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In this paper we propose a generalized condition for a sharp minimum,somewhat similar to the inexact oracle proposed recently byDevolder-Glineur-Nesterov. The proposed approach makes it possible to extendthe class of applicability of subgradient methods with the Polyak step-size, tothe situation of inexact information about the value of the minimum, as well asthe unknown Lipschitz constant of the objective function. Moreover, the use oflocal analogs of the global characteristics of the objective function makes itpossible to apply the results of this type to wider classes of problems. Weshow the possibility of applying the proposed approach to strongly convexnon-smooth problems, also, we make an experimental comparison with the knownoptimal subgradient method for such a class of problems. Moreover, there wereobtained some results connected to the applicability of the proposed techniqueto some types of problems with convexity relaxations: the recently proposednotion of weak $\beta$-quasi-convexity and ordinary quasi-convexity. Also inthe paper, we study a generalization of the described technique to thesituation with the assumption that the $\delta$-subgradient of the objectivefunction is available instead of the usual subgradient. For one of theconsidered methods, conditions are found under which, in practice, it ispossible to escape the projection of the considered iterative sequence onto thefeasible set of the problem.

Subgradient methods for non-smooth optimization problems with some relaxation of sharp minimum