Sets of efficient points in a normed space.(English)Zbl 0605.49020
The paper deals with efficient solutions of a multiobjective programming problem in a normed space; the objective functions to be minimized are distances to every point of a compact subset.
We give a geometrical description of the sets of strictly efficient, efficient and weakly efficient points which looks like a description of the convex hull. It makes use of a family of cones, which play the role of halfspaces in the Euclidean case. Topological properties of these cones are developed. The above description leads to hull and closure properties of sets of efficiency.
In addition the two-dimensional case is emphasized in connection with location theory where more precise results are obtained. In particular, when the space is strictly convex or polyhedral, the description of the sets of efficiency provides a geometrical construction of these sets.
We give a geometrical description of the sets of strictly efficient, efficient and weakly efficient points which looks like a description of the convex hull. It makes use of a family of cones, which play the role of halfspaces in the Euclidean case. Topological properties of these cones are developed. The above description leads to hull and closure properties of sets of efficiency.
In addition the two-dimensional case is emphasized in connection with location theory where more precise results are obtained. In particular, when the space is strictly convex or polyhedral, the description of the sets of efficiency provides a geometrical construction of these sets.
MSC:
| 49K27 | Optimality conditions for problems in abstract spaces |
| 52A07 | Convex sets in topological vector spaces (aspects of convex geometry) |
| 90C31 | Sensitivity, stability, parametric optimization |
| 46B99 | Normed linear spaces and Banach spaces; Banach lattices |
Cite
Full Text:DOI
References:
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