Non-Ergodic Convergence Algorithms for Distributed Consensus and Coupling-Constrained Optimization

Abstract--We study distributed convex optimization with two ubiquitous forms of coupling: consensus constraints and gl obal affine equalities. Without smooth ness or strong convexity, we establish non-ergodic sublinear ra tes of order O (1/ k) for both the objective optimality and the consensus violation. Leveraging duality, we then show that the eco nomic dispatch problem admits a dual consensus formulation, and t hat applying the same algorithm to the dual economic dispatch yi elds non-ergodic O (1/ k) decay for the error of the summation of the cost over the network and the equality-constraint resid ual under convexity and Slater's condition. Numerical results on the IEEE 118-bus system demonstrate faster reduction of bot h objective error and feasibility error relative to the state -of-the-art baselines, while the dual variables reach network-wide con sensus. This paper studies large-scale convex optimization proble ms formulated over networks, which frequently arise in engineering applications.