A pessimistic bilevel stochastic problem for elastic shape optimization

We consider pessimistic bilevel stochastic programs in which the follower maximizes over a fixed compact convex set a strictly convex quadratic function, whose Hessian depends on the leader’s decision. This results in a random upper level outcome which is evaluated by a convex risk measure. Under assumptions including real analyticity of the lower-level goal function, we prove the existence of optimal solutions. We discuss an alternate model, where the leader hedges against optimal lower-level solutions, and show that solvability can be guaranteed under weaker conditions in both, a deterministic and a stochastic setting. The approach is applied to a mechanical shape optimization problem in which the leader decides on an optimal material distribution to minimize a tracking-type cost functional, whereas the follower chooses forces from an admissible set to maximize a compliance objective. The material distribution is considered to be stochastically perturbed in the actual construction phase. Computational results illustrate the bilevel optimization concept and demonstrate the interplay of follower and leader in shape design and testing.