  This paper considers nonconvex distributed constrained optimization over
networks, modeled as directed (possibly time-varying) graphs. We introduce the
first algorithmic framework for the minimization of the sum of a smooth
nonconvex (nonseparable) function--the agent's sum-utility--plus a
Difference-of-Convex (DC) function (with nonsmooth convex part). This general
formulation arises in many applications, from statistical machine learning to
engineering. The proposed distributed method combines successive convex
approximation techniques with a judiciously designed perturbed push-sum
consensus mechanism that aims to track locally the gradient of the (smooth part
of the) sum-utility. Sublinear convergence rate is proved when a fixed
step-size (possibly different among the agents) is employed whereas asymptotic
convergence to stationary solutions is proved using a diminishing step-size.
Numerical results show that our algorithms compare favorably with current
schemes on both convex and nonconvex problems.
