Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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ON APPROXIMATED PROBLEMS FOR LOCALLY LIPSCHITZ OPTIMIZATION PROBLEMS

  • Kim, Moon-Hee (Department of Multimedia Engineering, Tongmyong University)
  • Published : 2010.01.30
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Abstract

In this paper, using nonsmooth analysis, we established equivalence results between a locally Lipschitz vector optimization problem and its associated approximated problem under the proper efficiency.

Keywords

References

  1. T. Antczak, A new approach to multiobjective progmmming with a modified objective function, J. Global Optimization 27(2003), 485-495. https://doi.org/10.1023/A:1026080604790
  2. T. Antczak, An n-appmximation appmach for nonlinear mathematical pmgmmming pmblems involving invex functions, Num. Functional Anal. Optimization 25(2004), 423-438. https://doi.org/10.1081/NFA-200042183
  3. T. Antczak, An n-appmximation method in nonlinear vector optimization, Nonlinear Analysis 63(2005), 225-236. https://doi.org/10.1016/j.na.2005.05.008
  4. S. Brumelle, Duality for multiple objective convex programs, Math. Oper. Res. 6(1981), 159-172. https://doi.org/10.1287/moor.6.2.159
  5. F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.
  6. B.D. Craven, Invex functions and constrained local minima, Bull. Aust. Math. Soc. 24(1981), 357-366. https://doi.org/10.1017/S0004972700004895
  7. B.D. Craven, Quasimin and quasi saddle point for vector optimization, Num. Functional Anal. Optimization 11(1990), 45-54. https://doi.org/10.1080/01630569008816360
  8. A. M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl. 22(1968), 618-630.
  9. G. Giorgi, A. Guerraggi, Various types of nonsmooth invexity, J. Inform. Optim. Sci. 17(1996), 137-150.
  10. M.A. Hanson, 0n sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80(1981), 545-550. https://doi.org/10.1016/0022-247X(81)90123-2
  11. X.X. Huang, X.Q. Yang, 0n charactenzations of proper efficiency for nonconvex multiobjective optimization, J. Global Optimization 23(2002), 213-231. https://doi.org/10.1023/A:1016522528364
  12. H. Isermann, Proper efficiency and linear vector maximum problem, Oper. Res. 22(1974), 189-191. https://doi.org/10.1287/opre.22.1.189
  13. V. Jeyakumar, B. Mond, On generalised convex mathematical programming, J. Austral Math. Soc. Ser. B 34(1992), 43-53. https://doi.org/10.1017/S0334270000007372
  14. D. S. Kim, Multiobjective fractional programming with a modified objective function, Commun. Korean Math. Soc. 20(2005), 837-847. https://doi.org/10.4134/CKMS.2005.20.4.837
  15. M.H. Kim, On linearized vector optimization problems with proper efficiency, J. Appl. Math. & Informatics 27(2009), 685-692.
  16. G.M. Lee, Optimality conditions in multiobejctive optimization problems, J. Inform. Optim. Sci. 13(1993), 39-41.
  17. Lun Li and Jun Li, Equivalence and existence of weak Pareto optima for multiobjective optimization problems with cone constraints, Appl. Math. Lett. 21(2008), 599-606. https://doi.org/10.1016/j.aml.2007.07.012
  18. Marko M. Makela and Pekka Neittaanmaki, Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control, World Scientific Publishing Co. Pte. Ltd.,1992.
  19. R.T. Rockafellar, Convex Analysis, Princeton, New Jersey, Princeton University Press, 1970.
  20. T. Weir, B. Mond, B.D. Craven, 0n duality for weakly minimized vector valued optimization problems, Optimization 17(1986), 711-721. https://doi.org/10.1080/02331938608843188