Ideal, weakly efficient solutions for vector optimization problems

We establish existence results for finite dimensional vector minimization problems allowing the solution set to be unbounded. The solutions to be referred are ideal(strong)/weakly efficient(weakly Pareto). Also several necessary and/or sufficient conditions for the solution set to be non-empty and compact are established. Moreover, some characterizations of the non-emptiness (boundedness) of the (convex) solution set in case the solutions are searched in a subset of the real line, are also given However, the solution set fails to be convex in general. In addition, special attention is addressed when the underlying cone is the non-negative orthant and when the semi-strict quasiconvexity of each component of the vector-valued function is assumed. Our approach is based on the asymptotic description of the functions and sets. Some examples illustrating such an approach are also exhibited.