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We consider pessimistic bilevel stochastic programs in which the followermaximizes over a fixed compact convex set a strictly convex quadratic function,whose Hessian depends on the leader's decision. The resulting random variableis evaluated by a convex risk measure. Under assumptions including realanalyticity of the lower-level goal function, we prove existence of optimalsolutions. We discuss an alternate model where the leader hedges againstoptimal lower-level solutions, and show that in this case solvability can beguaranteed under weaker conditions both in a deterministic and in a stochasticsetting. The approach is applied to a mechanical shape optimization problem inwhich the leader decides on an optimal material distribution to minimize atracking-type cost functional, whereas the follower chooses forces from anadmissible set to maximize a compliance objective. The material distribution isconsidered to be stochastically perturbed in the actual construction phase.Computational results illustrate the bilevel optimization concept anddemonstrate the interplay of follower and leader in shape design and testing.

A Pessimistic Bilevel Stochastic Problem for Elastic Shape Optimization