# On Modification of an Adaptive Stochastic Mirror Descent Algorithm for Convex Optimization Problems with Functional Constraints

## TL;DR

This paper introduces a modified adaptive stochastic mirror descent algorithm for constrained convex optimization, improving efficiency by selectively considering constraints and providing convergence analysis and numerical validation.

## Contribution

It proposes a new modification to existing algorithms that reduces computational time by not evaluating all constraints at every step, while maintaining optimal convergence rates.

## Key findings

- The modified algorithm achieves the same optimal complexity of O(ε^{-2})
- Numerical experiments demonstrate improved efficiency over standard methods
- The approach effectively handles multiple convex functional constraints in stochastic settings

## Abstract

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.