On minimax fractional programming problems involving generalized (H_p,r)-invex functions

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October 2014, 10(4): 1001-1018. doi: 10.3934/jimo.2014.10.1001

On minimax fractional programming problems involving generalized $(H_p,r)$-invex functions

Anurag Jayswal ^1, , Ashish Kumar Prasad ^1, and Izhar Ahmad ^2,

1.	Department of Applied Mathematics, Indian School of Mines, Dhanbad-826004, India, India
2.	Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran-31261, Saudi Arabia

Received May 2012 Revised July 2013 Published February 2014

Abstract
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In the present paper, we move forward in the study of minimax fractional programming problem and establish sufficient optimality conditions under the assumptions of generalized $(H_p,r)$-invexity. Weak, strong and strict converse duality theorems are also derived for two types of dual models related to minimax fractional programming problem involving aforesaid invex functions. In order to show the existence of introduced class of functions, examples are given.

Keywords: $(H_p, r)$-invexity, sufficiency, Minimax fractional programming, duality..

Mathematics Subject Classification: Primary: 90C32, 49K35; Secondary: 49N1.

Citation: Anurag Jayswal, Ashish Kumar Prasad, Izhar Ahmad. On minimax fractional programming problems involving generalized $(H_p,r)$-invex functions. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1001-1018. doi: 10.3934/jimo.2014.10.1001

References:

[1]	I. Ahmad, Optimality conditions and duality in fractional minimax programming involving generalized $\rho$-invexity, Inter. J. Manag. Syst., 19 (2003), 165-180.
[2]	T. Antczak, $(p, r)$-invex sets and functions, J. Math. Anal. Appl., 263 (2001), 355-379. doi: 10.1006/jmaa.2001.7574.
[3]	T. Antczak, Generalized fractional minimax programming with $B-(p,r)$-invexity, Comp. Math. Appl., 56 (2008), 1505-1525. doi: 10.1016/j.camwa.2008.02.039.
[4]	T. Antczak, Lipschitz $r$-invex functions and nonsmooth programming, Numer. Funct. Anal. Optim., 23 (2002), 265-283. doi: 10.1081/NFA-120006693.
[5]	C. Bajona-Xandri and J. E. Martinez-Legaz, Lower subdifferentiability in minimax fractional.programming, Optimization, 45 (1999), 1-12. doi: 10.1080/02331939908844423.
[6]	S. Chandra and V. Kumar, Duality in fractional minimax programming, J. Austral. Math. Soc. ser. A, 58 (1995), 376-386. doi: 10.1017/S1446788700038362.
[7]	M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981), 545-550. doi: 10.1016/0022-247X(81)90123-2.
[8]	Z. Liang and Z. 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.
[9]	J. C. Liu and C. S. Wu, On minimax fractional optimality conditions with invexity, J. Math. Anal. Appl., 219 (1998), 21-35. doi: 10.1006/jmaa.1997.5786.
[10]	X. Liu, D. Yuan, S. Yang and G. Lai, Multiple objective programming involving differentiable $(H_p,r)$-invex functions, CUBO A Mathematical Journal, 13 (2011), 125-136. doi: 10.4067/S0719-06462011000100008.
[11]	W. E. Schmittendorf, Necessary conditions and sufficient conditions for static minimax problem, J. Math. Anal. Appl., 57 (1977), 683-693. doi: 10.1016/0022-247X(77)90255-4.
[12]	R. G. Schroeder, Linear programming solutions to ratio games, Operations Research, 18 (1970), 300-305. doi: 10.1287/opre.18.2.300.
[13]	I. M. Stancu-Minasian and S. Tigan, On some fractional programming models occurring in minimum-risk problem, in Generalized Convexity and Fractional Programming with Economic Applications, (Eds. A. Cambini, E. Castagnoli, L. Martein, P. Mazzoleni, S. Schaible), Springer-Verlag, Berlin, (1990). doi: 10.1007/978-3-642-46709-7_22.
[14]	I. M. Stancu-Minasian, Fractional Programming: Theory, Methods and Applications, Kluwer, Dordrecht, 1997. doi: 10.1007/978-94-009-0035-6.
[15]	J. Von Neumann, A model of general economic equilibrium, Review of Economic Studies, 13 (1945), 1-9. doi: 10.2307/2296111.
[16]	S. R. Yadav and R. N. Mukherjee, Duality for fractional minimax programming problems, J. Australian Math. Soc. Ser. B, 31 (1990), 484-492. doi: 10.1017/S0334270000006809.
[17]	D. Yuan, X. Liu, S. Yang, N. Damdin and A. Chinchuluun, Optimality conditions and duality for nonlinear programming problems involving locally $(H_p,r,\alpha)$-pre-invex functions and $H_p$-invex sets, Int. J. Pure Appl. Math., 41 (2007), 561-576.

