Convergence rates of moment-sum-of-squares hierarchies for optimal control problems

TL;DR

This paper analyzes the convergence rate of moment-sum-of-squares hierarchies in polynomial optimal control problems, establishing an O(1/ log log d) rate for the approximation of the value function.

Contribution

It provides the first explicit convergence rate for these hierarchies in continuous-time optimal control, including infinite and finite horizon cases.

Findings

Convergence rate is O(1/ log log d) for polynomial under-approximations.

Results apply to continuous-time infinite-horizon discounted problems.

Method extends to finite-horizon and discrete-time problems.

Abstract

We study the convergence rate of moment-sum-of-squares hierarchies of semidefinite programs for optimal control problems with polynomial data. It is known that these hierarchies generate polynomial under-approximations to the value function of the optimal control problem and that these under-approximations converge in the L1 norm to the value function as their degree d tends to infinity. We show that the rate of this convergence is O(1/ log log d). We treat in detail the continuous-time infinite-horizon discounted problem and describe in brief how the same rate can be obtained for the finite-horizon continuous-time problem and for the discrete-time counterparts of both problems.