Optimality and Constraint Qualifications in Vector Optimization

We propose a unifying approach in deriving constraint qualifications and theorem of the alternative. We first introduce a separation theorem between a subspace and the non-positive orthant, and then we use it to derive a new constraint qualification for a smooth vector optimization problem with inequality constraints. The proposed condition is weaker than the existing conditions stated in the recent literature. According with the strict relationship between generalized convexity and constraint qualifications, we introduce a new class of generalized convex vector functions. This allows us to obtain some new constraint qualifications in a more general form than the ones related to componentwise generalized convexity. Finally, the introduced separation theorem allows us to derive some of the known theorems of the alternative which are used in the literature to get constraint qualifications.