A Fenchel-Rockafellar duality theorem for set-valued optimization

Optimization

Volumn 60, Issue 8-9, 2011, Pages 1023-1043

  Hamel, Andreas H ^a  

^a Princeton University   (United States)

Author keywords

conlinear space; duality; Fenchel conjugate; Fenchel Rockafellar theorem; set relations; set valued function; set valued risk measures; supremal convolution

Indexed keywords

EID: 84858176182     PISSN: 02331934     EISSN: 10294945     Source Type: Journal    
DOI: 10.1080/02331934.2010.534794     Document Type: Article

References (33)

1. Artzner, P, Delbaen, F, Eber, J-M and Heath, D. 1999. Coherent measures of risk. Math. Finance, 9: 203-228.
2. Borwein, JM. 1981. A Lagrange multiplier theorem and a Sandwich theorem for convex relations. Math. Scand., 48: 189-204.
3. Boţ, RI, Grad, S-M and Wanka, G. 2009. Duality in Vector Optimization, Berlin: Springer-Verlag.
4. Brumelle, S. 1981. Duality for multiple objective convex programs. Math. Oper. Res., 6: 159-172.
5. Fenchel, W. 1953. Convex Cones, Sets and Functions, Princeton: Lecture Notes, Princeton University Press.
6. Föllmer, H and Schied, A. 2004. Stochastic Finance. An Introduction in Discrete Time, 2, Berlin: De Gruyter.
7. Göpfert, A, Riahi, H, Tammer, C and Zǎlinescu, C. 2003. "Variational Methods in Partially Ordered Spaces". In CMS Books in Mathematics, Vol. 17, New York: Springer-Verlag.
8. Hamel, AH. 2005. "Variational principles on metric and uniform spaces". In Habilitation thesis, Martin-Luther-University Halle-Wittenberg. Available at http://sundoc.bibliothek.uni-halle.de/habil-online/05/05H167/habil.pdf
9. Hamel, AH. 2009. A duality theory for set-valued functions I: Fenchel conjugation theory. Set-Valued Var. Anal., 17: 153-182.
10. Hamel, AH and Heyde, F. 2010. Duality for set-valued measures of risk. SIAM J. Financial Math., 1: 66-95.
11. Hamel, AH, Heyde, F, Löhne, A, Tammer, C and Winkler, K. 2004. Closing the duality gap in linear vector optimization. J. Convex Analy., 11: 163-178.
12. Hernández, E and Rodríguez-Marín, L. 2007. Langrage duality in set-valued optimization. J. Optim. Theory Appl., 134: 119-134.
13. Heyde, F. 2009. Set-valued AV@R, Personal communication.
14. Heyde, F and Löhne, A. 2008. Geometric duality in multiple objective linear programming. SIAM J. Optim., 19: 836-845.
15. Heyde, F and Löhne, A. 2008. Solution concepts in vector optimization. A fresh look at an old story, Report 19, Halle, Germany: Institute for Mathematics, Martin-Luther-University, Halle-Wittenberg. Available at: http://www.mathematik.uni-halle.de/reports/sources/2008/08-19report.pdf
16. Heyde, F, Löhne, A and Tammer, C. 2008. Set-valued duality theory for multiple objective linear programs and applications to mathematical finance. Math. Meth. Oper. Res., 19: 836-845.
17. Ivanov, EH and Nehse, R. 1985. Some results on dual vector optimization problems. Optimization, 16: 505-517.
18. Jahn, J. 1983. Duality in vector optimization. Math. Program., 25: 342-353.
19. Jahn, J. 2004. Vector Optimization. Theory, Applications, and Extensions, Berlin: Springer-Verlag.
20. Kuroiwa, D, Tanaka, T and Truong, XDH. 1997. On cone convexity of set-valued maps. Nonlinear Anal.: Theory, Methods and Appl., 30: 1487-1496.
21. Löhne, A. 2005. "Optimization with set relations". In Ph.D. thesis, Halle, Germany: Martin-Luther-Universität Halle-Wittenberg.
22. Löhne, A. 2005. Optimization with set relations: conjugate duality. Optimization, 54: 265-282.
23. Löhne, A. 2010. "Vector optimization with infimum and supremum". In Habilitation thesis, Halle, Germany: Martin-Luther-Universität Halle-Wittenberg.
24. Löhne, A and Tammer, C. 2007. A new approach to duality in vector optimization. Optimization, 56: 221-239.
25. Nieuwenhuis, JW. 1980. Supremal points and generalized duality. Math. Operationsforsch. Stat. Ser. Optim., 11: 41-59.
26. Rockafellar, RT. 1966. An extension of Fenchel's duality theorem for convex functions. Duke Math. J., 33: 81-90.
27. Rockafellar, RT and Uryasev, S. 2000. Optimization of conditional value-at-risk. J. Risk, 2
28. Schrage, C. 2009. "Set-valued convex analysis". In Ph.D. thesis, Halle, Germany: Martin-Luther-University Halle-Wittenberg.
29. Song, W. 1998. A generalization of Fenchel duality in set-valued vector optimization. Math. Methods Oper. Res., 48: 259-272.
30. Yankova, M. 2010. "Set-valued average value at risk via linear programming duality". In Junior Project, ORFE, Princeton: Princeton University.
31. Zǎlinescu, C. 1983. "Duality for Vectorial Nonconvex Optimization by Convexification and Applications". Vol. XXIX, 16-34. Analele Stiintifice ale Universitatii 'Al. I. Cuza' DIN Iasi.
32. Zǎlinescu, C. 2002. Convex Analysis in General Vector Spaces, World Scientific, Singapore.
33. Zowe, J. 1975. A duality theorem for a convex programming problem in order complete vector spaces. J. Math. Anal. Appl., 50: 273-287.