The goal of this paper is to study the so-called worst-case or robust
optimal design problem for minimal compliance. In the context of linear
elasticity we seek an optimal shape which minimizes the largest, or worst,
compliance when the loads are subject to some unknown perturbations.
We first prove that, for a fixed shape, there exists indeed a worst
perturbation (possibly non unique) that we characterize as the maximizer
of a nonlinear energy. We also propose a stable algorithm to
compute it. Then, in the framework of Hadamard method, we
compute the directional shape derivative of this criterion which is
used in a numerical algorithm, based on the level set method,
to find optimal shapes that minimize the worst-case compliance.
Since this criterion is usually merely directionally differentiable,
we introduce a semidefinite programming approach to
select the best descent direction at each step of a
gradient method. Numerical examples are given in 2-d and 3-d.