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A New Approach to Multiobjective Programming with a Modified Objective Function

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Abstract

In this paper, optimality for multiobjective programming problems having invex objective and constraint functions (with respect to the same function η) is considered. An equivalent vector programming problem is constructed by a modification of the objective function. Furthermore, an η-Lagrange function is introduced for a constructed multiobjective problem and modified saddle point results are presented.

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References

  1. Beato-Moreno, A., Ruiz-Canales, P., Luque-Calvo, P.-L. and Blanquero-Bravo, R. (1998), Multiobjective quadratic problem: characterization of the efficient points, in: Crouzeix, J.P. et al.(eds.), Generalized Convexity, Generalized Monotonicity, Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  2. Bazaraa, M.S., Sherali, H.D. and Shetty, C.M. (1991). Nonlinear Programming: Theory and Algorithms, John Wiley and Sons, New York.

    Google Scholar 

  3. Ben-Israel, A. and Mond, B. (1986), What is invexity?, Journal of Australian Mathematical Society Ser. B28, 1–9.

    Google Scholar 

  4. Brumelle, S. (1981), Duality for multiple objective convex programs, Mathematics of Operations Research 6, 159–172.

    Google Scholar 

  5. Craven, B.D. (1981), Invex functions and constrained local minima, Bulletin of the Australian Mathematical Society 24, 357–366.

    Google Scholar 

  6. Craven, B.D. (1981), Vector-valued optimization. In: Schaible, S. and Ziemba, W.T. (eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, pp. 661–687.

    Google Scholar 

  7. Craven, B.D. and Glover, B.M. (1985), Invex functions and duality, Journal of Australian Mathematical Society Ser. A39, 1–20.

    Google Scholar 

  8. Das, L.N. and Nanda, S. (1995), Proper efficiency conditions and duality for multiobjective programming problems involving semilocally invex functions, Optimization 34, 43–51.

    Google Scholar 

  9. Egudo, R.R. and Hanson, M.A. (1987), Multi-objective duality with invexity, Journal of Mathematical Analysis and Applications 126, 469–477.

    Google Scholar 

  10. Geoffrion, M.A. (1968), Proper efficiency and the theory of vector maximization, Journal of Mathematical Analysis and Applications 22, 613–630.

    Google Scholar 

  11. Giorgi, G. and Guerraggio, A. The notion of invexity in vector optimization: smooth and nonsmooth case. In: Crouzeix, J.P., Martinez-Legaz, J.E. and Volle, M. (eds.), Generalized Convexity, Generalized Monotonicity, Procedings of the Fifth Symposium on Generalized Convexity, Luminy, France, 1997, Kluwer Academic Publishers.

  12. Hanson, M.A. (1981), On sufficiency of the Kuhn–Tucker conditions, Journal of Mathematical Analysis and Applications 80, 545–550.

    Google Scholar 

  13. Ivanov, E.H. and Nehse, R. (1985), Some results on dual vector optimization problems, Optimization 4, 505–517.

    Google Scholar 

  14. Jeyakumar, V. and Mond, B (1992), On generalized convex mathematical programming, Journal of Australian Mathematical Society Ser. B34 43–53.

    Google Scholar 

  15. Kanniappan, P. (1983), Necessary conditions for optimality of nondifferentiable convex multiobjective programming, Journal of Optimization Theory and Applications 40, 167–174.

    Google Scholar 

  16. Mangasarian, O.L. (1969), Nonlinear Programming, McGraw-Hill, New York.

    Google Scholar 

  17. Pareto, V. (1986), Cours de Economie Politique, Rouge, Lausanne, Switzerland.

    Google Scholar 

  18. Ruiz-Canales, P. and Rufián-Lizana, A. (1995), A characterization of weakly efficient points, Mathematical Programming 68, 205–212.

    Google Scholar 

  19. Singh, C. (1987), Optimality Conditions in multiobjective differentiable programming, Journal of Optimization Theory and Applications 53, 115–123.

    Google Scholar 

  20. Tanino, T. and Sawaragi, Y. (1979), Duality theory in multiobjective programming, Journal of Optimization Theory and Applications 27, 509–529.

    Google Scholar 

  21. Weir, T., Mond, B. and Craven, B.D. (1986), On duality for weakly minimized vector valued optimization problems, Optimization 17, 711–721.

    Google Scholar 

  22. Weir, T. (1988) A note on invex functions and duality in multiple objective optimization, Opsearch 25, 98–104.

    Google Scholar 

  23. Weir, T. and Mond, B. (1989), Generalized convexity and duality in multiple objective programming, Bulletin of the Australian Mathematical Society 39, 287–299.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, University of Łódź, Banacha 22, 90-238, Łódź, Poland

    Tadeusz Antczak

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  1. Tadeusz Antczak
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Antczak, T. A New Approach to Multiobjective Programming with a Modified Objective Function. Journal of Global Optimization 27, 485–495 (2003). https://doi.org/10.1023/A:1026080604790

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