Second-order conditions in C1,1 constrained vector optimization

Abstract

We consider the constrained vector optimization problem minC f(x), g(x) ∈ −K, where f:ℝn→ℝm and g:ℝn→ℝp are C1,1 functions, and C ** ℝm and K ** ℝp 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 x0 to be a w-minimizer and second-order sufficient conditions for x0 to be an i-minimizer of order two. We discuss the reversal of the sufficient conditions under suitable constraint qualifications of Kuhn-Tucker type. The obtained results improve the ones in Liu, Neittaanmaki, Křižek [21].