  This paper is devoted to some approaches for convex min-min problems with
smoothness and strong convexity in only one of the two variable groups. It is
shown that the proposed approaches, based on Vaidya's cutting plane method and
Nesterov's fast gradient method, achieve the linear convergence. The outer
minimization problem is solved using Vaidya's cutting plane method, and the
inner problem (smooth and strongly convex) is solved using the fast gradient
method. Due to the importance of machine learning applications, we also
consider the case when the objective function is a sum of a large number of
functions. In this case, the variance-reduced accelerated gradient algorithm is
used instead of Nesterov's fast gradient method. The numerical experiments'
results illustrate the advantages of the proposed procedures for logistic
regression with the prior on one of the parameter groups.
