  This paper studies the distributed optimization problem when the objective
functions might be nondifferentiable and subject to heterogeneous set
constraints. Unlike existing subgradient methods, we focus on the case when the
exact subgradients of the local objective functions can not be accessed by the
agents. To solve this problem, we propose a projected primal-dual dynamics
using only the objective function's approximate subgradients. We first prove
that the formulated optimization problem can generally be solved with an error
depending upon the accuracy of the available subgradients. Then, we show the
exact solvability of this distributed optimization problem when the accumulated
approximation error of inexact subgradients is not too large. After that, we
also give a novel componentwise normalized variant to improve the transient
behavior of the convergent sequence. The effectiveness of our algorithms is
verified by a numerical example.
