A Two‐Average‐Tracking Scheme in Distributed Optimization With a Small‐Gain Convergence Analysis

ABSTRACT

This paper studies distributed convex optimization for multi‐agent systems over undirected connected graphs and proposes a novel distributed algorithm based on a two‐average‐tracking mechanism. Unlike existing methods that rely solely on gradient tracking, the proposed design decentralizes two intrinsic averaging operations in the centralized dynamics by introducing separate trackers for gradient information and agents' states. This two‐average‐tracking architecture opens up a novel algorithmic avenue within the distributed optimization design paradigm. Beyond algorithmic innovations, this paper makes a theoretical contribution to convergence analysis by developing a novel small‐gain theorem‐based framework, in contrast to the conventional Lyapunov‐function approach. Unlike existing small‐gain analysis that is largely restricted to discrete‐time systems or depends on elaborate inequality manipulations, the proposed framework is tailored to continuous‐time distributed optimization and features a simplified gain verification procedure: finite gain of linear static subsystems is certified via the Kalman–Yakubovich–Popov (KYP) lemma, while that of nonlinear dynamic subsystems is ensured through the Lipschitz continuity of the gradient mapping. Furthermore, the small‐gain theorem is extended from the classical two‐subsystem setting to a three‐subsystem decomposition, enabling more targeted gain analysis and improving applicability to high‐dimensional and strongly coupled systems.