Abstract
In this paper, we investigate abstract convexity of non-positive increasing and radiant (IR) functions over a topological vector space. We characterize the essential results of abstract convexity such as support set, subdifferential and polarity of these functions. We also give some characterizations of a certain kind of polarity and separation property for non-convex radiant and co-radiant sets.
Similar content being viewed by others
References
Abasov T.M., Rubinov A.M.: Subdifferential of Some Classes of Non-smooth Functions, Mathematical Models of Analysis of Non-smooth Models. St. Petersburg University Press, Russia (1996)
Doagooei A.R., Mohebi H.: Monotonic analysis over ordered topological vector spaces: IV. J. Global Optim. 45, 355–369 (2009)
Dutta J., Martínez-Legaz J.E., Rubinov A.M.: Monotonic analysis over cones: I. Optimization 53, 165–177 (2004)
Dutta J., Martínez-Legaz J.E., Rubinov A.M.: Monotonic analysis over cones: II. Optimization 53, 529–547 (2004)
Dutta J., Martínez-Legaz J.-E., Rubinov A.M.: Monotonic analysis over cones: III. J. Convex Anal. 15, 581–592 (2008)
Martínez-Legaz J.E., Rubinov A.M., Schaible S.: Increasing quasi-concave co-radiant functions with applications in mathematical economics. Math. Methods Oper. Res. 61, 261–280 (2005)
Mohebi H., Doagooei A.R.: Abstract convexity of extended real valued increasing and positively homogeneous functions. J. DCDIS B 17, 659–674 (2010)
Mohebi H., Sadeghi H.: Monotonic analysis over non-convex cones. Numer. Funct. Anal. Optim. 26(7–8), 879–895 (2005)
Mohebi H., Sadeghi H.: Monotonic analysis over ordered topological vector apaces: I. Optimization 56(3), 305–321 (2007)
Pallaschke D., Rolewicez S.: Foundations of Mathematical Optimization (Convex Analysis without Linearity). Kluwer Academic Publishers, Boston (1997)
Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rubinov A.M.: Abstract Convexity and Global Optimization. Kluwer Academic Publishers, Dordrecht (2000)
Singer I.: Abstract Convex Analysis. Wiley-Interscience, New York (1997)
Zaffaroni A.: Superlinear separation of radiant and co-radiant sets. Optimization 56(1–2), 267–285 (2007)
Zaffaroni A.: Is every radiant function the sum of quasiconvex functions?. Math. Methods Oper. Res. 59, 221–233 (2004)
Zaffaroni A.: Superlinear separation and dual characterizations of radiant functions. Pac. J. Optim. 2(1), 181–202 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mohebi, H. Abstract convexity of radiant functions with applications. J Glob Optim 55, 521–538 (2013). https://doi.org/10.1007/s10898-012-9888-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-012-9888-1