A generalized Ekeland vector variational principle and its applications in optimization

In this paper, we give a generalized Ekeland vector variational principle. By using the principle, we extend and improve the related results in sharp efficiency. In the framework of locally convex spaces, we introduce two kinds of generalized sharp efficiencies and prove that they are equivalent. In particular, we show that a sharp efficient solution with respect to an interior point of the ordering cone is also one with respect to every interior point. Moreover, we introduce the generalized Takahashi’s condition and the generalized Hamel’s condition for vector-valued functions. From the generalized Ekeland principle we deduce that the two conditions are equivalent. From this, we discuss the relationship between the ‘distance’ of f(x)f(x) from E(f(X))E(f(X)) and the distance of xx from E(f)E(f), where E(f(X))E(f(X)) denotes the efficient point set of f(X)f(X) and E(f)E(f) denotes the efficient solution set.