Primal and dual approximation algorithms for convex vector optimization problems
 Andreas Löhne,
 Birgit Rudloff,
 Firdevs Ulus
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Two approximation algorithms for solving convex vector optimization problems (CVOPs) are provided. Both algorithms solve the CVOP and its geometric dual problem simultaneously. The first algorithm is an extension of Benson’s outer approximation algorithm, and the second one is a dual variant of it. Both algorithms provide an inner as well as an outer approximation of the (upper and lower) images. Only one scalar convex program has to be solved in each iteration. We allow objective and constraint functions that are not necessarily differentiable, allow solid pointed polyhedral ordering cones, and relate the approximations to an appropriate \(\epsilon \) solution concept. Numerical examples are provided.
 Ararat, Ç, Hamel, A.H., Rudloff, B.: Setvalued shortfall and divergence risk measures. submitted (2013)
 Benson, HP (1998) An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Glob. Optim. 13: pp. 124 CrossRef
 Bremner, D, Fukuda, K, Marzetta, A (1998) Primaldual methods for vertex and facet enumeration. Discrete Comput. Geom. 20: pp. 333357 CrossRef
 Csirmaz L.: Using multiobjective optimization to map the entropy region of four random variables. Preprint http://eprints.renyi.hu/66/2/globopt.pdf (2013)
 CVX Research. Inc.: CVX: Matlab software for disciplined convex programming, version 2.0 beta., September 2012
 Ehrgott, M, Löhne, A, Shao, L (2012) A dual variant of Benson’s outer approximation algorithm. J. Glob. Optim. 52: pp. 757778 CrossRef
 Ehrgott, M, Shao, L, Schöbel, A (2011) An approximation algorithm for convex multiobjective programming problems. J. Glob. Optim. 50: pp. 397416 CrossRef
 Ehrgott, M., Wiecek, M. M.: Multiobjective Programming. In: Figueira, J., Greco, S., Ehrgott, M., (eds.) Multiple Criteria Decision Analysis: State of the Art Surveys. Springer Science + Business Media, Berlin, pp. 667–722 (2005)
 Grant, M., Boyd, S.: Recent advances in learning and control, chapter Graph implementations for nonsmooth convex programs. Lecture Notes in Control and Information Sciences. Springer, Berlin, pp. 95–110 (2008)
 Hamel, AH, Löhne, A (2013) Lagrange duality in set optimization. J. Optim. Theory Appl..
 Hamel, AH, Löhne, A, Rudloff, B (2013) A Benson type algorithm for linear vector optimization and applications. J. Glob. Optim..
 Hamel, AH, Rudloff, B, Yankova, M (2013) Setvalued average value at risk and its computation. Math. Financ. Econ. 7: pp. 229246 CrossRef
 Heyde, F (2013) Geometric duality for convex vector optimization problems. J. Convex Anal. 20: pp. 813832
 Heyde, F, Löhne, A (2008) Geometric duality in multiple objective linear programming. SIAM J. Optim. 19: pp. 836845 CrossRef
 Heyde, F, Löhne, A (2011) Solution concepts in vector optimization: a fresh look at an old story. Optimization 60: pp. 14211440 CrossRef
 Jahn, J (2004) Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin CrossRef
 Kabanov, YM (1999) Hedging and liquidation under transaction costs in currency markets. Financ. Stoch. 3: pp. 237248 CrossRef
 Löhne, A (2011) Vector Optimization with Infimum and Supremum. Springer, Berlin CrossRef
 Löhne, A., Rudloff, B.: An algorithm for calculating the set of superhedging portfolios in markets with transaction costs. Int. J. Theor. Appl. Finance. (to appear)
 Luc, D.: Theory of vector optimization. In: Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)
 Rockafellar, RT (1970) Convex Analysis. Princeton University Press, Princeton
 Ruzika, S, Wiecek, MM (2005) Approximation methods in multiobjective programming. J. Optim. Theory Appl. 126: pp. 473501 CrossRef
 Shao, L, Ehrgott, M (2008) Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning. Math. Methods Oper. Res. 68: pp. 257276 CrossRef
 Shao, L, Ehrgott, M (2008) Approximating the nondominated set of an MOLP by approximately solving its dual problem. Math. Methods Oper. Res. 68: pp. 469492 CrossRef
 Title
 Primal and dual approximation algorithms for convex vector optimization problems
 Journal

Journal of Global Optimization
Volume 60, Issue 4 , pp 713736
 Cover Date
 20141201
 DOI
 10.1007/s1089801301360
 Print ISSN
 09255001
 Online ISSN
 15732916
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Vector optimization
 Multiple objective optimization
 Convex programming
 Duality
 Algorithms
 Outer approximation
 Industry Sectors
 Authors

 Andreas Löhne ^{(1)}
 Birgit Rudloff ^{(2)} ^{(3)}
 Firdevs Ulus ^{(2)}
 Author Affiliations

 1. Department of Mathematics, MartinLutherUniversität HalleWittenberg, NWF II, 06099 , Halle (Saale), Germany
 2. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, 08544, USA
 3. Bendheim Center for Finance, Princeton University, Princeton, NJ, 08544, USA