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We develop a novel unified randomized block-coordinate primal-dual algorithmto solve a class of nonsmooth constrained convex optimization problems, whichcovers different existing variants and model settings from the literature. Weprove that our algorithm achieves optimal $\mathcal{O}(n/k)$ and$\mathcal{O}(n^2/k^2)$ convergence rates (up to a constant factor) in twocases: general convexity and strong convexity, respectively, where $k$ is theiteration counter and n is the number of block-coordinates. Our convergencerates are obtained through three criteria: primal objective residual and primalfeasibility violation, dual objective residual, and primal-dual expected gap.Moreover, our rates for the primal problem are on the last iterate sequence.Our dual convergence guarantee requires additionally a Lipschitz continuityassumption. We specify our algorithm to handle two important special cases,where our rates are still applied. Finally, we verify our algorithm on twowell-studied numerical examples and compare it with two existing methods. Ourresults show that the proposed method has encouraging performance on differentexperiments.

A New Randomized Primal-Dual Algorithm for Convex Optimization with Optimal Last Iterate Rates