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

Linear Convergence of Randomized Primal-Dual Coordinate Method for Large-scale Linear Constrained Convex Programming