Domain decomposition for integer optimal control with total variation regularization

TL;DR

This paper introduces a globally convergent domain decomposition algorithm for large-scale integer optimal control problems with total variation regularization, significantly reducing computational costs compared to existing methods.

Contribution

It develops a novel domain decomposition approach with convergence guarantees for solving PDE-constrained integer optimal control problems efficiently.

Findings

The algorithm achieves faster solutions than state-of-the-art methods.

Global optimality conditions are equivalent to subdomain conditions with overlap.

The method effectively handles large-scale problems with total variation regularization.

Abstract

Total variation integer optimal control problems admit solutions and necessary optimality conditions via geometric variational analysis. In spite of the existence of said solutions, algorithms which solve the discretized objective suffer from high numerical cost associated with the combinatorial nature of integer programming. Hence, such methods are often limited to small- and medium-sized problems. We propose a globally convergent, coordinate descent-inspired algorithm that allows tractable subproblem solutions restricted to a partition of the domain. Our decomposition method solves relatively small trust-region subproblems that modify the control variable on a subdomain only. Given nontrivial subdomain overlap, we prove that a global first-order necessary optimality condition is equivalent to a first-order necessary optimality condition per subdomain. We additionally show that sufficient decrease is achieved on a single subdomain by way of a trust-region subproblem solver using geometric measure theoretic arguments, which we integrate with a greedy patch selection to prove convergence of our algorithm. We demonstrate the practicality of our algorithm on a benchmark large-scale, PDE-constrained integer optimal control problem, and find that our method is faster than the state-of-the-art.