Approximate Proper Efficiency in Vector Optimization via Benson's Approach

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, $\mathit{\text{An}}\mathit{\text{improved}}\mathit{\text{definition}}\mathit{\text{of}}\mathit{\text{proper}}\mathit{\text{efficiency}}\mathit{\text{for}}\mathit{\text{vector}}\mathit{\text{maximization}}\mathit{\text{with}}\mathit{\text{respect}}\mathit{\text{to}}\mathit{\text{cones}}$, J. Math. Anal. Appl. $\mathbf{\text{71}}$ (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.