  In this paper, we consider two variants of the concept of sharp minimum for
mathematical programming problems with quasiconvex objective function and
inequality constraints. It investigated the problem of describing a variant of
a simple subgradient method with switching along productive and non-productive
steps, for which, on a class of problems with Lipschitz functions, it would be
possible to guarantee convergence with the rate of geometric progression to the
set of exact solutions or its vicinity. It is important that to implement the
proposed method there is no need to know the sharp minimum parameter, which is
usually difficult to estimate in practice. To overcome this problem, the
authors propose to use a step djustment procedure similar to that previously
proposed by B.~T.~Polyak.
