Consider the setting where each vertex of a graph has a function, and communications can only occur between vertices connected by an edge. We wish to minimize the sum of these functions. For the case when each function is the sum of a strongly convex quadratic and a convex function, we propose a distributed version of Dykstra's algorithm. The computations to optimize the dual objective function can run asynchronously without a global clock, and in a distributed manner without a central controller. Convergence to the primal minimizer is deterministic instead of being probabilistic, and is guaranteed as long as in each cycle, the edges where two-way communications occur connects all vertices. We also look at an accelerated algorithm, and an algorithm for the case when the functions on the nodes are not strongly convex.

27 pages. Accepted in SIAM J. Optim. . Added Remark 2.6. Added references on the dual ascent perspective, and made other small changes. We thank the anonymous referees, associate editor and the journal staff for the quick processing of my submission