Necessary extremum conditions without a priori normality assumptions

In a linear space X, the following extremal problem is considered: f(x)tomin,; F_1(x)le 0,; F_2(x)=0,; xin C, where C is a closed set, the mappings F_1, F_2 have finite-dimensional images, and f, F_1, F_2 are twice smooth in any finite-dimensional subspace of X (i.e., with respect to the finite topology). No regularity assumptions are imposed on these mappings. The set C is approximated by using the cone of Mordukhovich. Second order necessary conditions for the local minimality in the finite topology are obtained.