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In this paper, the optimal convergence rate $O\left(N^{-1/2}\right)$ (where$N$ is the total number of iterations performed by the algorithm), without thepresence of a logarithmic factor, is proved for mirror descent algorithms withspecial time-varying step sizes, for solving both classical constrainednon-smooth problems and problems with the composite model. The proven result isan improvement on the well-known rate $O\left(\log (N) N^{-1/2}\right)$ for themirror descent algorithms with the time-varying step sizes under consideration.It was studied a new weighting scheme assigns smaller weights to the initialpoints and larger weights to the most recent points. This scheme improves theconvergence rate of the considered mirror descent methods, which in theconducted numerical experiments outperform the other methods providing a bettersolution in all the considered test problems.

Optimal Convergence Rate for Mirror Descent Methods with special Time-Varying Step Sizes Rules