  In this paper, we propose Distributed Mirror Descent (DMD) algorithm for
constrained convex optimization problems on a (strongly-)connected multi-agent
network. We assume that each agent has a private objective function and a
constraint set. The proposed DMD algorithm employs a locally designed Bregman
distance function at each agent, and thus can be viewed as a generalization of
the well-known Distributed Projected Subgradient (DPS) methods, which use
identical Euclidean distances at the agents. At each iteration of the DMD, each
agent optimizes its own objective adjusted with the Bregman distance function
while exchanging state information with its neighbors. To further generalize
DMD, we consider the case where the agent communication follows a
\emph{directed} graph and it may not be possible to design doubly-stochastic
weight matrices. In other words, we restrict the corresponding weight matrices
to be row-stochastic instead of doubly-stochastic. We study the convergence of
DMD in two cases: (i) when the constraint sets at the agents are the same; and,
(ii) when the constraint sets at the agents are different. By partially
following the spirit of our proof, it can be shown that a class of
consensus-based distributed optimization algorithms, restricted to
doubly-stochastic matrices, remain convergent with stochastic matrices.
