  Recently there were proposed some innovative convex optimization concepts,
namely, relative smoothness [1] and relative strong convexity [2,3]. These
approaches have significantly expanded the class of applicability of
gradient-type methods with optimal estimates of the convergence rate, which are
invariant regardless of the dimensionality of the problem. Later Yu. Nesterov
and H. Lu introduced some modifications of the Mirror Descent method for convex
minimization problems with the corresponding analogue of the Lipschitz
condition (so-called relative Lipschitz continuity). By introducing an
artificial inaccuracy to the optimization model, we propose adaptive methods
for minimizing a convex Lipschitz continuous function, as well as for the
corresponding class of variational inequalities. We also consider an adaptive
"universal" method, applicable to convex minimization problems both on the
class of relatively smooth and relatively Lipschitz continuous functionals with
optimal estimates of the convergence rate. The universality of the method makes
it possible to justify the applicability of the obtained theoretical results to
a wider class of convex optimization problems. We also present the results of
numerical experiments.
