Abstract
Given a multiobjective optimization problem with the components of the objective function as well as the constraint functions being composed convex functions, we introduce, by using the Fenchel-Moreau conjugate of the functions involved, a suitable dual problem. Under a standard constraint qualification and some convexity as well as monotonicity conditions we prove the existence of strong duality. Finally, some particular cases of this problem are presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. I. Boţ and G. Wanka, Duality for composed convex functions with applications in location theory, in: Multi-Criteria-und Fuzzy-Systeme in Theorie and Praxis, W. Habenicht, B. Scheubrein, R. Scheubrein (Eds.), Gabler Edition Wissenschaft (2003), pp. 1–18.
R. I. Boţ and G. Wanka, An analysis of some dual problems in multiobjective optimization (I), Optimization, 53 (2004), 281–300.
R. I. Boţ, and G. Wanka, An analysis of some dual problems in multiobjective optimization (II), Optimization, 53 (2004), 301–324.
J. V. Burke and R. A. Poliquin, Optimality conditions for non-finite valued convex composite functions, Mathematical Programming, 57, Ser. B (1992), 103–120.
C. Combari, M. Laghdir and L. Thibault, Sous-différentiels de fonctions convexes composées, Ann. Sci. Math. Québec, 18 (1994), 119–148.
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland Publishing Company (Amsterdam, 1976).
K. H. Elster, R. Reinhardt, M. Schäuble and G. Donath, Einführung in die Nichtlineare Optimierung, B. G. Teubner Verlag (Leipzig, 1977).
C. J. Goh and X. Q. Yang, Duality in Optimization and Variational Inequalities, Taylor and Francis Inc. (New York, 2002).
A. D. Ioffe, Necessary and sufficient conditions for a local minimum, 1: A reduction theorem and first-order conditions, SIAM Journal of Control Optimization, 17 (1979), 245–250.
A. D. Ioffe, Necessary and sufficient conditions for a local minimum, 2: Conditions of Levitin-Miljutin-Osmolovskii type, SIAM Journal of Control Optimization, 17 (1979), 251–265.
A. D. Ioffe, Necessary and sufficient conditions for a local minimum, 1: Second-order conditions and augmented duality, SIAM Journal of Control Optimization, 17 (1979), 266–288.
V. Jeyakumar, Composite nonsmooth programming with Gâteaux differentiability, SIAM Journal on Optimization, 1 (1991), 30–41.
V. Jeyakumar and X. Q. Yang, Convex composite multi-objective nonsmooth programming, Mathematical Programming, 59, Ser. A (1993), 325–343.
V. Jeyakumar and X. Q. Yang, Convex composite minimization with C 1,1 functions, Journal of Optimization Theory and Applications, 86 (1995), 631–648.
S. K. Mishra and R. N. Mukherjee, Generalized convex composite multi-objective nonsmooth programming and conditional proper efficiency, Optimization, 34 (1995), 53–66.
R. T. Rockafellar, Convex Analysis, Princeton University Press (Princeton, 1970).
G. Wanka and R. I. Boţ, Multiobjective duality for convex-linear problems, in: Operations Research Proceedings 1999, K. Inderfurth, G. Schwödiauer, W. Domschke, F. Juhnke, R. Kleinschmidt, G. Wäscher (Eds.), Springer Verlag (Berlin, 2000), pp. 36–40.
G. Wanka and R. I. Boţ, A new duality approach for multiobjective convex optimization problems, Journal of Nonlinear and Convex Analysis, 3 (2002), 41–57.
G. Wanka, R. I. Boţ and E. Vargyas, Duality for the multiobjective location model involving sets as existing facilities, in: Optimization and Optimal Control, P. M. Pardalos, I. Tsevendorj, R. Enkhbat (Eds.), World Scientific Publishing (2003), 307–333.
G. Wanka, R. I. Boţ and E. Vargyas, Duality for location problems with unbounded unit balls, European Journal of Operational Research, 179 (2007), 1252–1265.
G. Wanka, R. I. Boţ and E. Vargyas, On the relations between different duals assigned to composed optimization problems, to appear in Mathematical Methods of Operations Research.
X. Q. Yang and V. Jeyakumar, First and second-order optimality conditions for convex composite multiobjective optimization, Journal of Optimization Theory and Applications, 95 (1997), 209–224.
X. Q. Yang, Second-order optimality conditions for convex composite optimization, Mathematical Programming, 81 (1998), 327–347.
C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing (Singapore, 2002).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Boţ, R.I., Vargyas, E. & Wanka, G. Conjugate duality for multiobjective composed optimization problems. Acta Math Hung 116, 177–196 (2007). https://doi.org/10.1007/s10474-007-4273-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-007-4273-0