Minimax fractional programming problem involving nonsmooth generalized ?-univex functions
DOI:
https://doi.org/10.11121/ijocta.01.2013.00102Keywords:
Nondifferentiable minimax fractional programming, α-univexity, sufficient optimality conditions, dualityAbstract
In this paper, we introduce a new class of generalized ?-univex functions where the involved functions are locally Lipschitz. We extend the concept of ?-type I invex [S. K. Mishra, J. S. Rautela, On nondifferentiable minimax fractional programming under generalized ?-type I invexity, J. Appl. Math. Comput. 31 (2009) 317-334] to ?-univexity and an example is provided to show that there exist functions that are ?-univex but not ?-type I invex. Furthermore, Karush-Kuhn-Tucker-type sufficient optimality conditions and duality results for three different types of dual models are obtained for nondifferentiable minimax fractional programming problem involving generalized ?-univex functions. The results in this paper extend some known results in the literature.Downloads
References
C. R. Bector, B. L. Bhatia, Sufficient optimality and duality for a minimax problems. Utilitas Math., 27, 229-247 (1985).
C. R. Bector, S. K. Suneja, S. Gupta, Univex functions and univex nonlinear programming. Proceedings of the Administrative Sciences Association of Canada, 115-124, (1992).
S. Chandra, B. D. Craven, B. Mond, Generalized fractional programming duality: A ratio game approach. J. Austral. Math. Soc. Ser. B, 28, 170-180 (1986). CrossRef
A. Charnes, W. W. Copper, Goal programming and multi-objective optimization. Part I, European J. Oper. Res.1, 39-54 (1977). CrossRef
S. Chandra, V. Kumar, Duality in fractional minimax programming. J. Austral. Math. Soc. Ser. A, 58, 376-386 (1995). CrossRef
F. H. Clarke, Optimization and Nonsmooth Analysis. Wiley, New York, 1983.
L. Fan, Y. Guo, On strongly α-preinvex functions. J. Math. Anal. Appl., 330, 1412–1425 (2007). CrossRef
M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl., 80, 545-550 (1981). CrossRef
M. A. Hanson, B. Mond, Further generalizations of convexity in mathematical programming. J. Inform. Optim. Sci., 3, 25-32 (1982).
A. Jayswal, Non-differentiable minimax fractional programming with generalized -univexity. J. Comput. Appl. Math., 214, 121-135 (2008). CrossRef
H. C. Lai, J. C. Liu, K. Tanaka, Necessary and sufficient conditions for minimax fractional programming. J. Math. Anal. Appl., 230, 311-328 (1999). CrossRef
H. C. Lai, J. C. Lee, On duality theorems for a nondifferentiable minimax fractional programming. J. Comput. Appl. Math., 146, 115-126 (2002). CrossRef
J. C. Liu, Optimality and duality for multiobjective fractional programming involving nonsmooth -convex functions. Optimization, 36, 333-346 (1996). CrossRef
J. C. Liu, Optimality and duality for multiobjective fractional programming involving nonsmooth pseudoinvex functions. Optimization, 37, 27-39 (1996). CrossRef
X. J. Long, J. Quan, Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity, Numerical Algebra, Control and Optimization. 1, 361-370 (2011).
S. K. Mishra, R. P. Pant, J. S. Rautela, Generalized -invexity and nondifferentiable minimax fractional programming. J. Comput. Appl. Math., 206, 122-135 (2007). CrossRef
S. K. Mishra, J. S. Rautela, On nondifferentiable minimax fractional programming under generalized -type I invexity. J. Appl. Math. Comput., 31, 317-334 (2009). CrossRef
M. A. Noor, On generalized preinvex functions and monotonicities. J. Ineq. Pure Appl. Math., 5, 1-9 (2004).
W. E. Schmitendorf, Necessary conditions and sufficient conditions for static minimax problems, J. Math. Anal. Appl., 57, 683-693 (1977). CrossRef
I. M. Stancu Minasian, Fractional programming theory, methods and applications. Kluwer, Dordrecht, 1997.
S. Tigan, On some fractional programming models occurring in minimum-risk problem. in: A. Cambini, E. Castagnoli, L. Martine, P. Mazzoleni, S. Schaible (Eds.), Generalized Convexity and Fractional Programming with Economics Applications, Springer-link, Berlin, 1990.
J. Von Neumann, A model of general economic equilibrium. Review of Economics Studies, 13, 1-9 (1945). CrossRef
T. Weir, A dual for multiobjective fractional programming. J. Infor. Optim. Sci., 7, 261-269 (1986).
S. R. Yadav, R. N. Mukherjee, Duality for fractional minimax programming problems. J. Austral. Math. Soc. Ser. B., 31, 484-492 (1990). CrossRef
X. J. Zheng, L. Cheng, Minimax fractional programming under nonsmooth generalized -d-univex. J. Math. Anal. Appl., 328, 676-689 (2007). CrossRef
G. J. Zalmai, Optimality conditions and duality models for a class of nonsmooth constrained fractional variational problems. Optimization, 30, 15-51 (1994). CrossRef
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