2023/12/03 by Huong, Nguyen Thi Thu
#FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2312.01428
We present two criteria for checking approximate proper efficiency in vector optimization problems with the ordering cone being a nonnegative orthant. Although the criteria can be established by Benson's approach [H.P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones, J. Math. Anal. Appl. 71 (1979), 232--241], detailed proofs are given for the first time here. The two criteria are strong motivations to introduce the concept of e-properly efficient solution, where e is any nonzero vector taken from the closed pointed convex ordering cone. For an arbitrary linear vector optimization problem, we show that either the e-properly efficient solution set is empty or it coincides with the e-efficient solution set. This new result has no analogue in the literature.