Second- and First-Order Optimality Conditions in Vector Optimization

In this paper, we obtain second- and first-order optimality conditions of Kuhn–Tucker type and Fritz John one for weak efficiency in the vector problem with inequality constraints. In the necessary conditions, we suppose that the objective function and the active constraints are continuously differentiable. We introduce notions of KTSP-invex problem and second-order KTSP-invex one. We obtain that the vector problem is (second-order) KTSP-invex if and only if for every triple $(\bar{x},\bar{\lambda},\bar{\mu})$ with Lagrange multipliers $\bar{\lambda}$ and $\bar{\mu}$ for the objective function and constraints, respectively, which satisfies the (second-order) necessary optimality conditions, the pair $(\bar{x},\bar{\mu})$ is a saddle point of the scalar Lagrange function with a fixed multiplier $\bar{\lambda}$. We introduce notions second-order KT-pseudoinvex-I, second-order KT-pseudoinvex-II, second-order KT-invex problems. We prove that every second-order Kuhn–Tucker stationary point is a weak global Pareto minimizer (global Pareto minimizer) if and only if the problem is second-order KT-pseudoinvex-I (KT-pseudoinvex-II). It is derived that every second-order Kuhn–Tucker stationary point is a global solution of the weighting problem if and only if the vector problem is second-order KT-invex.