  A class of distributed optimization problem with a globally coupled equality
constraint and local constrained sets is studied in this paper. For its special
case where local constrained sets are absent, an augmented primal-dual gradient
dynamics is proposed and analyzed, but it cannot be implemented distributedly
since the violation of the coupled constraint needs to be used. Benefiting from
the brand-new comprehending of a classical distributed unconstrained
optimization algorithm, the novel implicit tracking approach is proposed to
track the violation distributedly, which leads to the birth of the
\underline{i}mplicit tracking-based \underline{d}istribut\underline{e}d
\underline{a}ugmented primal-dual gradient dynamics (IDEA). A projected variant
of IDEA, i.e., Proj-IDEA, is further designed to deal with the general case
where local constrained sets exist. With the aid of the Lyapunov stability
theory, the convergences of IDEA and Pro-IDEA over undigraphs and digraphs are
analyzed respectively. As far as we know, Proj-IDEA is the first constant
step-size distributed algorithm which can solve the studied problem without the
need of the strict convexity of local cost functions. Besides, if local cost
functions are strongly convex and smooth, IDEA can achieve exponential
convergence with a weaker condition about the coupled constraint. Finally,
numerical experiments are taken to corroborate our theoretical results.
