# American Institute of Mathematical Sciences

2011, 1(3): 361-370. doi: 10.3934/naco.2011.1.361

## Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity

 1 College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067 2 Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China

Received  April 2011 Revised  June 2011 Published  September 2011

Using a parametric approach, we establish necessary and sufficient optimality conditions and derive some duality theorems for a class of nonsmooth minmax fractional programming problems containing generalized univex functions. The results obtained in this paper extend and improve some corresponding results in the literature.
Citation: Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361
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##### References:
 [1] C. R. Bector, S. Chandra and M. K. Bector, Generalized fractional programming duality: a parametric approach, J. Optim. Theory Appl., 60 (1989), 243-260. doi: 10.1007/BF00940006.  Google Scholar [2] C. R. Bector, S. K. Duneja and S. Gupta, Univex functions and univex nonlinear programming, in: Proceedings of the Asministrative Sciences Association of Canada, 1992, 115-124. Google Scholar [3] S. Chandra, B. D. Craven and B. Mond, Generalized fractional programming duality: a ratio game approach, J. Austral. Math. Soc., 28 (1986), 170-180. doi: 10.1017/S0334270000005282.  Google Scholar [4] J. P. Crouzeix, J. A. Ferland and S. Schaible, Duality in generalized fractional programming, Math. Programming, 27 (1983), 342-354. doi: 10.1007/BF02591908.  Google Scholar [5] F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley-Interscience, New York, 1983.  Google Scholar [6] M. A. Hanson, On sufficiency of the Kuhn-Tucker condition, J. Math. Anal. Appl., 80 (1981), 545-550. doi: 10.1016/0022-247X(81)90123-2.  Google Scholar [7] I. Husain, M. A. Hanson and Z. Jabeen, On nondifferentiable fractional minimax programming, European J. Oper. Res., 160 (2005), 202-217. doi: 10.1016/S0377-2217(03)00437-5.  Google Scholar [8] V. Jeyakumar, Equivalence of saddle-points and optima, and duality for a class of nonsmooth non-convex problems, J. Math. Anal. Appl., 130 (1988), 334-343. doi: 10.1016/0022-247X(88)90309-5.  Google Scholar [9] V. Jeyakumar and B. Mond, On generalized convex mathematical programming, J. Austral. Math. Soc., 34 (1992), 43-53. doi: 10.1017/S0334270000007372.  Google Scholar [10] G. M. Lee, Nonsmooth invexity in multiobjective programming, J. Inform. Optim. Soc., 15 (1994), 127-136.  Google Scholar [11] Z. A. Liang and Z. W. Shi, Optimality conditions and duality for a minimax fractional programming with generalized convexity, J. Math. Anal. Appl., 277 (2003), 474-488. doi: 10.1016/S0022-247X(02)00553-X.  Google Scholar [12] J. C. Liu, Optimality and duality for generalized fractional programming involving nonsmooth $(F,\rho)$-convex functions, Comput. Math. Appl., 32 (1996), 91-102. doi: 10.1016/0898-1221(96)00106-X.  Google Scholar [13] J. C. Liu, Optimality and duality for generalized fractional programming involving nonsmooth pseudoinvex functions, J. Math. Anal. Appl., 202 (1996), 667-685. doi: 10.1006/jmaa.1996.0341.  Google Scholar [14] H. Z. Luo and H. X. Wu, On necessary conditions for a class of nondifferentiable minimax fractional programming, J. Comput. Appl. Math., 215 (2008), 103-113. doi: 10.1016/j.cam.2007.03.032.  Google Scholar [15] S. K. Mishra, R. P. Pant and J. S. Rautela, Generalized $\alpha$-invexity and nondifferentiable minimax fractional programming, J. Comput. Appl. Math., 206 (2007), 122-135. doi: 10.1016/j.cam.2006.06.009.  Google Scholar [16] S. K. Mishra, S. Y. Wang, K. K. Lai and J. M. Shi, Nondifferentiable minimax fractional programming under generalized univexity, J. Comput. Appl. Math., 158 (2007), 379-395. doi: 10.1016/S0377-0427(03)00455-2.  Google Scholar [17] B. Mond and T. Weir, Generalized concavity and duality, in "Generalized Concavity in optimization and economics" (eds. S. Schaible and W.T. Ziemba), Academic Press, (1981), 263-279. Google Scholar [18] T. W. Reiland, Nonsmooth invexity, Bull. Austral. Math. Soc., 42 (1990), 437-446. doi: 10.1017/S0004972700028604.  Google Scholar [19] W. E. Schmitendorf, Necessary conditions and sufficient conditions for static minimax problems, J. Math. Anal. Appl., 57 (1977), 683-693. doi: 10.1016/0022-247X(77)90255-4.  Google Scholar [20] G. J. Zalmai, Optimality conditions and duality models for generalized fractional programming problems containing locally subdifferentiable and $\rho$-convex functions, Optimization, 32 (1995), 95-124. doi: 10.1080/02331939508844040.  Google Scholar
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