  Linear-scaling techniques for Kohn-Sham density functional theory (KS-DFT)
are essential to describe the ground state properties of extended systems.
Still, these techniques often rely on the locality of the density matrix or on
accurate embedding approaches, limiting their applicability. In contrast,
stochastic density functional theory (sDFT) achieves linear- and
sub-linear-scaling by statistically sampling the ground state density without
relying on embedding or imposing localization. In return, ground state
observables, such as the forces on the nuclei, fluctuate in sDFT, making the
optimization of the nuclear structure a highly non-trivial problem. In this
work, we combine the most recent noise-reduction schemes for sDFT with
stochastic optimization algorithms to perform structure optimization within
sDFT. We compare the performance of the stochastic gradient descent (sGD)
approach and its variations (stochastic gradient descent with momentum (sGDM))
to stochastic optimization techniques that rely on the Hessian, such as the
stochastic Broyden-Fletcher-Goldfarb-Shanno (sBFGS) algorithm. We further
provide a detailed assessment of the computational efficiency and its
dependence on the optimization parameters for each methods for determining the
ground state structure of bulk silicon with varying supercell dimensions.
