Nonlinear strict cone separation theorems in real normed spaces

verfasst von: 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.
Organisationseinheit(en): Institut für Angewandte Mathematik
Externe Organisation(en): Martin-Luther-Universität Halle-Wittenberg
Typ: Artikel
Journal: Journal of Nonlinear and Variational Analysis
Band: 8
Seiten: 601-623
Anzahl der Seiten: 23
ISSN: 2560-6921
Publikationsdatum: 01.08.2024
Publikationsstatus: Veröffentlicht
Peer-reviewed: Ja
ASJC Scopus Sachgebiete: Analysis, Angewandte Mathematik
Elektronische Version(en): https://doi.org/10.48550/arXiv.2303.06392 (Zugang: Offen)
https://doi.org/10.23952/jnva.8.2024.4.08 (Zugang: Offen)

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",
}