Finding global solutions of some inverse optimal control problems using penalization and semismooth Newton methods

TL;DR

This paper introduces a novel approach combining penalization and semismooth Newton methods to find global solutions for certain inverse optimal control problems, addressing nonconvexity and regularity challenges.

Contribution

It proposes a relaxation-based decomposition method and employs a penalty approach with semismooth Newton techniques for global optimality in inverse control problems.

Findings

The method effectively solves a class of inverse optimal control problems.

Numerical results demonstrate the approach's convergence and practical viability.

Discussion of limitations highlights areas for future research.

Abstract

We present a method to solve a special class of parameter identification problems for an elliptic optimal control problem to global optimality. The bilevel problem is reformulated via the optimal-value function of the lower-level problem. The reformulated problem is nonconvex and standard regularity conditions like Robinson's CQ are violated. Via a relaxation of the constraints, the problem can be decomposed into a family of convex problems and this is the basis for a solution algorithm. The convergence properties are analyzed. It is shown that a penalty method can be employed to solve this family of problems while maintaining convergence speed. For an example problem, the use of the identity as penalty function allows for the solution by a semismooth Newton method. Numerical results are presented. Difficulties and limitations of our approach to solve a nonconvex problem to global optimality are discussed.