Destek fonksiyonları içeren varyasyonel problemler için simetrik dualite

Year 2010, Volume: 39 Issue: 1, 1 - 20, 03.12.2009

Iqbal Husain Rumana Mattoo

Abstract

Destek fonksiyonlu çok amaçlı varyasyonel problemler için Wolfe ve Mond-Weir t ipi simetrik dual modeller formüle edilmiştir. Bu çeşit problemler için; zayıf, güçlü ve karşıt dualite teoremleri fonksiyonellerin belirli kombinasyonları üzerine konvekslik-konkavlık ve pseudo-konvekslik, pseudo-konkavlık varsayımları altında geçerli kılınmıştır. İki çift için de öz dualite teoremleri kurulmuştur. Doğal sınır değerli problemler formüle edilmiştir. Ayrıca dualite sonuçlarımızın, destek fonksiyoları gibi diferansiyellenemeyen ifadelere sahip olan nonlineer programlama problemlerinin dinamik genelleştirmeleri olarak kabul edilebileceğine dikkat çekilmektedir.

Keywords

Çok amaçlı, varyasyonel problemler, destek fonksiyonlar, simetrik dualite, öz dualite, konvekslik-konkavlık, pseudokonvekslik- pseudokonkavlık

Symmetric duality for multiobjective variational problems containing support functions

Abstract

Wolfe and Mond-Weir type symmetric dual models for multiobjective variational problems with support functions are formulated. For these pairs of problems, weak, strong and converse d uality t heorems a re v alidated u nder convexity-concavity and pseudoconvexity, p seudo-concavity a ssumptions o n certain c ombination off unctionals. Self duality theorems for both pairs are established. The problems with natural boundary values are formulated. It is also pointed out that our duality results can be regarded as dynamic generalizations of nonlinear programming problems h aving nondifferentiable terms as support functions. 

Keywords

Multiobjective, variational problems, support functions, symmetric duality, self duality, convexity-concavity, pseudoconvexity-pseudoconcavity

References

• W.S. Dorn, “ Asymmetric d ual theorem f or q uadratic p rograms. Journal o f Operations Research Society of Japan. 2, 93-97 (1960).
• G.B. Dantzig, E isenberg a nd R. W.Cottle, “ Symmetric d ual n onlinear p rograms”. Pacific Journal of Mathematics. 15, 809-812 (1965).
• B. Mond, “A symmetric dual theorem for nonlinear programs”. Quaterly Jounal of Applied Mathematics. 23, 265-269 (1965). [4] M.S. Bazaraa, J.J. Goode, “On symmetric duality in nonlinear programming”. Operations Research. 21, (1), 1-9 (1973).
• B. Mond, R.W. Cottle, “Self duality i n m athematical pr ogramming”, SI AM. J.Appl.Math. 14, 420-423 (1966).
• B. Mond, T.Weir, “Generalized concavity and duality”, in: S.Sciable, W.T.Ziemba (Eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 1981.
• B. Mond, M.A. Hanson, “Symmetric Duality for Variational problems”. J.Math. Anal. Appl. 18, 161-172 (1967).
• C.R. Bector, S. Chandra, I. Husain, “Generalized concavity and duality in continuous programming. Utilitas Mathematica. 25, 171-190 (1984).
• S. Chandra, I. Husain, “Symmetric dual continuous fractional programming”. J. Inf. Opt. Sc. 10, 241-255 (1989).
• C.R. Bector, I. Husain, “ Duality f or mu ltiobjective v ariational p roblems. Journal of Math. Anal and Appl. 166, No.1, 214-224 (1992).
• Gulati, I . H usain, A. Ahmed, “ Multiobjective s ymmetric d uality w ith i nvexity”. Bulletin of the Australian Mathematical Society. 56, 25-36 (1997).
• I. Husain, Z. J abeen”, “Continuous p rogramming c ontaining s upport f unctions”. Journal of Applied Mathematics and Informatics. Vol.26, No 1-2, 75-106 (2008).
• B.D. Craven, “Lagrangian conditions and quasi-duality”. Bulletin of Australian Mathematical Society. 16, 325-339 (1977).
• I. Husain, A. Ahmed, R.G. Mattoo, “On Multiobjective nonlinear programming with support functions”. Journal of Applied Analysis. 2010 (article in press).
• S.K. Mishra, S.Y. Wang, K.K. Lai, “Symmetric duality for a class of nondifferentiable multiobjective fractional variational problems”. Journal of Math. Anal and Appl. 333, No. 2, 1093-1110 (2007).