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 -