American Institute of Mathematical Sciences

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

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

Xian-Jun Long 1, and  Jing Quan 2,

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.
Keywords: generalized univex functions, Minmax fractional programming, duality., optimality condition, nonsmooth.
Mathematics Subject Classification: Primary: 90C32, 49J3.
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
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

show all references

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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