  Distributed optimization has many applications, in communication networks,
sensor networks, signal processing, machine learning, and artificial
intelligence. Methods for distributed convex optimization are widely
investigated, while those for non-convex objectives are not well understood.
One of the first non-convex distributed optimization frameworks over an
arbitrary interaction graph was proposed by Di Lorenzo and Scutari [IEEE Trans.
on Signal and Information Processing over Network, 2 (2016), pp. 120-136],
which iteratively applies a combination of local optimization with convex
approximations and local averaging. We generalize the existing results in two
ways. In the case when the decision variables are separable such that there is
partial dependency in the objectives, we reduce the communication complexity of
the algorithm so that nodes only keep and communicate local variables instead
of the whole vector of variables. In addition, we relax the assumption that the
objectives' gradients are bounded and Lipschitz by means of successive proximal
approximations. Having developed the methodology, we then discuss many ways to
apply our algorithmic framework to resource allocation problems in
multi-cellular networks, where the two generalizations are found useful and
practical. Simulation results show the superiority of our resource allocation
algorithms over naive single cell methods, and furthermore, our approximation
framework lead to algorithms that are numerically more stable.
