A Pessimistic Bilevel Stochastic Problem for Elastic Shape Optimization

TL;DR

This paper introduces a novel pessimistic bilevel stochastic optimization framework for elastic shape design, analyzing existence of solutions and applying it to a mechanical shape optimization problem with stochastic material perturbations.

Contribution

It develops a new theoretical model for stochastic bilevel problems with convex risk measures and demonstrates its application to shape optimization under uncertainty.

Findings

Existence of optimal solutions under certain conditions.

A new model where the leader hedges against lower-level solutions.

Computational results illustrate the effectiveness of the approach.

Abstract

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. The resulting random variable is evaluated by a convex risk measure. Under assumptions including real analyticity of the lower-level goal function, we prove existence of optimal solutions. We discuss an alternate model where the leader hedges against optimal lower-level solutions, and show that in this case solvability can be guaranteed under weaker conditions both in a deterministic and in 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.