Randomized Coordinate Subgradient Method for Nonsmooth Optimization

Nonsmooth optimization finds wide applications in many engineering fields. In this work, we propose to utilize the {Randomized Coordinate Subgradient Method} (RCS) for solving both nonsmooth convex and nonsmooth nonconvex (nonsmooth weakly convex) optimization problems. At each iteration, RCS randomly selects one block coordinate rather than all the coordinates to update. Motivated by practical applications, we consider the {linearly bounded subgradients assumption} for the objective function, which is much more general than the Lipschitz continuity assumption. Under such a general assumption, we conduct thorough convergence analysis for RCS in both convex and nonconvex cases and establish both expected convergence rate and almost sure asymptotic convergence results. In order to derive these convergence results, we establish a convergence lemma and the relationship between the global metric subregularity properties of a weakly convex function and its Moreau envelope, which are fundamental and of independent interests. Finally, we conduct several experiments to show the possible superiority of RCS over the subgradient method.