Details
| Original language | English |
|---|---|
| Pages (from-to) | 1276-1298 |
| Number of pages | 23 |
| Journal | SIAM journal on optimization |
| Volume | 31 |
| Issue number | 2 |
| Publication status | Published - 10 May 2021 |
| Externally published | Yes |
Abstract
Convex cones play an important role in nonlinear analysis and optimization theory. In particular, specific normal cones and tangent cones are known to be convex cones, and it is a crucial fact that they are useful geometric objects for describing optimality conditions. As important applications (especially, in the fields of optimal control with PDE constraints, risk theory, duality theory, vector optimization, and order theory) show, there are many examples of convex cones with an empty (topological as well as algebraic) interior. In such situations, generalized interiority notions can be useful. In this article, we present new representations and properties of the relative algebraic interior (also known as intrinsic core) of relatively solid, convex cones in real linear spaces (which are not necessarily endowed with a topology) of both finite and infinite dimension. For proving our main results, we are using new separation theorems where relatively solid, convex sets (cones) are involved. For the intrinsic core of the dual cone of a relatively solid, convex cone, we also state new representations that involve the lineality space of the given convex cone. To emphasize the importance of the derived results, some applications in vector optimization are given.
Keywords
- Convex cone, Intrinsic core, Linear space, Linearity space, Pareto efficiency, Relative algebraic interior, Separation theorem, Vector optimization
ASJC Scopus subject areas
- Computer Science(all)
- Software
- Mathematics(all)
- Theoretical Computer Science
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In: SIAM journal on optimization, Vol. 31, No. 2, 10.05.2021, p. 1276-1298.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - On the Intrinsic Core of Convex Cones in Real Linear Spaces
AU - Khazayel, Bahareh
AU - Farajzadeh, Ali
AU - Günther, Christian
AU - Tammer, Christiane
N1 - Publisher Copyright: © 2021 Society for Industrial and Applied Mathematics
PY - 2021/5/10
Y1 - 2021/5/10
N2 - Convex cones play an important role in nonlinear analysis and optimization theory. In particular, specific normal cones and tangent cones are known to be convex cones, and it is a crucial fact that they are useful geometric objects for describing optimality conditions. As important applications (especially, in the fields of optimal control with PDE constraints, risk theory, duality theory, vector optimization, and order theory) show, there are many examples of convex cones with an empty (topological as well as algebraic) interior. In such situations, generalized interiority notions can be useful. In this article, we present new representations and properties of the relative algebraic interior (also known as intrinsic core) of relatively solid, convex cones in real linear spaces (which are not necessarily endowed with a topology) of both finite and infinite dimension. For proving our main results, we are using new separation theorems where relatively solid, convex sets (cones) are involved. For the intrinsic core of the dual cone of a relatively solid, convex cone, we also state new representations that involve the lineality space of the given convex cone. To emphasize the importance of the derived results, some applications in vector optimization are given.
AB - Convex cones play an important role in nonlinear analysis and optimization theory. In particular, specific normal cones and tangent cones are known to be convex cones, and it is a crucial fact that they are useful geometric objects for describing optimality conditions. As important applications (especially, in the fields of optimal control with PDE constraints, risk theory, duality theory, vector optimization, and order theory) show, there are many examples of convex cones with an empty (topological as well as algebraic) interior. In such situations, generalized interiority notions can be useful. In this article, we present new representations and properties of the relative algebraic interior (also known as intrinsic core) of relatively solid, convex cones in real linear spaces (which are not necessarily endowed with a topology) of both finite and infinite dimension. For proving our main results, we are using new separation theorems where relatively solid, convex sets (cones) are involved. For the intrinsic core of the dual cone of a relatively solid, convex cone, we also state new representations that involve the lineality space of the given convex cone. To emphasize the importance of the derived results, some applications in vector optimization are given.
KW - Convex cone
KW - Intrinsic core
KW - Linear space
KW - Linearity space
KW - Pareto efficiency
KW - Relative algebraic interior
KW - Separation theorem
KW - Vector optimization
UR - http://www.scopus.com/inward/record.url?scp=85106551374&partnerID=8YFLogxK
U2 - 10.1137/19m1283148
DO - 10.1137/19m1283148
M3 - Article
VL - 31
SP - 1276
EP - 1298
JO - SIAM journal on optimization
JF - SIAM journal on optimization
SN - 1052-6234
IS - 2
ER -