Primal and dual approximation algorithms for convex vector optimization problems
First Online: 12 January 2014 Received: 30 August 2013 Accepted: 20 December 2013 DOI:
Cite this article as: Löhne, A., Rudloff, B. & Ulus, F. J Glob Optim (2014) 60: 713. doi:10.1007/s10898-013-0136-0 Abstract
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. Keywords Vector optimization Multiple objective optimization Convex programming Duality Algorithms Outer approximation
B. Rudloff: Research supported by NSF award DMS-1007938.
Ararat, Ç, Hamel, A.H., Rudloff, B.: Set-valued shortfall and divergence risk measures. submitted (2013)
Benson, H.P.: An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Glob. Optim.
, 1–24 (1998)
Bremner, D., Fukuda, K., Marzetta, A.: Primal-dual methods for vertex and facet enumeration. Discrete Comput. Geom.
(3), 333–357 (1998)
Csirmaz L.: Using multiobjective optimization to map the entropy region of four random variables. Preprint
CVX Research. Inc.: CVX: Matlab software for disciplined convex programming, version 2.0 beta., September 2012
Ehrgott, M., Löhne, A., Shao, L.: A dual variant of Benson’s outer approximation algorithm. J. Glob. Optim.
(4), 757–778 (2012)
Ehrgott, M., Shao, L., Schöbel, A.: An approximation algorithm for convex multi-objective programming problems. J. Glob. Optim.
(3), 397–416 (2011)
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, A.H., Löhne, A.: Lagrange duality in set optimization. J. Optim. Theory Appl. (2013). doi:
Hamel, A.H., Löhne, A., Rudloff, B.: A Benson type algorithm for linear vector optimization and applications. J. Glob. Optim. (2013). doi:
Hamel, A.H., Rudloff, B., Yankova, M.: Set-valued average value at risk and its computation. Math. Financ. Econ.
(2), 229–246 (2013)
Heyde, F.: Geometric duality for convex vector optimization problems. J. Convex Anal.
20(3), 813–832 (2013)
Heyde, F., Löhne, A.: Geometric duality in multiple objective linear programming. SIAM J. Optim.
(2), 836–845 (2008)
Heyde, F., Löhne, A.: Solution concepts in vector optimization: a fresh look at an old story. Optimization
(12), 1421–1440 (2011)
Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2004)
Kabanov, Y.M.: Hedging and liquidation under transaction costs in currency markets. Financ. Stoch.
, 237–248 (1999)
Löhne, A.: Vector Optimization with Infimum and Supremum. Springer, Berlin (2011)
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, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Ruzika, S., Wiecek, M.M.: Approximation methods in multiobjective programming. J. Optim. Theory Appl.
(3), 473–501 (2005)
Shao, L., Ehrgott, M.: Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning. Math. Methods Oper. Res.
(2), 257–276 (2008)
Shao, L., Ehrgott, M.: Approximating the nondominated set of an MOLP by approximately solving its dual problem. Math. Methods Oper. Res.
(3), 469–492 (2008)
CrossRef Copyright information
© Springer Science+Business Media New York 2014