  This paper aims to develop distributed algorithms for nonconvex optimization
problems with complicated constraints associated with a network. The network
can be a physical one, such as an electric power network, where the constraints
are nonlinear power flow equations, or an abstract one that represents
constraint couplings between decision variables of different agents. Despite
the recent development of distributed algorithms for nonconvex programs, highly
complicated constraints still pose a significant challenge in theory and
practice. We first identify some difficulties with the existing algorithms
based on the alternating direction method of multipliers (ADMM) for dealing
with such problems. We then propose a reformulation that enables us to design a
two-level algorithm, which embeds a specially structured three-block ADMM at
the inner level in an augmented Lagrangian method (ALM) framework. Furthermore,
we prove the global and local convergence as well as iteration complexity of
this new scheme for general nonconvex constrained programs, and show that our
analysis can be extended to handle more complicated multi-block inner-level
problems. Finally, we demonstrate with computation that the new scheme provides
convergent and parallelizable algorithms for various nonconvex applications,
and is able to complement the performance of the state-of-the-art distributed
algorithms in practice by achieving either faster convergence in optimality gap
or in feasibility or both.
