We consider stochastic convex optimization problems with affine constraints and develop several methods using the either primal or dual approach to solve it. In the primal case, we use special penalization technique to make the initial problem more convenient for using optimization methods. We propose algorithms to solve it based on Similar Triangles Method Gasnikov and Nesterov (2018); Nesterov (2018) with Inexact Proximal Step for the convex smooth and strongly convex smooth objective functions and methods based on Gradient Sliding algorithm Lan (2012) to solve the same problems in the non-smooth case. We prove the convergence guarantees in the smooth convex case with deterministic first-order oracle.We propose and analyze three novel methods to handle stochastic convex optimization problems with affine constraints: SPDSTM, SSTM sc and R-RRMA-AC-SA 2 . We develop convergence analysis for these methods for the unbiased (for R-RRMA-AC-SA 2 ) and biased (for SPDSTM and SSTM sc) stochastic oracles.Finally, we apply all aforementioned results and approaches to solve the decentralized distributed optimization problem and discuss the optimality of the obtained results in terms of communication rounds and the number of oracle calls per node.

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