  Discovering relationships between materials' microstructures and mechanical
properties is a key goal of materials science. Here, we outline a strategy
exploiting Bayesian optimization to efficiently search the multidimensional
space of microstructures, defined here by the size distribution of precipitates
(fixed impurities or inclusions acting as obstacles for dislocation motion)
within a simple two-dimensional discrete dislocation dynamics model. The aim is
to design a microstructure optimizing a given mechanical property, e.g.,
maximizing the expected value of shear stress for a given strain. The problem
of finding the optimal discretized shape for a distribution involves a norm
constraint, and we find that sampling the space of possible solutions should be
done in a specific way in order to avoid convergence problems. To this end, we
propose a general mathematical approach that can be used to generate trial
solutions uniformly at random while enforcing an Euclidean norm constraint.
Both equality and inequality constraints are considered. A simple technique can
then be used to convert between Euclidean and other Lebesgue $p$-norm (the
1-norm in particular) constrained representations. Considering different
dislocation-precipitate interaction potentials, we demonstrate the convergence
of the algorithm to the optimal solution and discuss its possible extensions to
the more complex and realistic case of three-dimensional dislocation systems
where also the optimization of precipitate shapes could be considered.
