Dini Set-Valued Directional Derivative in Locally Lipschitz Vector Optimization

The present paper studies the following constrained vector optimization problem: min C f ( x ), g ( x )∈− K , h ( x )=0, where f :ℝ n →ℝ m , g :ℝ n →ℝ p and h :ℝ n →ℝ q are locally Lipschitz functions and C ⊂ℝ m , K ⊂ℝ p are closed convex cones. In terms of the Dini set-valued directional derivative, first-order necessary and first-order sufficient conditions are obtained for a point x 0 to be a w -minimizer (weakly efficient point) or an i -minimizer (isolated minimizer of order 1). It is shown that, under natural assumptions (given by a nonsmooth variant of the implicit function theorem for the equality constraints), the obtained conditions improve some given by Clarke and Craven. Further comparison is done with some recent results of Khanh, Tuan and of Jiiménez, Novo.