Abstract
We introduce a special class of monotonic functions with the help of support functions and polar sets, and use it to construct a scalarized problem and its dual for a vector optimization problem. The dual construction allows us to develop a new method for generating weak efficient solutions of a concave vector maximization problem and establish its convergence. Some numerical examples are given to illustrate the applicability of the method.
Similar content being viewed by others
References
Attouch H. (1984). Variational convergence for functions and operators. Pitman, London
Benson H.P. (1998). An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Global Optim. 13:1–24
Benson H.P., Sun E. (2000). Outcome space partition of the weight set in multiobjective linear programming. J. Optim. Theory Appl. 105:17–36
Chen P.C., Hansen P., Jaumard B. (1991). On-line and off-line vertex enumeration by adjacency lists. Operat. Res. Lett. 10:403–409
Das I., Dennis J.E. (1998). Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optimi. 8:631–657
Horst R., Tuy H. (1996). Global optimization. Springer-Verlag, Berlin
Holmes R.B. (1975). Geometric functional analysis and its applications. Springer, Berlin
Jahn J. (2004). Vector Optimization: theory, applications, and extensions. Springer, Berlin
Kim N.T.B., Luc D.T. (2000). Normal cones to a polyhedral convex set and generating efficient faces in linear multiobjective programming. Act. Math. Vietnamica 25:101–124
Luc, D.T.: On scalarizing method in vector optimization. In: Fandel, G., Grauer, M., Kurzhanski, A., Wiersbicki, A.P. (eds.) Large-Scale Modeling and Interactive Decision Analysis, Proceedings, Eisenach 1985. LNEMS 273, 46–51 (1986)
Luc D.T. (1987). Scalarization of vector optimization problems. J. Optim. Theory Appl. 55:85–102
Luc D.T. (1989). Theory of vector optimization. LNEMS 319. Springer-Verlag, Germany
Luc D.T., Jahn J. (1992). Axiomatic approach to duality in vector optimization. Numer. Funct. Anal. Optim. 13:305–326
Luc D.T., Phong T.Q., Volle M. (2005). Scalarizing functions for generating the weakly efficient solution set in convex mutiobjective problems. SIAM J. Optim. 15:987–1001
Miettinen K. (1999). Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston
Rakowska J., Hafka R.T., Watson L.T. (1991). Tracing the efficient curve for multi-objective control-structure optimization. Comput. Syst. Eng. 2:461–471
Singer I. (1979). A Fenchel-Rockafellar type duality theorem for maximization. Bull. Australian Math. Soc. 20:193–198
Stewart T.J., van den Honert R.C. (eds) (1997). Trends in Multicriteria Decision Making, Lectures Notes in Economics and Mathematical Systems vol. 465. Springer, Berlin
Toland J. (1979). A duality principle for nonconvex optimization and the calculus of variations. Arch. Rational Mech. Anal. 71: 41–61
Yu P.L. (1985). Multiple-criteria decision making: concepts, techniques and extensions. Plenum Press, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
The, L.D., Quynh, P.T. & Michel, V. A New Duality Approach to Solving Concave Vector Maximization Problems. J Glob Optim 36, 401–423 (2006). https://doi.org/10.1007/s10898-006-9018-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-006-9018-z