  Linear constrained convex programming has many practical applications,
including support vector machine and machine learning portfolio problems. We
propose the randomized primal-dual coordinate (RPDC) method, a randomized
coordinate extension of the first-order primal-dual method by Cohen and Zhu,
1984 and Zhao and Zhu, 2019, to solve linear constrained convex programming. We
randomly choose a block of variables based on a uniform distribution,
linearize, and apply a Bregman-like function (core function) to the selected
block to obtain simple parallel primal-dual decomposition. We then establish
almost surely convergence and expected O(1/t) convergence rate, and expected
linear convergence under global strong metric subregularity. Finally, we
discuss implementation details for the randomized primal-dual coordinate
approach and present numerical experiments on support vector machine and
machine learning portfolio problems to verify the linear convergence.
