Summary
We study topical functions in a class of ordered Banach spaces and show that these functions are abstract convex with respect to a certain set of elementary functions and obtain an explicit formula for their subdifferential. We give characterizations of the Fenchel-Moreau conjugate and the conjugate of type Lau of topical functions. We also present necessary and sufficient conditions for plus-weak Pareto points of a closed downward set in terms of separation from outside points.
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References
Gaubert, S., Gunawardena, J.: A non-linear hierarchy for discrete event dynamical systems. Proc. 4th Workshop on discrete event systems, Calgiari, Technical Report HPL-BRIMS-98-20, Hewlett-Packard Labs. (1998)
Gunawardena, J.: An introduction to idempotency. Cambridge University Press, Cambridge (1998)
Gunawardena, J.: Prom 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)
Gunawardena, J., Keane, M.: On the existence of cycle times for some non-expansive maps. Technical Report HPL-BRIMS-95-003, Hewlett-Packard Labs. (1995)
Martinez-Legaz, J.-E., Rubinov, A.M., Singer, I.: Downward sets and their separation and approximation properties. Journal of Global Optimization, 23, 111–137 (2002)
Mohebi, H., Rubinov, A.M.: Best approximation by downward sets with applications. Journal of Analysis in Theory and Applications, (to appear) (2005)
Rubinov, A.M.: Abstarct Convexity and Global Optimization. Kluwer Academic Publishers, Boston/Dordrecht/London (2000)
Rubinov, A.M., Singer, I.: Topical and sub-topical functions, downward sets and abstract convexity. Optimization, 50, 307–351 (2001)
Singer, L: The theory of best approximation and functional analysis. Regional Conference Series in Applied Mathematics, 13 (1974)
Singer, I.: Abstract Convex Analysis. Wiley-Interscience, New York (1987)
Singer, I.: Further application of the additive min-type coupling function. Optimization, 51, 471–485 (2002)
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Mohebi, H. (2005). Topical Functions and their Properties in a Class of Ordered Banach Spaces. In: Jeyakumar, V., Rubinov, A. (eds) Continuous Optimization. Applied Optimization, vol 99. Springer, Boston, MA. https://doi.org/10.1007/0-387-26771-9_12
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DOI: https://doi.org/10.1007/0-387-26771-9_12
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-26769-2
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