Abstract
The aim of this paper is to present dual optimality conditions for the difference of two extended real valued increasing co-radiant functions. We do this by first characterizing dual optimality conditions for the difference of two nonpositive increasing co-radiant functions. Finally, we present dual optimality conditions for the difference of two extended real valued increasing co-radiant functions. Our approach is based on the Toland–Singer formula.
Similar content being viewed by others
Data availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
References
Daryaei, M.H., Mohebi, H.: Abstract convexity of extended real valued ICR functions. Optimization 62(6), 835–855 (2013)
Daryaei, M.H., Mohebi, H.: Global minimization of the difference of strictly non-positive valued affine ICR functions. J. Global Optim. 61, 311–323 (2015)
Doagooei, A.R., Mohebi, H.: Monotonic analysis over ordered topological vector spaces: IV. J. Global Optim. 45, 355–369 (2009)
Doagooei, A.R., Mohebi, H.: Optimization of the difference of ICR functions. Nonlinear Anal. 71, 4493–4499 (2009)
Glover, B.M., Jeyakumar, V.: Nonlinear extensions of Farkas’ lemma with applications to global optimization and least squares. Math. Oper. Res. 20, 818–837 (1995)
Glover, B.M., Ishizuka, Y., Jeyakumar, V., Rubinov, A.M.: Inequality systems and global optimization. J. Math. Anal. Appl. 202, 900–919 (1996)
Hiriart-Urruty, J.B.: From convex to nonconvex minimization: necessary and sufficient conditions for global optimality. In: Nonsmooth Optimization and Related Topics, pp. 219–240. Plenum, New York (1989)
Mohebi, H.: Abstract convexity of radiant functions with applications. J. Global Optim. 55, 521–538 (2013)
Mohebi, H., Mirzadeh, S.: Abstract convexity of extended real valued increasing and radiant functions. Filomat 26(5), 1005–1022 (2012)
Mohebi, H., Sadeghi, H.: Monotonic analysis over ordered topological vector spaces: I. Optimization 56(3), 305–321 (2007)
Mohebi, H., Sadeghi, H.: Monotonic analysis over ordered topological vector spaces: II. Optimization 58(2), 241–249 (2009)
Nedić, A., Ozdaglar, A., Rubinov, A.M.: Abstract convexity for nonconvex optimization duality. Optimization 56(5), 655–674 (2007)
Rubinov, A.M.: Abstract Convexity and Global Optimization. Kluwer Academic Publishers, Dordrecht-Boston-London (2000)
Rubinov, A.M., Glover, B.M.: Increasing convex-along-rays functions with application to global optimization. J. Optim. Theory Appl. 102(3), 615–642 (1999)
Singer, I.: Abstract Convex Analysis. Wiley-Interscience, New York (1997)
Zaffaroni, A.: Is every radiant function the sum of quasiconvex functions? Math. Meth. Oper. Res. 59, 221–233 (2004)
Zaffaroni, A.: Superlinear separation and dual characterizations of radiant functions. Pacific J. Optim. 2, 181–202 (2006)
Acknowledgements
The authors are very grateful to the anonymous referee and the Associate Editor for their helpful comments and valuable suggestions and criticism regarding an earlier version of this paper. The comments of the referee and the Associate Editor were very useful and they helped us to improve the paper significantly. The second author was partially supported by Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Iran [Grant No: 99/3668].
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
This declaration is not applicable. Indeed, there is no any Conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Daryaei, M.H., Mohebi, H. Dual optimality conditions for the difference of extended real valued increasing co-radiant functions. J Glob Optim (2024). https://doi.org/10.1007/s10898-024-01404-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10898-024-01404-1
Keywords
- Global optimization
- Abstract convexity
- Increasing co-radiant function
- Abstract subdifferential
- Support set
- Toland–Singer formula