Optimality conditions for vector optimization problems (Q1035917)

The paper considers vector optimization problems (VOP) of the form \[ \min F(x) \text{ subject to } u_i(x) \leq 0, i\in\{1, \dots, m\}; v_j(x) =0, j\in\{1, \dots, n\}, \] where \(F:X \rightarrow \mathbb{R}^L\) and \(u_i, v_j: X\rightarrow \mathbb{R}\) and \(X\) is a Banach space. Using nondifferentiable Abadie or generalized Zangwill constraint qualifications the authors derive necessary KKT conditions for weakly effficient solutions of (VOP) and sufficient conditions under Michel-Penot pseudoconvexity of \(F\) and Michel-Penot quasiconvexity of the constraints. Results for the special case of linear problems are also explicitely stated. Finally, a partial calmness condition is introduced and it is shown that under this condition weakly efficient solutions of the (VOP) and local minimia of a scalar penalty problem related to (VOP) are equivalent.