Research ArticleOpen AccessGoogle Scholar indexed
Multiobjective Duality in Variational Problems with Higher Order Derivatives
- 1
- 2
Communications and Network·Volume 02 (2010)·Pages 138–144·Published 31 May 2010·DOI10.4236/cn.2010.22021
Copy link · social · email
Abstract
A multiobjective variational problem involving higher order derivatives is considered and optimality condi-tions for this problem are derived. A Mond-Weir type dual to this problem is constructed and various duality results are validated under generalized invexity. Some special cases are mentioned and it is also pointed out that our results can be considered as a dynamic generalization of the already existing results in nonlinear programming.
KeywordsMultiobjective Variational ProblemEfficiencyDualityPseudoinvexityQuasinvexityNonlinear Programming
- R. Courant and D. Hilbert, “Methods of Mathematical Physics,” Wiley, New York, Vol. 1, 1943.
- K. O. Friedrichs, “Ein Verfrahren der Variations-Rechung das Minimum eines Integrals Maximum eines Anderen Ausdruckes Dazustellan, Göttingen Nachrichten, 1929.
- M. A. Hanson, “Bonds for Functionally Convex Optimal Control Problems,” Journal of Mathematical Analysis and Applications, Vol. 8, No. 1, February 1964, pp. 84-89.
- B. Mond and M. A. Hanson, “Duality for Variational Problems”, Journal of Mathematical Analysis and Appli-cations, Vol. 18, No. 2, May 1967, pp. 355-364.
- F. A. Valentine, “The Problem of Lagrange with Diffe-rential Inequalities as Added Side Conditions,” Contribu-tions to the Calculus of Variations, 1933-1937, University of Chicago Press, 1937, pp. 407-448.
- C. R. Bector, S. Chandra and I. Husain, “Generalized Concavity and Duality in Continuous Programming,” Utilitas Mathematica, Vol. 25, 1984, pp. 171-190.
- S. Chandra, B. D. Craven and I. Husain, “A Class of Nondifferentiable Continuous Programming Problems,” Journal of Mathematical Analysis Applications, Vol. 107, No. 1, April 1985, pp. 122-131.
- S. Chandra, B. D. Craven and I. Husain, “Continuous Programming Containing Arbitrary Norms,” Journal of Australian Mathematical Society (Series A), Vol. 39, No. 1, 1985, pp. 28-38.
- I. Husain and Z. Jabeen, “On Variational Problems In-volving Higher Order Derivatives,” Journal of Applied Mathematics and Computing, Vol. 27, No. 1-2, March 2005, pp. 433-455.
- [B. Mond and S. Chandra and I. Husain, “Duality of Vari-ational Problems with Invexity”, Journal of Mathematical Analysis and Applications, Vol. 134, No. 2, September 1988, pp. 322-328.
- C. R. Bector and I. H. Husain, “Duality for Multiobjective Variational Problems,” Journal of Mathematical Analysis and Applications, Vol. 166, No. 1, 1 May 1992, pp. 214-224.
- X. H. Chen, “Duality for Multiobjective Variational Problems with Invexity” Journal of Mathematical Analysis and Appliactions, Vol. 203, No. 1, October 1996, pp. 236-253.
- B. Mond and I. Smart, “Duality with Invexity for a Class of Nondifferentiable Static and Continuous Programming Problems,” Journal of Mathematical Analysis and Appli-cations, Vol. 136, 1988, pp. 325-333.
- V. Chankong and Y. Y. Haimes, “Multiobjective Decision Making: Theory and Methodology,” North Holland, New York, 1983.