  Many realistic decision-making problems in networked scenarios, such as
formation control and collaborative task offloading, often involve
complicatedly entangled local decisions, which, however, have not been
sufficiently investigated yet. Motivated by this, we study a class of
decentralized optimization problems with a variable coupling structure that is
new to the literature. Specifically, we consider a network of nodes
collaborating to minimize a global objective subject to a collection of global
inequality and equality constraints, which are formed by the local objective
and constraint functions of the nodes. On top of that, we allow such local
functions of each node to depend on not only its own decision variable but the
decisions of its neighbors as well. To address this problem, we propose a
decentralized projected primal-dual algorithm. It first incorporates a
virtual-queue technique with a primal-dual-primal scheme, and then linearizes
the non-separable objective and constraint functions to enable decentralized
implementation. Under mild conditions, we derive $O(1/k)$ convergence rates for
both objective error and constraint violations. Finally, two numerical
experiments corroborate our theoretical results and illustrate the competitive
performance of the proposed algorithm.
