Characterization of Solutions of Nonsmooth Variational Problems and Duality

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.

• Juan Zhang
• Changzhao Li

• Variational problem
• Nonsmooth pseudo-invex
• Nonsmooth quasi-invex
• Critical point
• Duality

<p>[1] M. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal.
Appl., 80 (1981) 545-550.
[2] B. Craven, Nondifferential optimization by nonsmooth approximations,
Optimization, 17 (1986) 3-17.
[3] T. Reiland, Nonsmooth invexity, Bull. Austral. Math. Soc., 42 (1990)
437-446.
[4] D. Martin, The essence of invexity, J. Optim. Th. Appl., 47 (1986) 65-76.
[5] B. Mond, S. Chandra, I. Husain, Duality for variational problems with
invexity, J. Math. Anal. Appl., 134 (1988) 322-328.
[6] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York,
1983.
[7] F. H. Clarke, Nonsmooth Analysis and Control Theory, Berlin: Springer-
Verlag, 1998.
[8] 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.
[9] 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>