  This paper proposes a fast decentralized algorithm for solving a consensus
optimization problem defined in a directed networked multi-agent system, where
the local objective functions have the smooth+nonsmooth composite form, and are
possibly nonconvex. Examples of such problems include decentralized compressed
sensing and constrained quadratic programming problems, as well as many
decentralized regularization problems. We extend the existing algorithms
PG-EXTRA and ExtraPush to a new algorithm PG-ExtraPush for composite consensus
optimization over a directed network. This algorithm takes advantage of the
proximity operator like in PG-EXTRA to deal with the nonsmooth term, and
employs the push-sum protocol like in ExtraPush to tackle the bias introduced
by the directed network. With a proper step size, we show that PG-ExtraPush
converges to an optimal solution at a linear rate under some regular
assumptions. We conduct a series of numerical experiments to show the
effectiveness of the proposed algorithm. Specifically, with a proper step size,
PG-ExtraPush performs linear rates in most of cases, even in some nonconvex
cases, and is significantly faster than Subgradient-Push, even if the latter
uses a hand-optimized step size. The established theoretical results are also
verified by the numerical results.
