Geoffrion type characterization of higher-order properly efficient points in vector optimization

We consider the constrained vector optimization problem min(C) f (x), x is an element of A, where X and Y are normed spaces, A subset of X(0) subset of X are given sets, C subset of Y, C not equal Y, is a closed convex cone, and f : X(0) -> Y is a given function. We recall the notion of a properly efficient point (p-minimizer) for the considered problem and in terms of the so-called oriented distance we define also the notion of a properly efficient point of order n (p-minimizers of order n). We show that the p-minimizers of higher order generalize the usual notion of a properly efficient point. The main result is the characterization of the p-minimizers of higher order in terms of "trade-offs." In such a way we generalize the result of A.M. Geoffrion [A.M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl. 22 (3) (1968) 618-630] in two directions, namely for properly efficient points of higher order in infinite dimensional spaces, and for arbitrary closed convex ordering cones.