Bilevel Optimal Control: Existence Results and Stationarity Conditions

Mehlitz, Patrick; Wachsmuth, Gerd

doi:10.1007/978-3-030-52119-6_16

Cited by 8 publications

(4 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We finally mention that more general results on the existence of solutions for bilevel optimal control problems are given in [Mehlitz, Wachsmuth, 2020]. In particular, our result is covered by the second part of [Mehlitz, Wachsmuth, 2020, Theorem 16.3.5].…”

Section: Preliminary Resultsmentioning

confidence: 98%

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

Friedemann¹,

Harder²,

Wachsmuth³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Preliminary Resultsmentioning

confidence: 98%

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

Friedemann¹,

Harder²,

Wachsmuth³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the same vein, bilevel optimal control problems, which are bilevel optimization problems with control at least one level, have been considered because of its numerous applications; see, e.g., [1,2,3,4,5,6,7] and the references therein. Many methods have been used to investigate such problems, including theoretical studies [8,9,10] and numerical studies [4,11,12].…”

Section: Introductionmentioning

confidence: 99%

Pontryagin optimality conditions for generalized bilevel optimal control problems with pure state inequality constraints

2024

J. Appl. Numer. Optim.

View full text Add to dashboard Cite

In this paper, we study a generalized bilevel optimal control problem that has a variational inequality parametrized by the final state on the follower and pure state constraints on the leader. After reducing the problem with a gap function to an analogous single-level optimal control problem, we focus on the development of a necessary optimality condition of the Pontryagin type. We highlight some significant issues originating from the generalized bilevel structure and its pure state constraints on the leader, which give rise to a degenerated maximum principle in the absence of constraint qualifications. To ensure the nondegeneracy of the derived maximum principle, we employ a partial penalization strategy and a well-known regularity criterion for optimal control problems with pure state constraints.

show abstract

“…On the other hand, the characterization of optimal solutions through Pontryagin's maximum has long been of significant interest, aiming to refine first-order optimality conditions for many classic bilevel optimal control problems. Numerous studies, such as [3,5,13,14,21], provide a comprehensive survey of the theoretical landscape on bilevel optimal control problems. Additionally, studies on practical applications can be found in [2,[9][10][11].…”

Section: Introductionmentioning

confidence: 99%

“…The optimality conditions obtained were Pontryagin-weak and strong conditions. The same authors addressed in [14] an inverse optimal control problem. They initially established the existence of solutions, and subsequently, by employing the optimistic approach and the concept of relaxation, they established necessary optimality conditions for the given problem.…”

Section: Introductionmentioning

confidence: 99%

Optimality conditions for bilevel optimal control problems with non-convex quasi-variational inequalities

El Idrissi,

Lafhim,

Kalmoun

et al. 2024

RAIRO-Oper. Res.

View full text Add to dashboard Cite

We establish Pontryagin optimality conditions for a generalized bilevel optimal control problem in which the leader is subject to a pure state inequality constraint, while the follower is governed by a non-convex quasi-variational inequality parameterized by the final state. To simplify the problem at hand, we convert it into a single-level optimal control problem by mapping the solution set of the quasi-variational inequality to a parametric optimization problem and employing the value function reformulation. Furthermore, we introduce certain regularity conditions to ensure that the derived maximum principle remains non-degenerate. Finally, we provide an illustrative example to elucidate our research findings.

show abstract

Bilevel Optimal Control: Existence Results and Stationarity Conditions

Cited by 8 publications

References 48 publications

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

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

Pontryagin optimality conditions for generalized bilevel optimal control problems with pure state inequality constraints

Optimality conditions for bilevel optimal control problems with non-convex quasi-variational inequalities

Contact Info

Product

Resources

About