Proper or Weak Efficiency via Saddle Point Conditions in Cone-Constrained Nonconvex Vector Optimization Problems

Motivated by many applications (for instance, some production models in finance require infinity-dimensional commodity spaces, and the preference is defined in terms of an ordering cone having possibly empty interior), this paper deals with a unified model, which involves preference relations that are not necessarily transitive or reflexive. Our study is carried out by means of saddle point conditions for the generalized Lagrangian associated with a cone-constrained nonconvex vector optimization problem. We establish a necessary and sufficient condition for the existence of a saddle point in case the multiplier vector related to the objective function belongs to the quasi-interior of the polar of the ordering set. Moreover, exploiting suitable Slater-type constraints qualifications involving the notion of quasi-relative interior, we obtain several results concerning the existence of a saddle point, which serve to get efficiency, weak efficiency and proper efficiency. Such results generalize, to the nonconvex vector case, existing conditions in the literature. As a by-product, we propose a notion of properly efficient solution for a vector optimization problem with explicit constraints. Applications to optimality conditions for vector optimization problems are provided with particular attention to bicriteria problems, where optimality conditions for efficiency, proper efficiency and weak efficiency are stated, both in a geometric form and by means of the level sets of the objective functions.