Numerical solution of optimal control problems using quadratic transport regularization
Nicolas Borchard, Gerd Wachsmuth
TL;DR
This work addresses PDE-constrained optimal control where the control is a measure regularized by the quadratic Wasserstein distance $W_2^2(u_d,u)$ to a prior $u_d$. By discretizing the control as Dirac masses at fixed points, the problem becomes finite-dimensional, enabling a Brenier-structure based reformulation in terms of a transport map that is the gradient of a convex potential. The reformulation yields a semismooth equation $r(\xi)=0$ and a reduced problem in variables $\xi$, with theoretical results on semismoothness and convergence of fixed-point and semismooth Newton methods, complemented by numerical experiments. The approach leverages Kantorovich duality and Brenier's theorem to connect optimal transport to a tractable finite-dimensional optimization, offering a rigorous framework for semi-discrete OT problems with transport regularization and potential extensions to storage fees or queue penalization scenarios.
Abstract
We address optimal control problems on the space of measures for an objective containing a smooth functional and an optimal transport regularization. That is, the quadratic Monge-Kantorovich distance between a given prior measure and the control is penalized in the objective. We consider optimality conditions and reparametrize the problem using the celebrated structure theorem by Brenier. The optimality conditions can be formulated as a piecewise differentiable equation. This is utilized to formulate solution algorithms and to analyze their local convergence properties. We present a numerical example to illustrate the theoretical findings.