Abstract
In this chapter, we justify theoretically the original axiomatic approach to Pareto set reduction based on a finite collection of information quanta. Here the exposition seems most difficult in mathematical terms, but the readers with an insufficient background may skip it without losing the comprehension of further material. The whole essence of the results derived below can be expressed as follows. Information in the form of quanta is complete: for any multicriteria choice problem from a definite (rather wide) class, it is possible to find the unknown set of nondominated vectors (nondominated alternatives) with an arbitrary accuracy only based on such information. Moreover, if the number of feasible vectors is finite, then the set of nondominated vectors can be constructed precisely.
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Notes
- 1.
Recall that \(K\) is an acute convex cone of the preference relation \(\succ\).
- 2.
The closure of a set contains this set together with all its limit points.
- 3.
The convex hull of a given set A is the smallest convex set containing A.
- 4.
The K -boundedness of the set of feasible vectors can be omitted, since any finite set is bounded, ergo K -bounded.
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© 2018 Springer International Publishing AG
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Noghin, V.D. (2018). Completeness Property of Information Quanta. In: Reduction of the Pareto Set. Studies in Systems, Decision and Control, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-67873-3_6
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DOI: https://doi.org/10.1007/978-3-319-67873-3_6
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Publisher Name: Springer, Cham
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Online ISBN: 978-3-319-67873-3
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