  In this paper, we introduce some adaptive methods for solving variational
inequalities with relatively strongly monotone operators. Firstly, we focus on
the modification of the recently proposed, in smooth case [1], adaptive
numerical method for generalized smooth (with H\"older condition) saddle point
problem, which has convergence rate estimates similar to accelerated methods.
We provide the motivation for such an approach and obtain theoretical results
of the proposed method. Our second focus is the adaptation of widespread
recently proposed methods for solving variational inequalities with relatively
strongly monotone operators. The key idea in our approach is the refusal of the
well-known restart technique, which in some cases causes difficulties in
implementing such algorithms for applied problems. Nevertheless, our algorithms
show a comparable rate of convergence with respect to algorithms based on the
above-mentioned restart technique. Also, we present some numerical experiments,
which demonstrate the effectiveness of the proposed methods.
  [1] Jin, Y., Sidford, A., & Tian, K. (2022). Sharper rates for separable
minimax and finite sum optimization via primal-dual extragradient methods.
arXiv preprint arXiv:2202.04640.
