Parabolic optimal control problems with combinatorial switching constraints -- Part II: Outer approximation algorithm

TL;DR

This paper develops an outer approximation algorithm for binary control problems governed by PDEs with combinatorial switching constraints, proving convergence and demonstrating efficiency through a numerical example.

Contribution

It introduces an outer approximation algorithm for PDE control problems with combinatorial constraints, utilizing convex hull descriptions and semi-smooth Newton methods.

Findings

Algorithm converges strongly in L^2 to the global minimizer.

Semi-smooth Newton method efficiently solves subproblems.

Numerical example demonstrates practical effectiveness.

Abstract

We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon, they can thus be seen as dynamic switches. The switching patterns may be subject to combinatorial constraints such as, e.g., an upper bound on the total number of switchings or a lower bound on the time between two switchings. In a companion paper [arXiv:2203.07121], we describe the $L^p$-closure of the convex hull of feasible switching patterns as intersection of convex sets derived from finite-dimensional projections. In this paper, the resulting outer description is used for the construction of an outer approximation algorithm in function space, whose iterates are proven to converge strongly in $L^2$ to the global minimizer of the convexified optimal control problem. The linear-quadratic subproblems arising in each iteration of the outer approximation algorithm are solved by means of a semi-smooth Newton method. A numerical example in two spatial dimensions illustrates the efficiency of the overall algorithm.