Second-order conditions in C-1,C-1 constrained vector optimization

We consider the constrained vector optimization problem min(C) f(x), g(x) epsilon-K, where f : R-n -> R-m and g : R-n -> R-p are C-1,C-1 functions, and C subset of R-m and K subset of R-p are closed convex cones with nonempty interiors. Two type of solutions are important for our considerations, namely w-minimizers (weakly efficient points) and i-minimizers (isolated minimizers). We formulate and prove in terms of the Dini directional derivative second-order necessary conditions for a point x(0) to be a w-minimizer and second-order sufficient conditions for x(0) to be an i-minimizer of order two. We discuss the reversal of the sufficient conditions under suitable constraint qualifications of Kuhn-Tucker type.