  In this paper, we propose a distributed algorithm, called
Directed-Distributed Gradient Descent (D-DGD), to solve multi-agent
optimization problems over directed graphs. Existing algorithms mostly deal
with similar problems under the assumption of undirected networks, i.e.,
requiring the weight matrices to be doubly-stochastic. The row-stochasticity of
the weight matrix guarantees that all agents reach consensus, while the
column-stochasticity ensures that each agent's local gradient contributes
equally to the global objective. In a directed graph, however, it may not be
possible to construct a doubly-stochastic weight matrix in a distributed
manner. We overcome this difficulty by augmenting an additional variable for
each agent to record the change in the state evolution. In each iteration, the
algorithm simultaneously constructs a row-stochastic matrix and a
column-stochastic matrix instead of only a doubly-stochastic matrix. The
convergence of the new weight matrix, depending on the row-stochastic and
column-stochastic matrices, ensures agents to reach both consensus and
optimality. The analysis shows that the proposed algorithm converges at a rate
of $O(\frac{\ln k}{\sqrt{k}})$, where $k$ is the number of iterations.
