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Lagrangian Duality for Minimization of Nonconvex Multifunctions

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Abstract

Two alternative type theorems for nearly convexlike or * quasiconvex multifunctions are presented. They are used to derive Lagrangian conditions and duality results for vector optimization problems when the objectives and the constraints are nearly convexlike or * quasiconvex multifunctions.

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References

  1. Benson, H. P., An Improved Definition of Proper Efficiency for Vector Minimization with Respect to Cones, Journal of Mathematical Analysis and Applications, Vol. 71, pp. 232–241, 1979.

    Google Scholar 

  2. Borwein, J. M., Proper Efficient Points for Maximizations with Respect to Cones, SIAM Journal on Control and Optimization, Vol. 15, pp. 57–63, 1977.

    Google Scholar 

  3. Borwein, J. M., The Geometry of Pareto Efficiency over Cones, Mathematische Operationsforschung und Statistik, Series Optimization, Vol. 11, pp. 235–248, 1980.

  4. Hayashi, M., and Komiya, H., Perfect Duality for Convexlike Programs, Journal of Optimization Theory and Applications, Vol. 38, pp. 179–189, 1982.

    Google Scholar 

  5. Jeyakumar, V., Convexlike Alternative Theorems and Mathematical Programming, Optimization, Vol. 16, pp. 643–652, 1985.

    Google Scholar 

  6. Jeyakumar, V., A Generalization of a Minimax Theorem of Fan via a Theorem of the Alternative, Vol. 48, pp. 525–533, 1986.

    Google Scholar 

  7. Jeyakumar, V., and Gwinner, J., Inequality Systems and Optimization, Journal of Mathematical Analysis and Applications, Vol. 159, pp. 51–71, 1991.

    Google Scholar 

  8. Jeyakumar, V., Oettli, W., and Natividad, M., A Solvability Theorem for a Class of Quasiconvex Mapping with Applications to Optimization, Journal of Mathematical Analysis and Applications, Vol. 179, pp. 537–546, 1993.

    Google Scholar 

  9. Liu, S., and You, Z., Proper Efficient Solution of Nonsmooth and Nonconvex Vector Extreme Problems, Applied Mathematics, Vol. 4, pp. 34–39, 1990 (in Chinese).

    Google Scholar 

  10. Paeck, S., Convexlike and Concavelike Conditions in Alternative, Minimax, and Minimization Theorems, Journal of Optimization Theory and Applications, Vol. 74, pp. 317–332, 1992.

    Google Scholar 

  11. Jahn, J., Mathematical Vector Optimization in Partially-Ordered Linear Spaces, Peter Lang, Frankfurt, Germany, 1986.

    Google Scholar 

  12. Yu, P. L., Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjective, Journal of Optimization Theory and Applications, Vol. 14, pp. 319–377, 1974.

    Google Scholar 

  13. Corley, H. W., Existence and Lagrangian Duality for Maximizations of Set-Valued Functions, Journal of Optimization Theory and Applications, Vol. 54, pp. 489–501, 1987.

    Google Scholar 

  14. Luc, D. T., Theory of Vector Optimization, Springer Verlag, Berlin, Germany, 1989.

    Google Scholar 

  15. Sach, P. H., and Craven, B. D., Invex Multifunctions and Duality, Numerical Functional Analysis and Optimization, Vol. 12, pp. 575–591, 1991.

    Google Scholar 

  16. Sion, M., On General Minimax Theorems, Pacific Journal of Mathematics, Vol. 8, pp. 171–176, 1958.

    Google Scholar 

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Song, W. Lagrangian Duality for Minimization of Nonconvex Multifunctions. Journal of Optimization Theory and Applications 93, 167–182 (1997). https://doi.org/10.1023/A:1022658019642

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