  We are concerned with the convergence of NEAR-DGD$^+$ (Nested Exact
Alternating Recursion Distributed Gradient Descent) method introduced to solve
the distributed optimization problems. Under the assumption of the strong
convexity of local objective functions and the Lipschitz continuity of their
gradients, the linear convergence is established in \cite{BBKW - Near DGD}. In
this paper, we investigate the convergence property of NEAR-DGD$^+$ in the
absence of strong convexity. More precisely, we establish the convergence
results in the following two cases: (1) When only the convexity is assumed on
the objective function. (2) When the objective function is represented as a
composite function of a strongly convex function and a rank deficient matrix,
which falls into the class of convex and quasi-strongly convex functions.
Numerical results are provided to support the convergence results.
