Stochastic bilevel programming in structural optimization

We consider the mathematical modelling and solution of robust and cost-optimizing structural (topology) design problems. The setting is the optimal design of a linear-elastic structure, for example a truss topology, under unilateral frictionless contact, and under uncertainty in the data describing the load conditions, the material properties, and the rigid foundation. The resulting stochastic bilevel optimization model finds a structural design that responds the best to the given probability distribution in the data. This model is of special interest when a structural failure will lead to a reconstruction cost, rather than loss of life. For the mathematical model, we provide results on the existence of optimal solutions which allow for zero lower design bounds. We establish that the optimal solution is continuous in the lower design bounds, a result which validates the use of small but positive values of them, and for such bounds we also establish the locally Lipschitz continuity and directional differentiability of the implicit upper-level objective function. We also provide a heuristic algorithm for the solution of the problem, which makes use of its differentiability properties and parallelization strategies across the scenarios. A small set of numerical experiments illustrates the behaviour of the stochastic solution compared to all average-case deterministic one, establishing ail increased robustness.