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In the first part of the paper we consider accelerated first orderoptimization method for convex functions with $L$-Lipschitz-continuousgradient, that is able to automatically adapts to problems which satisfiesPolyak-{\L}ojasiewicz condition or which is strongly convex with the value ofparameter equal to $\mu$. In these cases method poses linear convergence withfactor $\frac{\mu}{L}$, if $\mu$ is unknown. If the problem is strongly convexand $\mu$ is known, than the method possesses linear convergence with thefactor $\sqrt{\frac{\mu}{L}}$. If that are not the cases, the method convergeswith a rate $O(1/k^2)$. The second part contains generalization of the methodto the problems, that allows alternating minimization and proofs of the sameasymptotic convergence rates. Also it is considered the approach calledAdaptive 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 thediscrete entropically regularized optimal transport problem. The result of thework is the attempt to explain the reason why Alternating AGM converge fasterthan AGM or Adaptive Catalyst despite of the asymptotic theoretical rate$O(1/k^2)$. The hypothesis relies on the fact that Alternating AGM adapts tostrong convexity. The hypothesis was tested on quadratic functions, on thatAlternating AGM also showed faster convergence.

Accelerated Alternating Minimization and Adaptability to Strong Convexity