Generalized Gerstewitz’s Functions and Vector Variational Principle for ϵ-Efficient Solutions in the Sense of Németh

Abstract

In this paper, we first generalize Gerstewitz’s functions from a single positive vector to a subset of the positive cone. Then, we establish a partial order principle, which is indeed a variant of the pre-order principle [Qiu, J. H.: A pre-order principle and set-valued Ekeland variational principle. J. Math. Anal. Appl., 419, 904–937 (2014)]. By using the generalized Gerstewitz’s functions and the partial order principle, we obtain a vector EVP for e-efficient solutions in the sense of Nemeth, which essentially improves the earlier results by completely removing a usual assumption for boundedness of the objective function. From this, we also deduce several special vector EVPs, which improve and generalize the related known results.