show all references

References:

[1]	I. Ahmad, Optimality conditions and duality in fractional minimax programming involving generalized $\rho$-invexity, Inter. J. Manag. Syst., 19 (2003), 165-180.
[2]	T. Antczak, $(p, r)$-invex sets and functions, J. Math. Anal. Appl., 263 (2001), 355-379. doi: 10.1006/jmaa.2001.7574.
[3]	T. Antczak, Generalized fractional minimax programming with $B-(p,r)$-invexity, Comp. Math. Appl., 56 (2008), 1505-1525. doi: 10.1016/j.camwa.2008.02.039.
[4]	T. Antczak, Lipschitz $r$-invex functions and nonsmooth programming, Numer. Funct. Anal. Optim., 23 (2002), 265-283. doi: 10.1081/NFA-120006693.
[5]	C. Bajona-Xandri and J. E. Martinez-Legaz, Lower subdifferentiability in minimax fractional.programming, Optimization, 45 (1999), 1-12. doi: 10.1080/02331939908844423.
[6]	S. Chandra and V. Kumar, Duality in fractional minimax programming, J. Austral. Math. Soc. ser. A, 58 (1995), 376-386. doi: 10.1017/S1446788700038362.
[7]	M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981), 545-550. doi: 10.1016/0022-247X(81)90123-2.
[8]	Z. Liang and Z. 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.
[9]	J. C. Liu and C. S. Wu, On minimax fractional optimality conditions with invexity, J. Math. Anal. Appl., 219 (1998), 21-35. doi: 10.1006/jmaa.1997.5786.
[10]	X. Liu, D. Yuan, S. Yang and G. Lai, Multiple objective programming involving differentiable $(H_p,r)$-invex functions, CUBO A Mathematical Journal, 13 (2011), 125-136. doi: 10.4067/S0719-06462011000100008.
[11]	W. E. Schmittendorf, Necessary conditions and sufficient conditions for static minimax problem, J. Math. Anal. Appl., 57 (1977), 683-693. doi: 10.1016/0022-247X(77)90255-4.
[12]	R. G. Schroeder, Linear programming solutions to ratio games, Operations Research, 18 (1970), 300-305. doi: 10.1287/opre.18.2.300.
[13]	I. M. Stancu-Minasian and S. Tigan, On some fractional programming models occurring in minimum-risk problem, in Generalized Convexity and Fractional Programming with Economic Applications, (Eds. A. Cambini, E. Castagnoli, L. Martein, P. Mazzoleni, S. Schaible), Springer-Verlag, Berlin, (1990). doi: 10.1007/978-3-642-46709-7_22.
[14]	I. M. Stancu-Minasian, Fractional Programming: Theory, Methods and Applications, Kluwer, Dordrecht, 1997. doi: 10.1007/978-94-009-0035-6.
[15]	J. Von Neumann, A model of general economic equilibrium, Review of Economic Studies, 13 (1945), 1-9. doi: 10.2307/2296111.
[16]	S. R. Yadav and R. N. Mukherjee, Duality for fractional minimax programming problems, J. Australian Math. Soc. Ser. B, 31 (1990), 484-492. doi: 10.1017/S0334270000006809.
[17]	D. Yuan, X. Liu, S. Yang, N. Damdin and A. Chinchuluun, Optimality conditions and duality for nonlinear programming problems involving locally $(H_p,r,\alpha)$-pre-invex functions and $H_p$-invex sets, Int. J. Pure Appl. Math., 41 (2007), 561-576.

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