  This paper considers a nested stochastic distributed optimization problem. In
it, approximate solutions to realizations of the inner-problem are leveraged to
obtain a Distributed Stochastic Cubic Regularized Newton (DiSCRN) update to the
decision variable of the outer problem. We provide an example involving
electric vehicle users with various preferences which demonstrates that this
model is appropriate and sufficiently complex for a variety of data-driven
multi-agent settings, in contrast to non-nested models. The main two
contributions of the paper are: (i) development of local stopping criterion for
solving the inner optimization problem which guarantees sufficient accuracy for
the outer-problem update, and (ii) development of the novel DiSCRN algorithm
for solving the outer-problem and a theoretical justification of its efficacy.
Simulations demonstrate that this approach is more stable and converges faster
than standard gradient and Newton outer-problem updates in a highly nonconvex
scenario, and we also demonstrate that the method extends to an EV charging
scenario in which resistive battery losses and a time-of-use pricing model are
considered over a time horizon.
