Details
| Original language | English |
|---|---|
| Pages (from-to) | 408-442 |
| Number of pages | 35 |
| Journal | Journal of Optimization Theory and Applications |
| Volume | 193 |
| Issue number | 1-3 |
| Early online date | 20 Dec 2021 |
| Publication status | Published - Jun 2022 |
| Externally published | Yes |
Abstract
In vector optimization, it is of increasing interest to study problems where the image space (a real linear space) is preordered by a not necessarily solid (and not necessarily pointed) convex cone. It is well-known that there are many examples where the ordering cone of the image space has an empty (topological/algebraic) interior, for instance in optimal control, approximation theory, duality theory. Our aim is to consider Pareto-type solution concepts for such vector optimization problems based on the intrinsic core notion (a well-known generalized interiority notion). We propose a new Henig-type proper efficiency concept based on generalized dilating cones which are relatively solid (i.e., their intrinsic cores are nonempty). Using linear functionals from the dual cone of the ordering cone, we are able to characterize the sets of (weakly, properly) efficient solutions under certain generalized convexity assumptions. Toward this end, we employ separation theorems that are working in the considered setting.
Keywords
- (Weak) Pareto efficiency, Generalized dilating cones, Henig proper efficiency, Intrinsic core, Relatively solid convex cone, Scalarization, Separation theorem, Vector optimization
ASJC Scopus subject areas
- Decision Sciences(all)
- Management Science and Operations Research
- Mathematics(all)
- Control and Optimization
- Mathematics(all)
- Applied Mathematics
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In: Journal of Optimization Theory and Applications, Vol. 193, No. 1-3, 06.2022, p. 408-442.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces
AU - Günther, Christian
AU - Khazayel, Bahareh
AU - Tammer, Christiane
N1 - Funding information: The authors are grateful to the anonymous referees for their valuable comments.
PY - 2022/6
Y1 - 2022/6
N2 - In vector optimization, it is of increasing interest to study problems where the image space (a real linear space) is preordered by a not necessarily solid (and not necessarily pointed) convex cone. It is well-known that there are many examples where the ordering cone of the image space has an empty (topological/algebraic) interior, for instance in optimal control, approximation theory, duality theory. Our aim is to consider Pareto-type solution concepts for such vector optimization problems based on the intrinsic core notion (a well-known generalized interiority notion). We propose a new Henig-type proper efficiency concept based on generalized dilating cones which are relatively solid (i.e., their intrinsic cores are nonempty). Using linear functionals from the dual cone of the ordering cone, we are able to characterize the sets of (weakly, properly) efficient solutions under certain generalized convexity assumptions. Toward this end, we employ separation theorems that are working in the considered setting.
AB - In vector optimization, it is of increasing interest to study problems where the image space (a real linear space) is preordered by a not necessarily solid (and not necessarily pointed) convex cone. It is well-known that there are many examples where the ordering cone of the image space has an empty (topological/algebraic) interior, for instance in optimal control, approximation theory, duality theory. Our aim is to consider Pareto-type solution concepts for such vector optimization problems based on the intrinsic core notion (a well-known generalized interiority notion). We propose a new Henig-type proper efficiency concept based on generalized dilating cones which are relatively solid (i.e., their intrinsic cores are nonempty). Using linear functionals from the dual cone of the ordering cone, we are able to characterize the sets of (weakly, properly) efficient solutions under certain generalized convexity assumptions. Toward this end, we employ separation theorems that are working in the considered setting.
KW - (Weak) Pareto efficiency
KW - Generalized dilating cones
KW - Henig proper efficiency
KW - Intrinsic core
KW - Relatively solid convex cone
KW - Scalarization
KW - Separation theorem
KW - Vector optimization
UR - http://www.scopus.com/inward/record.url?scp=85121513185&partnerID=8YFLogxK
U2 - 10.1007/s10957-021-01976-y
DO - 10.1007/s10957-021-01976-y
M3 - Article
VL - 193
SP - 408
EP - 442
JO - Journal of Optimization Theory and Applications
JF - Journal of Optimization Theory and Applications
SN - 0022-3239
IS - 1-3
ER -