Abstract
This paper focuses on characterizations via scalarization of several kinds of minimal
solutions of set-valued optimization problems, where the objective values are compared
through relations between sets (set optimization). For this aim we follow an axiomatic
approach based on general order representation and order preservation properties, which
works in any abstract set ordered by a quasi order (i.e., reflexive and transitive) relation. Then,
following this approach, we study a recent Gerstewitz scalarization mapping for set-valued
optimization problems with K-proper sets and a solid ordering cone K. In particular we
show a dual minimax reformulation of this scalarization. Moreover, in the setting of normed
spaces ordered by non necessarily solid ordering cones, we introduce a new scalarization
functional based on the so-called oriented distance. Using these scalarization mappings, we obtain necessary and sufficient optimality conditions in set optimization. Finally, whenever
the ordering cone is solid, by considering suitable generalized Chebyshev norms with
appropriate parameters, we show that the three scalarizations studied in the present work are
coincident.
Lingua originale | English |
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pagine (da-a) | 525-552 |
Numero di pagine | 28 |
Rivista | Journal of Global Optimization |
Volume | 61 |
DOI | |
Stato di pubblicazione | Pubblicato - 2014 |
Keywords
- Cone Proper set
- Gerstewitz scalarization
- Nonlinear scalarization
- Order preservation property
- Order representation property
- Oriented distance
- Quasi ordering
- Set optimization
- Set relations