Nonconvex Distributed Composite Optimization With Coupled Inequality Constraints

This article focuses on nonconvex distributed composite optimization over time-varying multiagent networks, where each agent possesses a local objective function, composed of a nonconvex and smooth function plus a nonsmooth function. The network aims to minimize the sum of all local functions subject to local set constraints and global nonconvex coupled inequality constraints. The inherent nonconvex and nonlinear characteristics of the objective and constraint functions pose formidable challenges in developing efficient distributed algorithms with convergence guarantees. To tackle this intricate problem, a novel distributed linearized augmented primal-dual algorithm is designed by incorporating distributed tracking and dynamic consensus techniques. It is theoretically shown that, with appropriately chosen parameters, the proposed algorithm can find an $\epsilon $ -Karush-Kuhn-Tucker (KKT) point. Specifically, the sequences of average optimality, constraint violation, and complementary slackness measure converge to zero at sublinear rates. Finally, a numerical application is presented to validate the effectiveness of the proposed algorithm.