  This work focuses on a class of general decentralized constraint-coupled
optimization problems. We propose a novel nested primal-dual gradient algorithm
(NPGA), which can achieve linear convergence under the weakest known condition,
and its theoretical convergence rate surpasses all known results. More
importantly, NPGA serves not only as an algorithm but also as a unified
algorithmic framework, encompassing various existing algorithms as special
cases. By designing different network matrices, we can derive numerous versions
of NPGA and analyze their convergences by leveraging the convergence results of
NPGA conveniently, thereby enabling the design of more efficient algorithms.
Finally, we conduct numerical experiments to compare the convergence rates of
NPGA and existing algorithms, providing empirical evidence for the superior
performance of NPGA.
