Abstract
In finite-dimensional spaces, cone-monotone functions – a special case of which are coordinate-wise nondecreasing functions – possess several regularity properties like almost everywhere continuity and differentiability. Such facts carry over to a separable Banach space, provided that the cone has interior. This chapter shows that further generalizations are not readily possible. We display several examples of cone–monotone functions on various Banach spaces, lacking the regularity expected from their finite-dimensional counterparts.
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Acknowledgements
The first author’s research was partially supported by NSERC and by the Canada Research Chair Programme. The second author performed this research at the Centre for Experimental and Constructive Mathematics at Simon Fraser University and at the Department of Mathematics at University of British Columbia.
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Borwein, J., Goebel, R. (2009). On the nondifferentiability of cone-monotone functions in Banach spaces. In: Pearce, C., Hunt, E. (eds) Optimization. Springer Optimization and Its Applications, vol 32. Springer, New York, NY. https://doi.org/10.1007/978-0-387-98096-6_1
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DOI: https://doi.org/10.1007/978-0-387-98096-6_1
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