A Decentralised Asynchronous Optimisation Algorithm with an Application to Phase Retrieval

This paper tackles the challenge of decentralised, nonconvex optimisation in situations where agents work asynchronously. Our main contribution is a new algorithm, partially asynchronous ADMM, designed to solve decentralised optimisation problems like phase retrieval. Importantly, it does not require a central coordinator and can work with arbitrary connected network setups. We also prove that our algorithm is equivalent to the randomised block coordinate Douglas-Rachford Splitting method. To illustrate the algorithm's effectiveness, we provide numerical results for the distributed phase retrieval problem, demonstrating its correctness and performance.