  We propose a solution strategy for linear systems arising in interior method
optimization, which is suitable for implementation on hardware accelerators
such as graphical processing units (GPUs). The current gold standard for
solving these systems is the LDL^T factorization. However, LDL^T requires
pivoting during factorization, which substantially increases communication cost
and degrades performance on GPUs. Our novel approach solves a large indefinite
system by solving multiple smaller positive definite systems, using an
iterative solve for the Schur complement and an inner direct solve (via
Cholesky factorization) within each iteration. Cholesky is stable without
pivoting, thereby reducing communication and allowing reuse of the symbolic
factorization. We demonstrate the practicality of our approach and show that on
large systems it can efficiently utilize GPUs and outperform LDL^T
factorization of the full system.
