  This paper is devoted to a new modification of a recently proposed adaptive
stochastic mirror descent algorithm for constrained convex optimization
problems in the case of several convex functional constraints. Algorithms,
standard and its proposed modification, are considered for the type of problems
with non-smooth Lipschitz-continuous convex objective function and convex
functional constraints. Both algorithms, with an accuracy $\varepsilon$ of the
approximate solution to the problem, are optimal in the terms of lower bounds
of estimates and have the complexity $O\left( \varepsilon^{-2} \right)$. In
both algorithms, the precise first-order information, which connected with
(sub)gradient of the objective function and functional constraints, is replaced
with its unbiased stochastic estimates. This means that in each iteration, we
can still use the value of the objective function and functional constraints at
the research point, but instead of their (sub)gradient, we calculate their
stochastic (sub)gradient. Due to the consideration of not all functional
constraints on non-productive steps, the proposed modification allows saving
the running time of the algorithm. Estimates for the rate of convergence of the
proposed modified algorithm is obtained. The results of numerical experiments
demonstrating the advantages and the efficient of the proposed modification for
some examples are also given.
