Two-Group Distributed Optimization Under Cooperative-Collaborative Networks With Linear Convergence

This manuscript considers distributed optimization problems in systems with cooperative-collaborative relationships, involving two groups of nodes, each with its own optimization problem, but with a coupled communication topology. For the signed graph representing the cooperation and collaboration between agents, this manuscript introduces DIG-JOR, a discrete-time distributed algorithm that consists of three key modules: an inexact consensus and gradient descent module, a group gradient-tracking module, and a dynamic Jacobi over-relaxation (JOR) inverse-tracking module. To support the convergence analysis of the distributed optimization algorithm, this manuscript proposes the Multi-Loop Small Gain Theorem. Under the assumption of strong convexity and with appropriately chosen step sizes, it is proved that the DIG-JOR algorithm converges to the optimal solutions of both groups at an R-linear rate. The theoretical results are validated through a simulation example.