Integer optimal control problems with total variation regularization: Optimality conditions and fast solution of subproblems

TL;DR

This paper develops optimality conditions and a fast, polynomial-time algorithm for solving integer optimal control problems with total variation regularization, combining theoretical insights with practical numerical methods.

Contribution

It introduces new first and second order optimality conditions and a proximal-gradient algorithm leveraging Bellman's principle for efficient solutions.

Findings

Derived equivalence of local optimality conditions for control problems and switching point problems.

Proposed a polynomial-time algorithm for discretized subproblems using Bellman's optimality principle.

Demonstrated computational effectiveness of the method through numerical experiments.

Abstract

We investigate local optimality conditions of first and second order for integer optimal control problems with total variation regularization via a finite-dimensional switching point problem. We show the equivalence of local optimality for both problems, which will be used to derive conditions concerning the switching points of the control function. A non-local optimality condition treating back-and-forth switches will be formulated. For the numerical solution, we propose a proximal-gradient method. The emerging discretized subproblems will be solved by employing Bellman's optimality principle, leading to an algorithm which is polynomial in the mesh size and in the admissible control levels. An adaption of this algorithm can be used to handle subproblems of the trust-region method proposed in Leyffer, Manns, 2021. Finally, we demonstrate computational results.