  Topological photonic crystals have received considerable attention for their
ability to manipulate and guide light in unique ways. They are typically
designed by hand based on careful analysis of their bands and mode profiles,
but recent theoretical advances have revealed new and powerful insights into
the connection between band symmetry, connectivity, and topology. Here we
propose a combined global and local optimization framework that integrates a
flexible symmetry-constrained level-set parameterization with standard
gradient-free optimization algorithms to optimize topological photonic
crystals, a problem setting where the objective function may be highly
non-convex and non-continuous. Our framework can be applied to any
symmetry-identifiable band topology, and we demonstrate its applicability to
several prominent kinds of three-dimensional band topology, namely
$\Gamma$-enforced nodal lines, Weyl points, and Chern insulators. Requiring no
prior examples of topological photonic crystals or prior knowledge on the
connection between structure and band topology, our approach indicates a path
towards the automated discovery of novel topological photonic crystal designs.
