Globally-Constrained Decentralized Optimization with Variable Coupling

Abstract: 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.