Distributed double proximal splitting algorithm for global constraint-coupled optimization with asynchrony and delays

This paper studies a distributed constraint-coupled optimization problem in networked systems, where local objectives comprise three cost functions, two of which exhibit nonsmooth characteristics. To overcome the intractability of the summation of these nonsmooth functions, we first derive a novel local first-order sufficient condition using Lagrange duality theory. Building this foundation, we present a synchronous full-distributed double proximal splitting algorithm, in which network agents maintain private variables and collaboratively reach solutions while satisfying globally coupled linear constraints through localized information exchange. Given the issues of asynchrony and delays are non-negligible, we additionally develop an asynchronous distributed algorithm where agents independently execute computations and communications using old information for different durations. Compared to conventional synchronous approaches, this asynchronous implementation mitigates idle time caused by delays or heterogeneous node speeds. Theoretically, the convergence of the synchronous algorithm is established under local Lipschitz continuity assumption and uncoordinated constant step-size criteria. For the asynchronous variant, we prove almost sure convergence in expectation under time-varying yet bounded delays. Extensive numerical simulations on signal processing applications corroborate the theoretical findings and demonstrate algorithmic efficacy.