  We study nonconvex distributed optimization in multi-agent networks with
time-varying (nonsymmetric) connectivity. We introduce the first algorithmic
framework for the distributed minimization of the sum of a smooth (possibly
nonconvex and nonseparable) function - the agents' sum-utility - plus a convex
(possibly nonsmooth and nonseparable) regularizer. The latter is usually
employed to enforce some structure in the solution, typically sparsity. The
proposed method hinges on successive convex approximation techniques while
leveraging dynamic consensus as a mechanism to distribute the computation among
the agents: each agent first solves (possibly inexactly) a local convex
approximation of the nonconvex original problem, and then performs local
averaging operations. Asymptotic convergence to (stationary) solutions of the
nonconvex problem is established. Our algorithmic framework is then customized
to a variety of convex and nonconvex problems in several fields, including
signal processing, communications, networking, and machine learning. Numerical
results show that the new method compares favorably to existing distributed
algorithms on both convex and nonconvex problems.
