  This paper explores two condensed-space interior-point methods to efficiently
solve large-scale nonlinear programs on graphics processing units (GPUs). The
interior-point method solves a sequence of symmetric indefinite linear systems,
or Karush-Kuhn-Tucker (KKT) systems, which become increasingly ill-conditioned
as we approach the solution. Solving a KKT system with traditional sparse
factorization methods involve numerical pivoting, making parallelization
difficult. A solution is to condense the KKT system into a symmetric
positive-definite matrix and solve it with a Cholesky factorization, stable
without pivoting. Although condensed KKT systems are more prone to
ill-conditioning than the original ones, they exhibit structured
ill-conditioning that mitigates the loss of accuracy. This paper compares the
benefits of two recent condensed-space interior-point methods, HyKKT and
LiftedKKT. We implement the two methods on GPUs using MadNLP.jl, an
optimization solver interfaced with the NVIDIA sparse linear solver cuDSS and
with the GPU-accelerated modeler ExaModels.jl. Our experiments on the PGLIB and
the COPS benchmarks reveal that GPUs can attain up to a tenfold speed increase
compared to CPUs when solving large-scale instances.
