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.
Keywords: Constraint qualifications, generalized convex vector functions, multiobjective
programming, optimality conditions.
