This article is cited in 2 scientific papers (total in 2 papers)
Primal-dual mirror descent method for constraint stochastic optimization problems
A. S. Bayandinaa, A. V. Gasnikovab, E. V. Gasnikovaa, S. V. Matsievskiic
a Moscow Institute of Physics and Technology, Dolgoprudnyi, Russia
b Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia
c Kant Baltic Federal University, Kaliningrad, Russia
Extension of the mirror descent method developed for convex stochastic optimization problems to constrained convex stochastic optimization problems (subject to functional inequality constraints) is studied. A method that performs an ordinary mirror descent step if the constraints are insignificantly violated and performs a mirror descent step with respect to the violated constraint if this constraint is significantly violated is proposed. If the method parameters are chosen appropriately, a bound on the convergence rate (that is optimal for the given class of problems) is obtained and sharp bounds on the probability of large deviations are proved. For the deterministic case, the primal-dual property of the proposed method is proved. In other words, it is proved that, given the sequence of points (vectors) generated by the method, the solution of the dual method can be reconstructed up to the same accuracy with which the primal problem is solved. The efficiency of the method as applied for problems subject to a huge number of constraints is discussed. Note that the bound on the duality gap obtained in this paper does not include the unknown size of the solution to the dual problem.
Mirror descent method, convex stochastic optimization, constrained optimization, probability of large deviations, randomization.
Computational Mathematics and Mathematical Physics, 2018, 58:11, 1728–1736
A. S. Bayandina, A. V. Gasnikov, E. V. Gasnikova, S. V. Matsievskii, “Primal-dual mirror descent method for constraint stochastic optimization problems”, Zh. Vychisl. Mat. Mat. Fiz., 58:11 (2018), 1794–1803; Comput. Math. Math. Phys., 58:11 (2018), 1728–1736
Citation in format AMSBIB
\by A.~S.~Bayandina, A.~V.~Gasnikov, E.~V.~Gasnikova, S.~V.~Matsievskii
\paper Primal-dual mirror descent method for constraint stochastic optimization problems
\jour Zh. Vychisl. Mat. Mat. Fiz.
\jour Comput. Math. Math. Phys.
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This publication is cited in the following articles:
Bayandina A., Dvurechensky P., Gasnikov A., Stonyakin F., Titov A., “Mirror Descent and Convex Optimization Problems With Non-Smooth Inequality Constraints”, Large-Scale and Distributed Optimization, Lect. Notes Math., Lecture Notes in Mathematics, 2227, eds. Giselsson P., Rantzer A., Springer International Publishing Ag, 2018, 181–213
F. S. Stonyakin, M. Alkousa, A. N. Stepanov, A. A. Titov, “Adaptive mirror descent algorithms for convex and strongly convex optimization problems with functional constraints”, J. Appl. Industr. Math., 13:3 (2019), 557–574
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