problem and allows for much larger and more complex problems to be addressed with improved performance.