A Coordinate Descent Primal-Dual Algorithm and Application to Distributed Asynchronous Optimization

Based on the idea of randomized coordinate descent of $\alpha$-averaged operators, a randomized primal-dual optimization algorithm is introduced, where a random subset of coordinates is updated at each iteration. The algorithm builds upon a variant of a recent (deterministic) algorithm proposed by V\~u and Condat that includes the well known ADMM as a particular case. The obtained algorithm is used to solve asynchronously a distributed optimization problem. A network of agents, each having a separate cost function containing a differentiable term, seek to find a consensus on the minimum of the aggregate objective. The method yields an algorithm where at each iteration, a random subset of agents wake up, update their local estimates, exchange some data with their neighbors, and go idle. Numerical results demonstrate the attractive performance of the method. The general approach can be naturally adapted to other situations where coordinate descent convex optimization algorithms are used with a random choice of the coordinates.