Dini set-valued directional derivative in locally Lipschitz vector optimization (Q1039353)

The authors deal with the local solutions of the constrained vector optimization problem: \(\min_C f(x)\), \(g(x)\in -K\), \(h(x)=0\), where \(f:\mathbb{R}^n\rightarrow \mathbb{R}^m\), \(g:\mathbb{R}^n\rightarrow \mathbb{R}^p\) and \(h:\mathbb{R}^n\rightarrow \mathbb{R}^q\) are locally Lipschitz functions and \(C\subset \mathbb{R}^m\) and \(K\subset \mathbb{R}^p\) 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 \(x^o\) to be a weakly efficient point or an isolated minimizer of order \(1\). It is shown that, under natural assumptions (given by a nonsmooth variant of the implicit function theorem for equality constraint), the obtained conditions improve those given by Clarke and Craven. Further comparison is done with some recent results of Khan, Tuan and Jiimenez, Novo.