World Academy of Science, Engineering and Technology
Published September 6, 2013 | Version 16970
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Characterization of Solutions of Nonsmooth Variational Problems and Duality

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In this paper, we introduce a new class of nonsmooth pseudo-invex and nonsmooth quasi-invex functions to non-smooth variational problems. By using these concepts, numbers of necessary and sufficient conditions are established for a nonsmooth variational problem wherein Clarke’s generalized gradient is used. Also, weak, strong and converse duality are established.

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References

  • <p>
  • M. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981) 545-550.
  • B. Craven, Nondifferential optimization by nonsmooth approximations, Optimization, 17 (1986) 3-17.
  • T. Reiland, Nonsmooth invexity, Bull. Austral. Math. Soc., 42 (1990) 437-446.
  • D. Martin, The essence of invexity, J. Optim. Th. Appl., 47 (1986) 65-76.
  • B. Mond, S. Chandra, I. Husain, Duality for variational problems with invexity, J. Math. Anal. Appl., 134 (1988) 322-328.
  • F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
  • F. H. Clarke, Nonsmooth Analysis and Control Theory, Berlin: Springer- Verlag, 1998.
  • M. Arana-Jim´enez, R. Osuna-G´omez, G. Ruiz-Garz´on, M. Rojas-Medar, On variational problems: Characterization of solutions and duality, J. Math. Anal. Appl., 311 (2005) 1-12.
  • S. K. Suneja, S. Khurana, Vani, Generalized nonsmooth invexity over cones in vector optimization, European Journal of Operational Research, 186 (2008) 28-40. [10] C. Bector, S. Chandra, I. Husain, Generalized concavity and duality in continuous programming, Util. Math., 25 (1984) 171-190. [11] N. Yen, P. Sach, On locally Lipschitz vector valued invex functions, Bull. Austral. Math. Soc., 47 (1993) 259-271. [12] M. Hanson, B. Mond, Necessary and sufficient conditions in constrainted optimization, Math. Programming, 37 (1987) 51-58. [13] M. Hanson, Invexity and the Kuhn-Tucker theorem, J. Math. Anal. Appl., 236 (1999) 594-604. [14] B. Craven, X. Yang, A nonsmooth version of alternative theorem and nonsmooth multiobjective programming, Utilitas Mathematica, 40 (1991) 117-128. [15] M. Bazaraa, C. Shetty, Nonlinear Programming: Theory and Algorithms, Wiley, New York, 1979.</p>