Nonlinear strict cone separation theorems in real normed spaces

authored by: Christian Günther, Christiane Tammer, Bahareh Khazayel
Abstract: In this paper, we derive some new results for the separation of two not necessarily convex cones by a (convex) cone / conical surface in real (reflexive) normed spaces. In essence, we follow the nonlinear and nonsymmetric separation approach developed by Kasimbeyli (2010, SIAM J. Optim. 20), which is based on augmented dual cones and Bishop-Phelps type (normlinear) separating functions. Compared to Kasimbeyli's separation theorem, we formulate our theorems for the separation of two cones under weaker conditions (concerning convexity and closedness requirements) with respect to the involved cones. By a new characterization of the algebraic interior of augmented dual cones in real normed spaces, we are able to establish relationships between our cone separation results and the results derived by Kasimbeyli (2010, SIAM J. Optim. 20) and by Garcia-Castaño, Melguizo-Padial and Parzanese.
Organisation(s): Institute of Applied Mathematics
External Organisation(s): Martin Luther University Halle-Wittenberg
Type: Article
Journal: Journal of Nonlinear and Variational Analysis
Volume: 8
Pages: 601-623
No. of pages: 23
ISSN: 2560-6921
Publication date: 01.08.2024
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Analysis, Applied Mathematics
Electronic version(s): https://doi.org/10.48550/arXiv.2303.06392 (Access: Open)
https://doi.org/10.23952/jnva.8.2024.4.08 (Access: Open)

BibTeX

@article{c694a8ce31bf4d0085db63c3dc09869c,
title = "Nonlinear strict cone separation theorems in real normed spaces",
abstract = "In this paper, we derive some new results for the separation of two not necessarily convex cones by a (convex) cone / conical surface in real (reflexive) normed spaces. In essence, we follow the nonlinear and nonsymmetric separation approach developed by Kasimbeyli (2010, SIAM J. Optim. 20), which is based on augmented dual cones and Bishop-Phelps type (normlinear) separating functions. Compared to Kasimbeyli's separation theorem, we formulate our theorems for the separation of two cones under weaker conditions (concerning convexity and closedness requirements) with respect to the involved cones. By a new characterization of the algebraic interior of augmented dual cones in real normed spaces, we are able to establish relationships between our cone separation results and the results derived by Kasimbeyli (2010, SIAM J. Optim. 20) and by Garcia-Casta{\~n}o, Melguizo-Padial and Parzanese.",
keywords = "Augmented dual cone, Bishop-Phelps cone, Cone separation, Nonconvex cone, Separation theorem",
author = "Christian G{\"u}nther and Christiane Tammer and Bahareh Khazayel",
note = "Publisher Copyright: {\textcopyright} 2024 Journal of Nonlinear and Variational Analysis.",
year = "2024",
month = aug,
day = "1",
doi = "10.48550/arXiv.2303.06392",
language = "English",
volume = "8",
pages = "601--623",
journal = "Journal of Nonlinear and Variational Analysis",
issn = "2560-6921",
publisher = "Biemdas Academic Publishers",
number = "4",
}