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This paper is devoted to some approaches for convex min-min problems withsmoothness and strong convexity in only one of the two variable groups. It isshown that the proposed approaches, based on Vaidya's cutting plane method andNesterov's fast gradient method, achieve the linear convergence. The outerminimization problem is solved using Vaidya's cutting plane method, and theinner problem (smooth and strongly convex) is solved using the fast gradientmethod. Due to the importance of machine learning applications, we alsoconsider the case when the objective function is a sum of a large number offunctions. In this case, the variance-reduced accelerated gradient algorithm isused instead of Nesterov's fast gradient method. The numerical experiments'results illustrate the advantages of the proposed procedures for logisticregression with the prior on one of the parameter groups.

On solving convex min-min problems with smoothness and strong convexity in one variable group and small dimension of the other