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Optimality and Wolfe Duality for Multiobjective Programming Problems Involving n-set Functions

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Generalized Convexity and Generalized Monotonicity

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 502))

Abstract

In this paper we establish some optimality and Wolfe duality results under generalized convexity assumptions for a multiobjective programming problem involving d-type-I and related n-set functions.

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References

  1. C.R. BECTOR, D. BHATIA and S. PANDEY, Duality for multiobjective fractional programming involving n-set functions, J. Math. Anal. Appl. 186 (1994) 3, 747–768.

    Article  Google Scholar 

  2. C.R. BECTOR and M. SINGH, Duality for multiobjective B-vex programming involving n-set functions, J. Math. Anal. Appl. 202 (1996) 3, 701–726.

    Article  Google Scholar 

  3. D. BEGIS and R. GLOWINSKI, Application de la méthode des éléments finis à l’approximation d’une probléme de domaine optimal: Méthodes de résolution de problémes approchés, Appl. Math. Optim., 2 (1975) 2, 130–169.

    Article  Google Scholar 

  4. J. CEA, A. GIOAN and J. MICHEL, Quelques resultats sur l’identification de domaines, Calcolo, 10 (1973), 133–145.

    Article  Google Scholar 

  5. J.H. CHOU, W.S. HSIA and T.Y. LEE, On multiple objective programming problems with set functions, J. Math. Anal. Appl., 105 (1985) 2, 383–394.

    Article  Google Scholar 

  6. J.H. CHOU, W.S. HSIA and T.Y. LEE, Second order optimality conditions for mathematical programming with set functions, J. Austral. Math. Soc. Ser. B, 26 (1985), 284–292.

    Article  Google Scholar 

  7. J.H. CHOU, W.S. HSIA and T.Y. LEE, Epigraphs of convex set functions, J. Math. Anal. Appl., 118 (1986) 1, 247–254.

    Article  Google Scholar 

  8. H.W. CORLEY, Optimization theory for n-set functions, J. Math. Anal. Appl., 127 (1987) 1, 193–205.

    Article  Google Scholar 

  9. H.W. CORLEY and S.D. ROBERTS, A partitioning problem with applications in regional design, Oper. Res., 20(1972), 1010–1019.

    Article  Google Scholar 

  10. G. DANTZIG and A. WALD, On the fundamental lemma of Neyman and Pearson, Ann. Math. Stat., 22 (1951), 87–93.

    Article  Google Scholar 

  11. R.R. EGUDO Multiobjective fractional duality, Bull. Austral. Math. Soc., 37 (1988) 3, 367–378.

    Article  Google Scholar 

  12. R.V. HOGG and A.T. CRAIG, Introduction to Mathematical Statistics, Macmillan Co., New Jork, 1978.

    Google Scholar 

  13. W.S. HSIA and T.Y. LEE, Proper D-solutions of multiobjective programming problems with set functions, J. Optim. Theory Appl., 53(1987), 247–258.

    Article  Google Scholar 

  14. V. JEYAKUMAR and B. MOND, On generalised convex mathematical programming, J. Austral. Math. Soc, Ser. B, 34(1992) 1, 43–53.

    Article  Google Scholar 

  15. D.S. KIM, C.L. JO and G.M. LEE, Optimality and duality for multiobjective fractional programming involving n-set functions, J. Math. Anal. Appl., 224(1998) 1, 1–13.

    Article  Google Scholar 

  16. H.C. LAI and S.S. YANG, Saddlepoint and duality in the optimization theory of convex set functions, J. Austral. Math. Soc. Ser. B, 24 (1982), 130–137.

    Article  Google Scholar 

  17. H.C. LAI, S.S. YANG and G.R. HWANG, Duality in mathematical programming of set functions: on Fenchel duality theorem, J. Math. Anal. Appl., 95 (1983) 1, 223–234.

    Article  Google Scholar 

  18. L.J. LIN, Optimality of differentiable, vector-valued n-set functions, J. Math. Anal. Appl., 149 (1990) 1, 255–270.

    Article  Google Scholar 

  19. L.J. LIN, On the optimality conditions of vector-valued n-set functions, J. Math. Anal. Appl., 161 (1991) 2, 367–387.

    Article  Google Scholar 

  20. L.J. LIN, Duality theorems of vector valued n-set functions, Comput. Math. Appl., 21 (1991), 165–175.

    Article  Google Scholar 

  21. L.J. LIN, On optimality of differentiable nonconvex n-set functions, J. Math. Anal. Appl., 168 (1992), 351–366.

    Article  Google Scholar 

  22. O.L. MANGASARIAN, Nonlinear Programming, McGraw-Hill, New York, 1969.

    Google Scholar 

  23. P. MAZZOLENI, On constrained optimization for convex set functions, In: Survey of Mathematical Programming, Vol.1, A. PREKOPA (ed.), North-Holland, Amsterdam, 1979, pp.273–290.

