The present paper deals with sufficient conditions for a system of convex incqualities to be a local approximation of a given arbitrary system in the following sense: the solution set of the first system is tangential to the solution set of the second one at the point under consideration. a criterion is proposed. For the case, in which the given system is of then form x ∈ D, H(x) ∈ N, where D is a subset of a certain linear topological space X, H a mapping from D into R^k, N a closed convex cone in R_k. Form this eriterion an extremum principle is deduced, which contains as particular cases certain well-known results such as the clasical KUHN-TUCKER theorem, the Halkin's theorein [2], the Mangasarian-Fromowitz's theorem [3]. Some other results (containing a well-known theorem of Ljusternik [7]) are obtained in the case where H is a mapping from D into a Banach space Y.