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.

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

IVANOV, IVAN GINCHEV;GUERRAGGIO, ANGELO;ROCCA, MATTEO
2007-01-01

Abstract

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.
2007
vector optimization, properly efficient points, properly efficient points of order k, characterization of properly efficient points
Ivanov, IVAN GINCHEV; Guerraggio, Angelo; Rocca, Matteo
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/1668981
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