  In this paper, we revisit a well-known distributed projected subgradient
algorithm which aims to minimize a sum of cost functions with a common set
constraint. In contrast to most of existing results, weight matrices of the
time-varying multi-agent network are assumed to be more general, i.e., they are
only required to be row stochastic instead of doubly stochastic. We focus on
analyzing convergence properties of this algorithm under general graphs. We
first show that there generally exists a graph sequence such that the algorithm
is not convergent when the network switches freely within finitely many general
graphs. Then to guarantee the convergence of this algorithm under any uniformly
jointly strongly connected general graph sequence, we provide a necessary and
sufficient condition, i.e., the intersection of optimal solution sets to all
local optimization problems is not empty. Furthermore, we surprisingly find
that the algorithm is convergent for any periodically switching general graph
sequence, and the converged solution minimizes a weighted sum of local cost
functions, where the weights depend on the Perron vectors of some product
matrices of the underlying periodically switching graphs. Finally, we consider
a slightly broader class of quasi-periodically switching graph sequences, and
show that the algorithm is convergent for any quasi-periodic graph sequence if
and only if the network switches between only two graphs.
