Vector Optimization Problem Is Weakly Efficient Solution Sets Is Not Empty And Compactness Characterizations

In this paper, the vector optimization problem that the object mapping is cone-paraconvex in finite-dimensional space Rn(n>1) are considered. The characteriza-tion for the nonemptiness and compactness of the set of weakly efficient solutions of vector optimization problem is given in terms of the O-coercivity of the scalar functions and the extreme direction of dual cone. In addition, an existence theorem about the weakly efficient solution of the vector optimization problem in Hausdorff space is given, and an counter-example is given to show that the conjecture which is put forward by Flores-Bazan and Vera [Characterization of the nonemptiness and compactness of solution sets in convex and nonconvex vector optimization. J. Optim. Theory Appl.130 (2006) 185-207] is false.