  This paper deals with an optimization problem over a network of agents, where
the cost function is the sum of the individual objectives of the agents and the
constraint set is the intersection of local constraints. Most existing methods
employing subgradient and consensus steps for solving this problem require the
weight matrix associated with the network to be column stochastic or even
doubly stochastic, conditions that can be hard to arrange in directed networks.
Moreover, known convergence analyses for distributed subgradient methods vary
depending on whether the problem is unconstrained or constrained, and whether
the local constraint sets are identical or nonidentical and compact. The main
goals of this paper are: (i) removing the common column stochasticity
requirement; (ii) relaxing the compactness assumption, and (iii) providing a
unified convergence analysis. Specifically, assuming the communication graph to
be fixed and strongly connected and the weight matrix to (only) be row
stochastic, a distributed projected subgradient algorithm and its variation are
presented to solve the problem for cost functions that are convex and Lipschitz
continuous. Based on a regularity assumption on the local constraint sets, a
unified convergence analysis is given that can be applied to both unconstrained
and constrained problems and without assuming compactness of the constraint
sets or an interior point in their intersection. Further, we also establish an
upper bound on the absolute objective error evaluated at each agent's available
local estimate under a nonincreasing step size sequence. This bound allows us
to analyze the convergence rate of both algorithms.
