Characterizing Efficiency on Infinite-dimensional Commodity Spaces with Ordering Cones Having Possibly Empty Interior
- 462 Downloads
Some production models in finance require infinite-dimensional commodity spaces, where efficiency is defined in terms of an ordering cone having possibly empty interior. Since weak efficiency is more tractable than efficiency from a mathematical point of view, this paper characterizes the equality between efficiency and weak efficiency in infinite-dimensional spaces without further assumptions, like closedness or free disposability. This is obtained as an application of our main result that characterizes the solutions to a unified vector optimization problem in terms of its scalarization. Standard models as efficiency, weak efficiency (defined in terms of quasi-relative interior), weak strict efficiency, strict efficiency, or strong solutions are carefully described. In addition, we exhibit two particular instances and compute the efficient and weak efficient solution set in Lebesgue spaces.
KeywordsVector optimization Scalarization Efficiency Infinite-dimensional commodity space Quasi-relative interior
Mathematics Subject Classication (2000)90C26 90C29 90C30 90C46
This research, for the first author, was supported in part by CONICYT-Chile through FONDECYT 112-0980 and BASAL projects, CMM, Universidad de Chile. Support by Ecos-Conicyt CE08-2010 and MATH-AmSud 13math-01 are also greatly acknowledged. The authors also wants to express their great gratitude to Nicolas Hadjisavvas for very stimulating discussions during the last stage of this final version.
- 4.Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 2. Springer, Berlin (1989)Google Scholar
- 21.Göpfert, A., Riahi, H., Tammer, Ch., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books in Mathematics. Springer, New york (2003)Google Scholar
- 22.Chen, G-Y.; Huang, X.; Yang, X., “Vector optimization. Set-valued and variational analysis”. Lecture Notes in Economics and Mathematical Systems, 541. Springer-Verlag, Berlin, 2005.Google Scholar
- 41.Magill, M., Quinzii, M.: Theory of incomplete markets, vol. 1. MIT Press, Cambridge (1996)Google Scholar