  In this paper, we consider the unconstrained distributed optimization
problem, in which the exchange of information in the network is captured by a
directed graph topology, thus, nodes can only communicate with their neighbors.
Additionally, in our problem, the communication channels among the nodes have
limited bandwidth. In order to alleviate this limitation, quantized messages
should be exchanged among the nodes. For solving this distributed optimization
problem, we combine a gradient descent method with a distributed quantized
consensus algorithm (which requires the nodes to exchange quantized messages
and converges in a finite number of steps). Specifically, at every optimization
step, each node (i) performs a gradient descent step (i.e., subtracts the
scaled gradient from its current estimate), and (ii) performs a finite-time
calculation of the quantized average of every node's estimate in the network.
As a consequence, this algorithm approximately mimics the centralized gradient
descent algorithm. We show that our algorithm asymptotically converges to a
neighborhood of the optimal solution with linear convergence rate. The
performance of the proposed algorithm is demonstrated via simple illustrative
examples.
