Benson type algorithms for linear vector optimization and applications

Journal of Global Optimization

Volumn 59, Issue 4, 2014, Pages 811-836

  Hamel, Andreas H ^a   Löhne, Andreas ^b   Rudloff, Birgit ^c  

^a YESHIVA UNIVERSITY   (United States)

^b Princeton University ^*

^c PRINCETON UNIVERSITY   (United States)

Author keywords

Algorithms; Duality; Linear programming; Multiple objective optimization; Outer approximation; Set valued risk measure; Transaction costs; Vector optimization

Indexed keywords

ALGORITHMS; ITERATIVE METHODS; LINEAR PROGRAMMING; MULTIOBJECTIVE OPTIMIZATION;

DUALITY; MULTIPLE-OBJECTIVE OPTIMIZATION; OUTER APPROXIMATION; RISK MEASURES; TRANSACTION COST; VECTOR OPTIMIZATIONS;

APPROXIMATION ALGORITHMS;

EID: 84904728128     PISSN: 09255001     EISSN: 15732916     Source Type: Journal    
DOI: 10.1007/s10898-013-0098-2     Document Type: Article

References (33)

1. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203-228 (1999)
2. Barber, C., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22(4), 469-483 (1996)
3. Benson, H.: Further analysis of an outcome set-based algorithm for multiple-objective linear programming. J. Optimiz. Theory Appl. 97(1), 1-10 (1998a)
4. Benson, H.: An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Global Optimiz. 13, 1-24 (1998b)
5. Bremner, D., Fukuda, K., Marzetta, A.: Primal - dual methods for vertex and facet enumeration. Discrete Comput. Geom. 20(3), 333-357 (1998)
6. Csirmaz, L.: Using multiobjective optimization to map the entropy region of four random variables. http://eprints.renyi.hu/66/ (2013)
7. Ehrgott, M.: Solving multiobjective linear programmes - from primal methods in decision space to dual methods in outcome space. In: Book of Abstracts, Second South Pacific Conference on Mathematics, Noumea, New Caledonia, August 30th to September 3rd, (2010). http://pages.univ-nc.nc/bonnel/ spcm-2010/confspcm10.htm
8. Ehrgott, M., Löhne, A., Shao, L.: A dual variant of Benson's outer approximation algorithm. Report 654, University of Auckland School of Engineering (2007)
9. Ehrgott, M., Shao, L., Schöbel, A.: An approximation algorithm for convex multi-objective programming problems. J. Global Optimiz. 50(3), 397-416 (2011)
10. Ehrgott, M., Löhne, A., Shao, L.: A dual variant of Benson's outer approximation algorithm. J. Global Optimiz. 52(4), 757-778 (2012)
11. Föllmer, H., Schied, A.: Stochastic Finance, Extended Edition. Walter de Gruyter, Berlin (2011)
12. Hamel, A.H., Löhne, A.: Lagrange duality in set optimization (submitted) (2012). arXiv:1207.4433.
13. Hamel, A.H.: A duality theory for set-valued functions I: Fenchel conjugation theory. J Set-Valued Var. Anal. 17, 153-182 (2009)
14. Hamel, A.H.: A Fenchel-Rockafellar duality theorem for set-valued optimization. Optimization 60(7-9), 1023-1043 (2011)
15. Hamel, A.H., Heyde, F.: Duality for set-valued measures of risk. SIAM J. Financ. Math. 1, 66-95 (2010)
16. Hamel, A.H., Heyde, F., Löhne, A., Tammer, C., Winkler, K.: Closing the duality gap in linear vector optimization. J. Convex Anal. 11(1), 163-178 (2004)
17. Hamel, A.H., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Math. Financ. Econ. 5, 1-28 (2011)
18. Hamel, A.H., Rudloff, B., Yankova, M.: Set-valued average value at risk and its computation. Math. Financ. Econ. 7(2), 229-246 (2013)
19. Heyde, F.: Geometric duality for convex vector optimization problems. (submitted) (2011). arXiv:1109.3592v1.
20. Heyde, F., Löhne, A.: Geometric duality in multiple objective linear programming. SIAM J. Optimiz. 19(2), 836-845 (2008)
21. Heyde, F., Löhne, A.: Solution concepts in vector optimization: A fresh look at an old story. Optimization 60(12), 1421-1440 (2011)
22. Heyde, F., Löhne, A., Tammer, C.: The attainment of the solution of the dual program in vertices for vectorial linear programs. In: Barichard, V. et al. (Eds.) Multiobjective programming and goal programming. Theoretical results and practical applications. Lecture Notes in Economics and Mathematical Systems vol. 618. Springer, pp. 13-24 (2009a)
23. Heyde, F., Löhne, A., Tammer, C.: Set-valued duality theory for multiple objective linear programs and application to mathematical finance. Math. Methods Oper. Res. 69(1), 159-179 (2009b)
24. Jouini, E., Meddeb, M., Touzi, N.: Vector-valued coherent risk measures. Finance Stoch. 8, 531-552 (2004)
25. Korn, R., Müller, S.: The decoupling approach to binomial pricing of multi-asset options. J. Comput. Finance 12(3), 1-30 (2009)
26. Löhne, A.: Vector Optimization with Infimum and Supremum. Springer, Berlin (2011)
27. Löhne, A., Rudloff, B.: An algorithm for calculating the set of superhedging portfolios and strategies in markets with transaction costs. (submitted) (2011). arXiv:1107.5720v1.
28. Löhne, A., Schrage, C.: An algorithm to solve polyhedral convex set optimization problems. Optimization 62(1), 131-141 (2013)
29. Luc, D.T.: On duality in multiple objective linear programming. Eur. J. Oper. Res. 210, 158-168 (2011)
30. Rockafellar, R.T., Uryasev, S.P.: Optimization of conditional value-at-risk. J. Risk 2, 21-42 (2000)
31. Ruszczyński, A., Vanderbei, R.J.: Frontiers of stochastically nondominated portfolios. Econometrica 71(4), 1287-1297 (2003)
32. Shao, L., Ehrgott, M.: Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning. Math. Methods Oper. Res. 68(2), 257-276 (2008a)
33. Shao, L., Ehrgott, M.: Approximating the nondominated set of an MOLP by approximately solving its dual problem. Math. Methods Operat. Res. 68(3), 469-492 (2008b)