Relaxed multibang regularization for the combinatorial integral approximation

TL;DR

This paper introduces a new class of polyhedral functions that improve the regularization of relaxations in integer optimal control, with an extended algorithmic framework ensuring convergence to optimal solutions.

Contribution

It develops a novel approach combining multibang regularization with combinatorial integral approximation, enhancing convergence properties for integer control problems.

Findings

Polyhedral functions have beneficial properties for regularization.

The extended algorithm guarantees convergence of discrete controls.

Improved methods for solving relaxed integer optimal control problems.

Abstract

Multibang regularization and combinatorial integral approximation decompositions are two actively researched techniques for integer optimal control. We consider a class of polyhedral functions that arise particularly as convex lower envelopes of multibang regularizers and show that they have beneficial properties with respect to regularization of relaxations of integer optimal control problems. We extend the algorithmic framework of the combinatorial integral approximation such that a subsequence of the computed discrete-valued controls converges to the infimum of the regularized integer control problem.