Algorithms of Inertial Mirror Descent in Stochastic Convex Optimization Problems.

The goal is to modify the known method of mirror descent (MD) in convex optimization, which having been proposed by Nemirovsky and Yudin in 1979 and generalized the standard gradient method. To start, the paper shows the idea of a new, so-called inertial MD method with the example of a deterministic optimization problem in continuous time. In particular, in the Euclidean case, the heavy ball method by Polyak is realized. It is noted that the new method does not use additional averaging of the points. Then, a discrete algorithm of inertial MD is described and the upper bound on error in objective function is proved. Finally, inertial MD randomized algorithm for finding a principal eigenvector of a given stochastic matrix (i.e., for solving a well known PageRank problem) is treated. Particular numerical example illustrates the general decrease of the error in time and corroborates theoretical results.