Necessary Optimality Conditions for a Class of Bilevel Problems Necessary Optimality Conditions for a Class of Bilevel Problems

This paper addresses a class of bilevel optimization problems involving an upper level problem which is a static optimization problem aimed at minimizing a first performance criterion over optimal trajectories . These trajectories are parameterized by a set of parameters (which has the structure of...

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Bibliographic Details
Published in:Journal of optimization theory and applications Vol. 208; no. 1; p. 30
Main Authors: Abdel Wahab, Abdallah, Bettiol, Piernicola
Format: Journal Article
Language:English
Published: New York Springer US 01.01.2026
Springer Nature B.V
Springer Verlag
Subjects:
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Constraints
Engineering
Euclidean space
Mathematics
Mathematics and Statistics
Metric space
Operations Research/Decision Theory
Optimal control
Optimization
Parameter uncertainty
Perturbation methods
Theory of Computation
Trajectory optimization
Necessary Optimality Conditions
AMS subject classifications: 49K15
49K21
49J52
Uncertainty Parameters
Nonsmooth Analysis
Multiprocesses
Bilevel Problems
Necessary Optimality Conditions · Bilevel Problems · Uncertainty Parameters · Nonsmooth Analysis · Multiprocesses
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Summary:This paper addresses a class of bilevel optimization problems involving an upper level problem which is a static optimization problem aimed at minimizing a first performance criterion over optimal trajectories . These trajectories are parameterized by a set of parameters (which has the structure of a metric space) and evaluated at a finite set of prescribed time instants. The optimal trajectories are the minimizers of a lower level problem , which is a dynamic optimization problem where the objective is to minimize a second cost functional over trajectory/control pairs on a given time interval. We consider intermediate-point problems (i.e. performance criteria take into account the values of the trajectories in the middle of the underlying time interval) including problems with free final and intermediate-point constraints, as well as problems with more general endpoint and intermediate-point constraints. We introduce, to the best of our knowledge, a new notion of solution for the reference bilevel problem: this is a pair composed of a control function and a family of associated trajectories that are optimal for both the upper and lower problems. We establish necessary conditions of optimality for these classes of bilevel problems by transforming them into single ‘min-min’ optimal control problems with uncertainty parameters. To derive our results, we employ perturbation methods to construct, with the help of Ekeland’s variational principle, a sequence of multiprocess optimization problems, on which we apply Clarke and Vinter’s multiprocesses theory together with techniques recently developed for optimal control problems with uncertainty parameters which can be adapted to the bilevel problem studied in the present paper.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-025-02855-6