    Google Scholar 

  24. B. MOND and T. WEIR, Generalized concavity and duality, In: Generalized Concavity in Optimization and Economics, S. Schaible and W.T. Ziemba (eds.), Academic Press, New York 1981, 263–279.

    Google Scholar 

  25. R.J.T. MORRIS, Optimization Problem Involving Set Functions, Ph. D. dissertation, University of California, Los Angeles, 1978.

    Google Scholar 

  26. R.J.T. MORRIS, Optimal constrained selection of a measurable subset, J. Math. Anal. Appl. 70(1979) 2, 546–562.

    Article  Google Scholar 

  27. R.N. MUKHERJEE, Generalized convex duality for multiobjective fractional programs, J. Math. Anal. Appl. 162 (1991), 309–316.

    Article  Google Scholar 

  28. J. NEYMANN and E.S. PEARSON, On the problem of the most efficient tests of statistical hypotheses, Philos. Trans. Roy. Soc. London, Ser.A, 231 (1933), 289–337.

    Article  Google Scholar 

  29. V. PREDA, On minmax programming problems containing n-set functions, Optimization, 22(1991) 4, 527–537.

    Article  Google Scholar 

  30. V. PREDA, On duality of multiobjective fractional measurable subset selection problems, J. Math. Anal. Appl., 196 (1995), 514–525.

    Article  Google Scholar 

  31. V. PREDA and I.M. STANCU-MINASIAN, Mond-Weir duality for multiobjective mathematical programming with n-set functions, Analele Universitatii Bucuresti, Matematica-Informatica 46 (1997), 89–97.

    Google Scholar 

  32. V. PREDA and I.M. STANCU-MINASIAN, Mond-Weir duality for multiobjective mathematical programming with n-set functions, Rev. Roumaine Math. Pures Appl. 44(1999) 4, 629–644.

    Google Scholar 

  33. J. ROSENMULLER, Some properties of convex set functions, Arch. Math., 22 (1971), 420–430.

    Article  Google Scholar 

  34. J. ROSENMULLER and H.G. WEIDNER, A class of extreme convex set functions with finite carrier, Adv. Math., 10(1973), 1–38.

    Article  Google Scholar 

  35. J. ROSENMULLER and H.G. WEIDNER, Extreme convex set functions with finite carrier: General theory, Discrete Math, 10 (1974), 343–382.

    Article  Google Scholar 

  36. S.K. SUNEJA and M.K. SRIVASTAVA, Optimality and duality in nondiffer-entiable multiobjective optimization involving d-type I and related functions, J. Math. Anal. Appl., 206(1997) 2, 465–479.

    Article  Google Scholar 

  37. K. TANAKA and Y. MARUYAMA, The multiobjective optimization problem of set function, J. Inform. Optim. Sci., 5 (1984) 3, 293–306.

    Google Scholar 

  38. P.K.C. WANG, On a class of optimization problems involving domain variations. Lectures Notes in Control and Information Sciences, Vol.2., Springer-Verlag, Berlin, 1977, 49–60.

    Google Scholar 

  39. T. WEIR and B. MOND, Generalized convexity and duality in multiple objective programming, Bull. Austral. Math. Soc, 39(1989), 287–299.

    Article  Google Scholar 

  40. Y.L. YE, D-invexity and optimality conditions, J. Math. Anal. Appl., 162(1991), 242–249.

    Article  Google Scholar 

  41. G.J. ZALMAI, Optimality conditions and duality for constrained measurable subset selection problems with minmax objective functions, Optimization, 20(1989) 4, 377–395.

    Article  Google Scholar 

  42. G.J. ZALMAI, Optimality conditions and duality for multiobjective measurable subset selection problems, Optimization, 22 (1991) 2, 221–238.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Bucharest, Romania

    Vasile Preda

  2. Centre for Mathematical, Statistics The Romanian Academy, UK

    Ion M. Stancu-Minasian

Authors
  1. Vasile Preda
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  2. Ion M. Stancu-Minasian
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of the Aegean, 83200, Karlovassi/Samos, Greece

    Nicolas Hadjisavvas

  2. CODE and Department of Economics, Universitat Autonoma de Barcelona, 08193, Bellaterra, Spain

    Juan Enrique Martínez-Legaz

  3. Department of Mathematics Faculty of Sciences, Université de Pau, 64000, Pau, France

    Jean-Paul Penot

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© 2001 Springer-Verlag Berlin Heidelberg

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Preda, V., Stancu-Minasian, I.M. (2001). Optimality and Wolfe Duality for Multiobjective Programming Problems Involving n-set Functions. In: Hadjisavvas, N., Martínez-Legaz, J.E., Penot, JP. (eds) Generalized Convexity and Generalized Monotonicity. Lecture Notes in Economics and Mathematical Systems, vol 502. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56645-5_25

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  • DOI: https://doi.org/10.1007/978-3-642-56645-5_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41806-1

  • Online ISBN: 978-3-642-56645-5

  • eBook Packages: Springer Book Archive

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