  The problem of minimizing a separable convex function under linearly coupled
constraints arises from various application domains such as economic systems,
distributed control, and network flow. The main challenge for solving this
problem is that the size of data is very large, which makes usual
gradient-based methods infeasible. Recently, Necoara, Nesterov, and Glineur
[Journal of Optimization Theory and Applications, 173 (2017) 227-2254] proposed
an efficient randomized coordinate descent method to solve this type of
optimization problems and presented an appealing convergence analysis. In this
paper, we develop new techniques to analyze the convergence of such algorithms,
which are able to greatly improve the results presented there. This refined
result is achieved by extending Nesterov's second technique developed by
Nesterov [SIAM J. Optim. 22 (2012) 341-362] to the general optimization
problems with linearly coupled constraints. A novel technique in our analysis
is to establish the basis vectors for the subspace of the linearly constraints.
