  Recently some specific classes of non-smooth and non-Lipschitz convex
optimization problems were selected by Yu.~Nesterov along with H.~Lu. We
consider convex programming problems with similar smoothness conditions for the
objective function and functional constraints. We introduce a new concept of an
inexact model and propose some analogues of switching subgradient schemes for
convex programming problems for the relatively Lipschitz-continuous objective
function and functional constraints. Some class of online convex optimization
problems is considered. The proposed methods are optimal in the class of
optimization problems with relatively Lipschitz-continuous objective and
functional constraints.
