  In this article we consider min-min type of problems or minimization by two
groups of variables. Min-min problems may occur in case if some groups of
variables in convex optimization have different dimensions or if these groups
have different domains. Such problem structure gives us an ability to split the
main task to subproblems, and allows to tackle it with mixed oracles. However
existing articles on this topic cover only zeroth and first order oracles, in
our work we consider high-order tensor methods to solve inner problem and fast
gradient method to solve outer problem. We assume, that outer problem is
constrained to some convex compact set, and for the inner problem we consider
both unconstrained case and being constrained to some convex compact set. By
definition, tensor methods use high-order derivatives, so the time per single
iteration of the method depends a lot on the dimensionality of the problem it
solves. Therefore, we suggest, that the dimension of the inner problem variable
is not greater than 1000. Additionally, we need some specific assumptions to be
able to use mixed oracles. Firstly, we assume, that the objective is convex in
both groups of variables and its gradient by both variables is Lipschitz
continuous. Secondly, we assume the inner problem is strongly convex and its
gradient is Lipschitz continuous. Also, since we are going to use tensor
methods for inner problem, we need it to be $p$-th order Lipschitz continuous
($p > 1$). Finally, we assume strong convexity of the outer problem to be able
to use fast gradient method for strongly convex functions. We need to
emphasize, that we use superfast tensor method to tackle inner subproblem in
unconstrained case. And when we solve inner problem on compact set, we use
accelerated high-order composite proximal method.
