TY - JOUR
T1 - Scalarization in set optimization with solid and nonsolid ordering cones
AU - Gutiérrez, C.
AU - Gutierrez Vaquero, Cesar
AU - Jiménez, B.
AU - Miglierina, Enrico
AU - Molho, E.
PY - 2014
Y1 - 2014
N2 - 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.
AB - 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.
KW - Cone Proper set
KW - Gerstewitz scalarization
KW - Nonlinear scalarization
KW - Order preservation property
KW - Order representation property
KW - Oriented distance
KW - Quasi ordering
KW - Set optimization
KW - Set relations
KW - Cone Proper set
KW - Gerstewitz scalarization
KW - Nonlinear scalarization
KW - Order preservation property
KW - Order representation property
KW - Oriented distance
KW - Quasi ordering
KW - Set optimization
KW - Set relations
UR - http://hdl.handle.net/10807/55522
U2 - 10.1007/s10898-014-0179-x
DO - 10.1007/s10898-014-0179-x
M3 - Article
SN - 0925-5001
VL - 61
SP - 525
EP - 552
JO - Journal of Global Optimization
JF - Journal of Global Optimization
ER -