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Global Parametric Sufficient Optimality Conditions for Semi-infinite Discrete Minmax Fractional Programming Problems Involving Generalized (η,ρ)-invex Functions

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Abstract

In this paper, we discuss a large number of sets of global parametric sufficient optimality conditions under various generalized (η, ρ)-invexity assumptions for a semi-infinite minmax fractional programming problem.

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References

  1. Ben-Israel, A., Mond, B. What is invexity? J. Austral. Math. Soc. (Series B), 28: 1–9 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brosowski, B. Parametric Semi-infinite Optimization, Peter Lang, Frankfurt a. M., 1982

  3. Craven, B.D. Invex functions and constrained local minima, Bull. Austral. Math. Soc., 24: 357–366 (1981)

    MATH  MathSciNet  Google Scholar 

  4. Fiacco, A.V., Kortanek, K.O. (Eds.) Semi-infinite Programming and Applications. Lecture Notes in Economics and Mathematical Systems, Vol.215, Springer-Verlag, Berlin, 1983

  5. Fiacco, A.V., McCormick, G.P. Nonlinear Programming: Sequential Unconstrained Minimization Techniques. SIAM, Philadelphia, 1990

  6. Gehner, K.R. Necessary and sufficient conditions for the Fritz John problem with linear equality constraints. SIAM J. Control, 12: 140–149 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  7. Giorgi, G., Guerraggio, A. Various types of nonsmooth invex functions. J. Inform. Optim. Sci., 17: 137–150 (1996)

    MATH  MathSciNet  Google Scholar 

  8. Giorgi, G., Mititelu, S. Convexités généralisées et propriétés. Rev. Roumaine Math. Pures Appl., 38: 125–172 (1993)

    MATH  MathSciNet  Google Scholar 

  9. Glashoff, K., Gustafson, S.A. Linear Optimization and Approximation, Springer-Verlag, Berlin, 1983

  10. Goberna, M.A., López, M.A. Linear Semi-Infinite Optimization, Wiley, New York, 1998

  11. Goberna, M.A., López, M.A. (Eds.) Semi-infinite Programming-Recent Advances, Kluwer, Dordrecht, 2001

  12. Gribik, P.R. Selected applications of semi-infinite programming. In: Constructive Approaches to Mathematical Models, ed. by Coffman, C.V., Fix, G.J., Academic Press, New York, 1979, 171–187

  13. Gustafson, S.A., Kortanek, K.O. Semi-infinite programming and applications. In: Mathematical Programming: The State of the Art, ed. by Bachem, A. et al., Springer-Verlag, Berlin, 1983, 132–157

  14. Hanson, M.A. On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl., 80: 545–550 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hanson, M.A., Mond, B. Further generalizations of convexity in mathematical programming. J. Inform. Optim. Sci., 3: 25–32 (1982)

    MATH  MathSciNet  Google Scholar 

  16. Henn, R., Kischka, P. ¨Uber einige Anwendungen der semi-infiniten Optimierung. Zeitschrift Oper. Res., 20: 39–58 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hettich, R. (Ed.) Semi-infinite Programming. Lecture Notes in Control and Information Sciences, Vol.7, Springer-Verlag, Berlin, 1976

  18. Hettich, R., Kortanek, K.O. Semi-infinite programming: theory, methods, and applications. SIAM Review, 35: 380–429 (1983)

    Article  MathSciNet  Google Scholar 

  19. Hettich, R., Zencke, P. Numerische Methoden der Approximation und semi-infinite Optimierung. Teubner, Stuttgart, 1982

  20. Jeyakumar, V. Strong and weak invexity in mathematical programming. Methods Oper. Res., 55: 109–125 (1985)

    MATH  MathSciNet  Google Scholar 

  21. Kanniappan, P., Pandian, P. On generalized convex functions in optimization theory - A survey. Opsearch, 33: 174–185 (1996)

    MathSciNet  Google Scholar 

  22. Maeda, T. Constraint qualifications in multiobjective optimization problems: Differentiable case. J. Optim. Theory Appl., 80: 483–500 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  23. Martin, D.H. The essence of invexity. J. Optim. Theory Appl., 47: 65–76 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mititelu, S., Stancu-Minasian, I.M. Invexity at a point: Generalizations and classification. Bull. Austral. Math. Soc., 48: 117–126 (1993)

    MATH  MathSciNet  Google Scholar 

  25. Mond, B., Weir, T. Generalized concavity and duality. In: Generalized Concavity in Optimization and Economics ed. by Schaible, S., Ziemba, W.T., Academic Press, New York, 1981, 263–279

  26. Pini, R. Invexity and generalized convexity. Optimization, 22: 513–525 (1991)

    MATH  MathSciNet  Google Scholar 

  27. Pini, R., Singh, C. A survey of recent [1985–1995] advances in generalized convexity with applications to duality theory and optimality conditions. Optimization, 39: 311–360 (1997)

    MATH  MathSciNet  Google Scholar 

  28. Reemtsen, R., Rückmann, J.J. (Eds.) Semi-Infinite Programming, Kluwer, Boston, 1998

  29. Reiland, T.W. Nonsmooth invexity. Bull. Austral. Math. Soc., 42: 437–446 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  30. Stein, O. Bilevel Strategies in Semi-infinite Programming, Kluwer, Boston, 2003

  31. Weber, G.W. Generalized semi-infinite optimization: theory and applications in optimal control and discrete optimization. J. Stat. Management Syst., 5: 359–388 (2002)

    MATH  Google Scholar 

  32. Zalmai, G.J. Optimality principles and duality models for a class of continuous-time generalized fractional programming problems with operator constraints. J. Stat. Management Syst., 1: 61–100 (1998)

    MATH  MathSciNet  Google Scholar 

  33. Zalmai, G.J., Zhang, Q.H. Global nonparametric sufficient optimality conditions for semi-infinite minmax fractional programming problems involving generalized (η, ρ)-invex functions, to appear in Numerical Functional Analysis and Optimization.

  34. Zalmai, G.J., Zhang, Q.H. Parametric duality models for semi-infinite discrete minmax fractional programming problems involving generalized (η, ρ)-invex functions, to appear in Acta Mathematicae Applicatae Sinica, 23(3): 2007

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  1. Department of Mathematics and Computer Science, Northern Michigan University, Marquette, Michigan, 49855, USA

    G. J. Zalmai & Qing-hong Zhang

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  1. G. J. Zalmai
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  2. Qing-hong Zhang
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Correspondence to Qing-hong Zhang.

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Zalmai, G.J., Zhang, Qh. Global Parametric Sufficient Optimality Conditions for Semi-infinite Discrete Minmax Fractional Programming Problems Involving Generalized (η,ρ)-invex Functions. Acta Mathematicae Applicatae Sinica, English Series 23, 217–234 (2007). https://doi.org/10.1007/s10255-007-0365-4

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  • DOI: https://doi.org/10.1007/s10255-007-0365-4

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