  We develop optimization-based structure-preserving model order reduction
(MOR) methods for port-Hamiltonian (pH) descriptor systems of differentiation
index one. Descriptor systems in pH form permit energy-based modeling and
intuitive coupling of physical systems across different physical domains,
scales, and accuracies. This makes pH models well-suited building-blocks for
component-wise modeling of large system networks. In this context, it is often
necessary to preserve the pH structure during MOR. We discuss current
projection-based and structure-preserving MOR algorithms for pH systems and
present a new optimization-based framework for that task. The benefits of our
method include a simplified treatment of algebraic constraints and often a
higher accuracy of the resulting reduced-order model, which is demonstrated by
several numerical examples.
