  In this note, we extend the algorithms Extra and subgradient-push to a new
algorithm ExtraPush for consensus optimization with convex differentiable
objective functions over a directed network. When the stationary distribution
of the network can be computed in advance}, we propose a simplified algorithm
called Normalized ExtraPush. Just like Extra, both ExtraPush and Normalized
ExtraPush can iterate with a fixed step size. But unlike Extra, they can take a
column-stochastic mixing matrix, which is not necessarily doubly stochastic.
Therefore, they remove the undirected-network restriction of Extra.
Subgradient-push, while also works for directed networks, is slower on the same
type of problem because it must use a sequence of diminishing step sizes.
  We present preliminary analysis for ExtraPush under a bounded sequence
assumption. For Normalized ExtraPush, we show that it naturally produces a
bounded, linearly convergent sequence provided that the objective function is
strongly convex.
  In our numerical experiments, ExtraPush and Normalized ExtraPush performed
similarly well. They are significantly faster than subgradient-push, even when
we hand-optimize the step sizes for the latter.
