Sequential Linear Integer Programming for Integer Optimal Control with Total Variation Regularization

TL;DR

This paper introduces a trust-region method that solves a sequence of linear integer programs to address integer optimal control problems with total variation regularization, ensuring existence of solutions and convergence to stationary points.

Contribution

It presents a novel algorithm combining linear integer programming with total variation regularization for optimal control, with theoretical convergence guarantees.

Findings

Proves existence of minimizers for the regularized problem

Establishes convergence of the algorithm to stationary points in 1D cases

Demonstrates effectiveness through a computational example

Abstract

We propose a trust-region method that solves a sequence of linear integer programs to tackle integer optimal control problems regularized with a total variation penalty. The total variation penalty allows us to prove the existence of minimizers of the integer optimal control problem. We introduce a local optimality concept for the problem, which arises from the infinite-dimensional perspective. In the case of a one-dimensional domain of the control function, we prove convergence of the iterates produced by our algorithm to points that satisfy first-order stationarity conditions for local optimality. We demonstrate the theoretical findings on a computational example.