Open AccessTWO-LEVEL PROBLEMS AND TWO-STAGE PROXIMAL ALGORITHM
Author(s) -
V. V. Semenov,
AUTHOR_ID,
Ya. I. Vedel,
С. В. Денисов,
AUTHOR_ID,
AUTHOR_ID
Publication year - 2021
Publication title -
žurnal občislûvalʹnoï ta prikladnoï matematiki
Language(s) - English
Resource type - Journals
eISSN - 2706-9699
pISSN - 2706-9680
DOI - 10.17721/2706-9699.2021.2.07
Subject(s) - variational inequality , lipschitz continuity , monotone polygon , mathematics , convergence (economics) , hilbert space , algorithm , strongly monotone , regularization (linguistics) , mathematical optimization , computer science , pure mathematics , geometry , artificial intelligence , economics , economic growth
In this paper, a two-level problem is considered: a variational inequality on the set of solutions to the equilibrium problem. An example of such a problem is the search for the normal Nash equilibrium. To solve this problem, two algorithms are proposed. The first combines the ideas of a two-step proximal method and iterative regularization. And the second algorithm is an adaptive version of the first with a parameter update rule that does not use the values of the Lipschitz constants of the bifunction. Theorems on strong convergence of algorithms are proved for monotone bifunctions of Lipschitz type and strongly monotone Lipschitz operators. It is shown that the proposed algorithms can be applied to monotone two-level variational inequalities in Hilbert spaces.