Geodesic
Numerical Methods for Some Classes of Variational Inequalities with Relatively Strongly Monotone Operators
Matematičeskie zametki, Tome 112 (2022) no. 6, pp. 879-894

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The paper deals with a significant extension of the recently proposed class of relatively strongly convex optimization problems in spaces of large dimension. In the present paper, we introduce an analog of the concept of relative strong convexity for variational inequalities (relative strong monotonicity) and study estimates for the rate of convergence of some numerical first-order methods for problems of this type. The paper discusses two classes of variational inequalities depending on the conditions related to the smoothness of the operator. The first of these classes of problems contains relatively bounded operators, and the second, operators with an analog of the Lipschitz condition (known as relative smoothness). For variational inequalities with relatively bounded and relatively strongly monotone operators, a version of the subgradient method is studied and an optimal estimate for the rate of convergence is justified. For problems with relatively smooth and relatively strongly monotone operators, we prove the linear rate of convergence of an algorithm with a special organization of the restart procedure of a mirror prox method for variational inequalities with monotone operators.
Keywords: variational inequality, relatively strongly convex function, strongly monotone operator, relatively bounded operator, relative smoothness, subgradient method, mirror prox method, adaptive method, restart procedure, saddle point problem. 
@article{MZM_2022_112_6_a7,
     author = {F. S. Stonyakin and A. A. Titov and D. V. Makarenko and M. S. Alkousa},
     title = {Numerical {Methods} for {Some} {Classes} of {Variational} {Inequalities} with {Relatively} {Strongly} {Monotone} {Operators}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {879--894},
     publisher = {mathdoc},
     volume = {112},
     number = {6},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_112_6_a7/}
}
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F. S. Stonyakin; A. A. Titov; D. V. Makarenko; M. S. Alkousa. Numerical Methods for Some Classes of Variational Inequalities with Relatively Strongly Monotone Operators. Matematičeskie zametki, Tome 112 (2022) no. 6, pp. 879-894. http://geodesic.mathdoc.fr/item/MZM_2022_112_6_a7/