Second- and First-Order Optimality Conditions in Vector Optimization
- Vsevolod I. Ivanov
- International Journal of Information Technology…
In this paper we obtain second-order optimality conditions of Fritz John and Karush–Kuhn–Tucker types for the problem with inequality constraints in nonsmooth settings using a new second-order directional derivative of Hadamard type. We derive necessary and sufficient conditions for a point x̄ to be a local minimizer and an isolated local one of order two. In the primal necessary conditions we suppose that all functions are locally Lipschitz, but in all other conditions the data are locally Lipschitz, regular in the sense of Clarke, Gâteaux differentiable at x̄ , and the constraint functions are second-order Hadamard differentiable at x̄ in every direction. It is shown by an example that regularity and Gâteaux differentiability cannot be removed from the sufficient conditions.