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In this thesis we consider bilevel optimization problems in infinite-dimensional spaces. In particular, we are interested in providing first-order necessary optimality conditions. We consider bilevel optimization problems both in abstract Banach spaces and in some special situations. This includes the optimal control of the obstacle problem, which is a typical bilevel optimization problem in Sobolev spaces, as well as a class of inverse optimal control problems. We obtain optimality conditions for these more specific optimization problems by applying our results from the abstract setting.
Our main approach for deriving optimality conditions in the abstract setting utilizes the relaxation of a reformulation of the bilevel optimization problem via the optimal value function. We also introduce the so-called normal-cone-preserving operators and show how this concept can be applied.
We also consider other topics that arise in this context. For instance, we investigate the so-called limiting normal cone to a complementarity set in Sobolev spaces. This complementarity set plays a central role in the context of the optimal control of the obstacle problem. The limiting normal cone is a concept which appears in the area of variational analysis and generalizes the usual normal cone from convex analysis. We also investigate in which spaces Legendre forms and Legendre-* forms can exist. We show that if a Legendre-* form exists in a reflexive Banach space or a space with a separable predual space, then this space is already isomorphic to a Hilbert space. We also consider a discretization of a bilevel optimization problem in Lebesgue spaces. We present both theoretical error estimates and numerical experiments.
The new results in this thesis are illustrated by examples and counterexamples. In order to present the topics in a self-contained way, we review some known concepts and their basic properties. A particular focus for this is on the definitions and properties from the area of capacity theory.