  We introduce a notion of inexact model of a convex objective function, which
allows for errors both in the function and in its gradient. For this situation,
a gradient method with an adaptive adjustment of some parameters of the model
is proposed and an estimate for the convergence rate is found. This estimate is
optimal on a class of sufficiently smooth problems in the presence of errors.
We consider a special class of convex nonsmooth optimization problems. In order
to apply the proposed technique to this class, an artificial error should be
introduced. We show that the method can be modified for such problems to
guarantee a convergence in the function with a nearly optimal rate on the class
of convex nonsmooth optimization problems. An adaptive gradient method is
proposed for objective functions with some relaxation of the Lipschitz
condition for the gradient that satisfy the Polyak--Lojasievicz gradient
dominance condition. Here, the objective function and its gradient can be given
inexactly. The adaptive choice of the parameters is performed during the
operation of the method with respect to both the Lipschitz constant of the
gradient and a value corresponding to the error of the gradient and the
objective function. The linear convergence of the method is justified up to a
value associated with the errors.
