Bilevel Optimal Control: Theory, Algorithms, and Applications

TL;DR

This paper investigates inverse optimal control problems, focusing on linear-quadratic models with control constraints, establishing optimality conditions, reviewing algorithms, and comparing their numerical performance.

Contribution

It introduces necessary optimality conditions for inverse control problems reformulated as MPCCs and compares two recent algorithms for solving these problems.

Findings

M-stationarity provides necessary optimality conditions.

Augmented Lagrangian and nonsmooth Newton methods are effective.

Numerical comparison shows differences in algorithm performance.

Abstract

In this chapter, we are concerned with inverse optimal control problems, i.e., optimization models which are used to identify parameters in optimal control problems from given measurements. Here, we focus on linear-quadratic optimal control problems with control constraints where the reference control plays the role of the parameter and has to be reconstructed. First, it is shown that pointwise M-stationarity, associated with the reformulation of the hierarchical model as a so-called mathematical problem with complementarity constraints (MPCC) in function spaces, provides a necessary optimality condition under some additional assumptions on the data. Second, we review two recently developed algorithms (an augmented Lagrangian method and a nonsmooth Newton method) for the computational identification of M-stationary points of finite-dimensional MPCCs. Finally, a numerical comparison of these methods, based on instances of the appropriately discretized inverse optimal control problem of our interest, is provided.