Abstract
In this paper, we first obtain some properties of topical (increasing and plus-homogeneous) functions in the framework of abstract convexity. Next, we use the Toland–Singer formula to characterize the dual problem for the difference of two topical functions. Finally, we present necessary and sufficient conditions for the global minimum of the difference of two strictly topical functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Doagooei A.R., Mohebi H.: Optimization of the difference of ICR functions. Nonlinear Anal. Theory Methods Appl. 71, 4493–4499 (2009)
Gaubert, S., Gunawardena, J.: A non-linear hierarchy for discrete event dynamical systems. In: Proceedings of the 4th Workshop on Discrete Event Systems, Calgiari, Technical Report HPL-BRIMS-98-20, Hewlett-Packard Labs. (1998)
Glover B.M., Rubinov A.M.: Toland-Singer formula cannot distinguish a global minimizer from a choice of stationary points. Numer. Funct. Anal. Optim. 20, 99–119 (1999)
Glover B.M., Rubinov A.M.: Increasing convex along rays functions with applications to global optimization. J. Optim. Theory Appl. 102, 615–642 (1999)
Gunawardena J.: An Introduction to Idempotency. Cambridge University Press, Cambridge (1998)
Gunawardena, J.: From Max-Plus Algebra to Non-expansive Mappings: A Non-linear Theory for Discrete Event Systems. Theoretical Computer Science, Technical Report HPL-BRIMS-99-07, Hewlett-Packard Labs. (1999)
Mohebi, H.: Topical functions and their properties in a class of ordered Bananch spaces. In: Jeyakumar, V., Rubinov, A., Continuous Optimization, Current Trends and Modern Applications, 99:343–361. Springer, USA (2005)
Mohebi H.: Optimization of the difference of IR functions. J. Appl. Math. Sci. 6(67), 3339–3345 (2012)
Rubinov A.: Abstract Convexity and Global Optimization. Kluwer, Boston (2000)
Toland J.F.: Duality in nonconvex optimization. J. Math. Anal. Appl. 66, 399–415 (1978)
Rubinov A.M.: Singer I. Topical and sub-topical functions, downward sets and abstract convexity. Optimization 50, 307–351 (2001)
Singer I.: Further application of the additive min-type coupling function. Optimization 51, 471–485 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Doagooei, A.R., Mohebi, H. Optimization of the difference of topical functions. J Glob Optim 57, 1349–1358 (2013). https://doi.org/10.1007/s10898-012-9991-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-012-9991-3
Keywords
- Global optimization
- Monotonic analysis
- Abstract convexity
- Topical function
- Toland–Singer formula
- Support set
- Subdifferential set