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Recent several years have witnessed the surge of asynchronous (async-)parallel computing methods due to the extremely big data involved in manymodern applications and also the advancement of multi-core machines andcomputer clusters. In optimization, most works about async-parallel methods areon unconstrained problems or those with block separable constraints.  In this paper, we propose an async-parallel method based on block coordinateupdate (BCU) for solving convex problems with nonseparable linear constraint.Running on a single node, the method becomes a novel randomized primal-dual BCUwith adaptive stepsize for multi-block affinely constrained problems. For theseproblems, Gauss-Seidel cyclic primal-dual BCU needs strong convexity to haveconvergence. On the contrary, merely assuming convexity, we show that theobjective value sequence generated by the proposed algorithm converges inprobability to the optimal value and also the constraint residual to zero. Inaddition, we establish an ergodic $O(1/k)$ convergence result, where $k$ is thenumber of iterations. Numerical experiments are performed to demonstrate theefficiency of the proposed method and significantly better speed-up performancethan its sync-parallel counterpart.

Asynchronous parallel primal-dual block coordinate update methods for affinely constrained convex programs