# BILEVEL PROBLEMS AND TWO-STAGE PROXIMAL ALGORITHM

### Abstract

In this paper, we consider bilevel problem: variational inequality problem over the set of solutions the equilibrium problems. To solve this problem, an iterative algorithm is proposed that combines the ideas of a two-stage proximal method and iterative regularization. In addition, an adaptive version of the algorithm with a rule for updating parameters without using the values of the Lipschitz constants of the bifunction was studied. For monotone bifunctions of Lipschitz type and strongly monotone Lipschitz continuous operators, the theorem on strong convergence of sequences generated by the algorithms is proved.

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