  In this paper we propose a generalized condition for a sharp minimum,
somewhat similar to the inexact oracle proposed recently by
Devolder-Glineur-Nesterov. The proposed approach makes it possible to extend
the class of applicability of subgradient methods with the Polyak step-size, to
the situation of inexact information about the value of the minimum, as well as
the unknown Lipschitz constant of the objective function. Moreover, the use of
local analogs of the global characteristics of the objective function makes it
possible to apply the results of this type to wider classes of problems. We
show the possibility of applying the proposed approach to strongly convex
non-smooth problems, also, we make an experimental comparison with the known
optimal subgradient method for such a class of problems. Moreover, there were
obtained some results connected to the applicability of the proposed technique
to some types of problems with convexity relaxations: the recently proposed
notion of weak $\beta$-quasi-convexity and ordinary quasi-convexity. Also in
the paper, we study a generalization of the described technique to the
situation with the assumption that the $\delta$-subgradient of the objective
function is available instead of the usual subgradient. For one of the
considered methods, conditions are found under which, in practice, it is
possible to escape the projection of the considered iterative sequence onto the
feasible set of the problem.
