Accelerated primal–dual proximal block coordinate updating methods for constrained convex optimization

Block coordinate update (BCU) methods enjoy low per-update computational complexity because every time only one or a few block variables would need to be updated among possibly a large number of blocks. They are also easily parallelized and thus have been particularly popular for solving problems involving large-scale dataset and/or variables. In this paper, we propose a primal–dual BCU method for solving linearly constrained convex program with multi-block variables. The method is an accelerated version of a primal–dual algorithm proposed by the authors, which applies randomization in selecting block variables to update and establishes an O(1 / t) convergence rate under convexity assumption. We show that the rate can be accelerated to O(1 / t²) if the objective is strongly convex. In addition, if one block variable is independent of the others in the objective, we then show that the algorithm can be modified to achieve a linear rate of convergence. The numerical experiments show that the accelerated method performs stably with a single set of parameters while the original method needs to tune the parameters for different datasets in order to achieve a comparable level of performance.