On Decomposition and Convergence of Distributed Optimization Algorithms

This paper analyzes distributed optimization algorithms from a frequency-domain perspective. We propose a general class of gradient-based distributed algorithms that can be characterized as Lur'e systems, thereby enabling the analysis and synthesis of algorithms following a robust control approach facilitated by the Zames-Falb criterion. By identifying algorithmic convergence with the absolute stability of a corresponding Lur'e system and decomposing the optimization objective into two canonical control problems, namely tracking and servomechanism, the problem of optimizing convergence rate is recast as a Nevanlinna-Pick interpolation problem. The solutions to such analytic interpolation problems lead to a parameterization of distributed optimization algorithms that achieve specified convergence rates.