  In this paper, we consider two types of problems that have some similarity in
their structure, namely, min-min problems and min-max saddle-point problems.
Our approach is based on considering the outer minimization problem as a
minimization problem with inexact oracle. This inexact oracle is calculated via
inexact solution of the inner problem, which is either minimization or a
maximization problem. Our main assumptions are that the problem is smooth and
the available oracle is mixed: it is only possible to evaluate the gradient
w.r.t. the outer block of variables which corresponds to the outer minimization
problem, whereas for the inner problem only zeroth-order oracle is available.
To solve the inner problem we use accelerated gradient-free method with
zeroth-order oracle. To solve the outer problem we use either inexact variant
of Vaydya's cutting-plane method or a variant of accelerated gradient method.
As a result, we propose a framework that leads to non-asymptotic complexity
bounds for both min-min and min-max problems. Moreover, we estimate separately
the number of first- and zeroth-order oracle calls which are sufficient to
reach any desired accuracy.
