  In this work we consider a possibility to use the conception of $(\delta,
L)$-model of a function for optimization tasks, whereby solving a primal
problem there is a necessity to recover a solution of a dual problem. The
conception of $(\delta, L)$-model is based on the conception of $(\delta,
L)$-oracle which was proposed by Devolder-Glineur-Nesterov, herewith the
authors proposed approximate a function with an upper bound using a convex
quadratic function with some additive noise $\delta$. They managed to get
convex quadratic upper bounds with noise even for nonsmooth functions. The
conception of $(\delta, L)$-model continues this idea by using instead of a
convex quadratic function a more complex convex function in an upper bound.
Possibility to recover the solution of a dual problem gives great benefits in
different problems, for instance, in some cases, it is faster to find a
solution in a primal problem than in a dual problem. Note that primal-dual
methods are well studied, but usually each class of optimization problems has
its own primal-dual method. Our goal is to develop a method which can find
solutions in different classes of optimization problems. This is realized
through the use of the conception of $(\delta, L)$-model and adaptive structure
of our methods. Thereby, we developed primal-dual adaptive gradient method and
fast gradient method with $(\delta, L)$-model and proved convergence rates of
the methods, moreover, for some classes of optimization problems the rates are
optimal. The main idea is the following: we find a dual solution to an
approximation of a primal problem using the conception of $(\delta, L)$-model.
It is much easier to find a solution to an approximated problem, however, we
have to do it in each step of our method, thereby the principle of "divide and
conquer" is realized.
