Bilevel stochastic linear programming with integer variables in the lower level problem

Bilevel models in the presence of stochastic uncertainty are among the newer research objects in leader-follower optimization. An extension to a scheme of alternating decision and observation seems obvious against the background of traditional two-stage stochastic problems. Thus, bilevel stochastic problems arise from the interaction of two decision makers on different levels of hierarchy and the realization of a random vector before the follower makes a choice (data uncertainty). While the optimal objective value of the recourse problem is returned to the first stage in a two-stage stochastic optimization problem, the set of optimal lower level solutions is returned to the leader in a bilevel (stochastic) problem. In the present work, we add lower level (equal to second stage) integrality constraints to a bilevel stochastic problem. This model extension is known to have drastic consequences for the analytical properties of the upper level objective function as a function of the leader's decision, even in the optimistic formulation of a bilevel linear problem. For example, the lower semicontinuity of this mapping cannot be maintained in general, and the existence of an optimal solution cannot be guaranteed under standard assumptions. These consequences are the subject of this thesis, which examines the optimistic (as well as the pessimistic) formulation of a bilevel stochastic linear model whose lower level solutions may not be unique. In addition, only the right-hand side of the system of lower level constraints is affected by both the decision of the leader and the realization of a random vector. Prior to this, we study the function obtained by optimizing a linear function over the set of minimizers of an integer linear problem, and characterize the set of its discontinuity points. Assuming that the follower is the only one acting with complete information, a convex risk measure is used to estimate and evaluate the upper level (equal to first stage) outcome. Sufficient conditions for the (local) Hölder continuity of the leader's risk functional are established, and conclusions on the existence of optimal solutions are drawn. Then, the qualitative stability of the model under weak convergence of the underlying probability measure is investigated. The joint continuity of the leader's risk functional in dependence on the upper level decision and the probability measure is verified. In the analysis, two types of ground sets of the lower level feasibility set are treated with different assumptions and using different techniques: Initially the set is finite, later the convex hull of the set is a polyhedron with integer vertices.