  In the first part of the paper we consider accelerated first order
optimization method for convex functions with $L$-Lipschitz-continuous
gradient, that is able to automatically adapts to problems which satisfies
Polyak-{\L}ojasiewicz condition or which is strongly convex with the value of
parameter equal to $\mu$. In these cases method poses linear convergence with
factor $\frac{\mu}{L}$, if $\mu$ is unknown. If the problem is strongly convex
and $\mu$ is known, than the method possesses linear convergence with the
factor $\sqrt{\frac{\mu}{L}}$. If that are not the cases, the method converges
with a rate $O(1/k^2)$. The second part contains generalization of the method
to the problems, that allows alternating minimization and proofs of the same
asymptotic convergence rates. Also it is considered the approach called
Adaptive Catalyst, which allows to increase convergence rate up to $O(1/k^2)$
and also it is provided an experimental comparison of the approach with AGM,
Alternating AGM, APDAGD and Sinkhorn's algorithm for the dual problem to the
discrete entropically regularized optimal transport problem. The result of the
work is the attempt to explain the reason why Alternating AGM converge faster
than AGM or Adaptive Catalyst despite of the asymptotic theoretical rate
$O(1/k^2)$. The hypothesis relies on the fact that Alternating AGM adapts to
strong convexity. The hypothesis was tested on quadratic functions, on that
Alternating AGM also showed faster convergence.
