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 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.