  In this paper, we address the challenging problem of optimal experimental
design (OED) of constrained inverse problems. We consider two OED formulations
that allow reducing the experimental costs by minimizing the number of
measurements. The first formulation assumes a fine discretization of the design
parameter space and uses sparsity promoting regularization to obtain an
efficient design. The second formulation parameterizes the design and seeks
optimal placement for these measurements by solving a small-dimensional
optimization problem. We consider both problems in a Bayes risk as well as an
empirical Bayes risk minimization framework. For the unconstrained inverse
state problem, we exploit the closed form solution for the inner problem to
efficiently compute derivatives for the outer OED problem. The empirical
formulation does not require an explicit solution of the inverse problem and
therefore allows to integrate constraints efficiently. A key contribution is an
efficient optimization method for solving the resulting, typically
high-dimensional, bilevel optimization problem using derivative-based methods.
To overcome the lack of non-differentiability in active set methods for
inequality constraints problems, we use a relaxed interior point method. To
address the growing computational complexity of empirical Bayes OED, we
parallelize the computation over the training models. Numerical examples and
illustrations from tomographic reconstruction, for various data sets and under
different constraints, demonstrate the impact of constraints on the optimal
design and highlight the importance of OED for constrained problems.
