Topology-Aware Distributed Optimization for Weakly Connected Digraphs

This article investigates distributed optimization over weakly connected digraphs (WCDGs), where unidirectional information flow structurally violates global optimality conditions and renders most existing optimization algorithms ineffective. To address this, we develop a topology-aware optimization framework that restores the feasibility of global optimality conditions through explicit topology compensation. Key contributions include: (i) Introduction of SCH partition for characterizing topology-induced gradient deviations in WCDGs, along with a quotient graph representation that preserves optimization equivalence while enabling topology analysis and controller design at a low-dimensional cell level. (ii) Design of an inverse edge controller on the quotient graph that restores a valid optimality structure via a minimal parallel compensation layer implemented as embedded control inputs, enabling direct implementation without native topology modifications and supporting rigorous analysis of both stability and convergence. (iii) Co-design of fully distributed PID optimization algorithms with adaptive coupling gains under this topological controller, guaranteeing exponential convergence for both second-order and heterogeneous systems over WCDGs. The presented results extend existing distributed optimization methods to WCDGs, relaxing the conventional strong connectivity assumptions.