On the Necessity of the Sufficient Conditions in Cone-Constrained Vector Optimization

Abstract: The object of investigation in this paper are vector nonlinear programming problems with cone constraints. We introduce the notion of a Fritz John pseudoinvex cone-constrained vector problem. We prove that a problem with cone constraints is Fritz John pseudoinvex if and only if every vector critical point of Fritz John type is a weak global minimizer. Thus, we generalize several results, where the Paretian case have been studied. We also introduce a new Frechet differentiable pseudoconvex problem. We derive that a problem with quasiconvex vector-valued data is pseudoconvex if and only if every Fritz John vector critical point is a weakly efficient global solution. Thus, we generalize a lot of previous optimality conditions, concerning the scalar case and the multiobjective Paretian one. Additionally, we prove that a quasiconvex vector-valued function is pseudoconvex with respect to the same cone if and only if every vector critical point of the function is a weak global minimizer, a result, which is a natural extension of a known characterization of pseudoconvex scalar functions.