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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

 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

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.
Citation: Anurag Jayswal, Ashish Kumar Prasad, Izhar Ahmad. On minimax fractional programming problems involving generalized $(H_p,r)$-invex functions. Journal of Industrial & 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. Google Scholar [2] T. Antczak, $(p, r)$-invex sets and functions,, J. Math. Anal. Appl., 263 (2001), 355. doi: 10.1006/jmaa.2001.7574. Google Scholar [3] T. Antczak, Generalized fractional minimax programming with $B-(p,r)$-invexity,, Comp. Math. Appl., 56 (2008), 1505. doi: 10.1016/j.camwa.2008.02.039. Google Scholar [4] T. Antczak, Lipschitz $r$-invex functions and nonsmooth programming,, Numer. Funct. Anal. Optim., 23 (2002), 265. doi: 10.1081/NFA-120006693. Google Scholar [5] C. Bajona-Xandri and J. E. Martinez-Legaz, Lower subdifferentiability in minimax fractional.programming,, Optimization, 45 (1999), 1. doi: 10.1080/02331939908844423. Google Scholar [6] S. Chandra and V. Kumar, Duality in fractional minimax programming,, J. Austral. Math. Soc. ser. A, 58 (1995), 376. doi: 10.1017/S1446788700038362. Google Scholar [7] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions,, J. Math. Anal. Appl., 80 (1981), 545. doi: 10.1016/0022-247X(81)90123-2. Google Scholar [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. doi: 10.1016/S0022-247X(02)00553-X. Google Scholar [9] J. C. Liu and C. S. Wu, On minimax fractional optimality conditions with invexity,, J. Math. Anal. Appl., 219 (1998), 21. doi: 10.1006/jmaa.1997.5786. Google Scholar [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. doi: 10.4067/S0719-06462011000100008. Google Scholar [11] W. E. Schmittendorf, Necessary conditions and sufficient conditions for static minimax problem,, J. Math. Anal. Appl., 57 (1977), 683. doi: 10.1016/0022-247X(77)90255-4. Google Scholar [12] R. G. Schroeder, Linear programming solutions to ratio games,, Operations Research, 18 (1970), 300. doi: 10.1287/opre.18.2.300. Google Scholar [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, (1990). doi: 10.1007/978-3-642-46709-7_22. Google Scholar [14] I. M. Stancu-Minasian, Fractional Programming: Theory, Methods and Applications,, Kluwer, (1997). doi: 10.1007/978-94-009-0035-6. Google Scholar [15] J. Von Neumann, A model of general economic equilibrium,, Review of Economic Studies, 13 (1945), 1. doi: 10.2307/2296111. Google Scholar [16] S. R. Yadav and R. N. Mukherjee, Duality for fractional minimax programming problems,, J. Australian Math. Soc. Ser. B, 31 (1990), 484. doi: 10.1017/S0334270000006809. Google Scholar [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. Google Scholar

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##### References:
 [1] I. Ahmad, Optimality conditions and duality in fractional minimax programming involving generalized $\rho$-invexity,, Inter. J. Manag. Syst., 19 (2003), 165. Google Scholar [2] T. Antczak, $(p, r)$-invex sets and functions,, J. Math. Anal. Appl., 263 (2001), 355. doi: 10.1006/jmaa.2001.7574. Google Scholar [3] T. Antczak, Generalized fractional minimax programming with $B-(p,r)$-invexity,, Comp. Math. Appl., 56 (2008), 1505. doi: 10.1016/j.camwa.2008.02.039. Google Scholar [4] T. Antczak, Lipschitz $r$-invex functions and nonsmooth programming,, Numer. Funct. Anal. Optim., 23 (2002), 265. doi: 10.1081/NFA-120006693. Google Scholar [5] C. Bajona-Xandri and J. E. Martinez-Legaz, Lower subdifferentiability in minimax fractional.programming,, Optimization, 45 (1999), 1. doi: 10.1080/02331939908844423. Google Scholar [6] S. Chandra and V. Kumar, Duality in fractional minimax programming,, J. Austral. Math. Soc. ser. A, 58 (1995), 376. doi: 10.1017/S1446788700038362. Google Scholar [7] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions,, J. Math. Anal. Appl., 80 (1981), 545. doi: 10.1016/0022-247X(81)90123-2. Google Scholar [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. doi: 10.1016/S0022-247X(02)00553-X. Google Scholar [9] J. C. Liu and C. S. Wu, On minimax fractional optimality conditions with invexity,, J. Math. Anal. Appl., 219 (1998), 21. doi: 10.1006/jmaa.1997.5786. Google Scholar [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. doi: 10.4067/S0719-06462011000100008. Google Scholar [11] W. E. Schmittendorf, Necessary conditions and sufficient conditions for static minimax problem,, J. Math. Anal. Appl., 57 (1977), 683. doi: 10.1016/0022-247X(77)90255-4. Google Scholar [12] R. G. Schroeder, Linear programming solutions to ratio games,, Operations Research, 18 (1970), 300. doi: 10.1287/opre.18.2.300. Google Scholar [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, (1990). doi: 10.1007/978-3-642-46709-7_22. Google Scholar [14] I. M. Stancu-Minasian, Fractional Programming: Theory, Methods and Applications,, Kluwer, (1997). doi: 10.1007/978-94-009-0035-6. Google Scholar [15] J. Von Neumann, A model of general economic equilibrium,, Review of Economic Studies, 13 (1945), 1. doi: 10.2307/2296111. Google Scholar [16] S. R. Yadav and R. N. Mukherjee, Duality for fractional minimax programming problems,, J. Australian Math. Soc. Ser. B, 31 (1990), 484. doi: 10.1017/S0334270000006809. Google Scholar [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. Google Scholar
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