  This paper studies the distributed optimization problem where 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 only be solved with an approximate
error depending upon the accuracy of the available subgradients. Then, we show
the exact solvability of this optimization problem if the accumulated
approximation error 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.
