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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 |
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. |
[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. |
[4] |
T. Antczak, Lipschitz $r$-invex functions and nonsmooth programming,, Numer. Funct. Anal. Optim., 23 (2002), 265.
doi: 10.1081/NFA-120006693. |
[5] |
C. Bajona-Xandri and J. E. Martinez-Legaz, Lower subdifferentiability in minimax fractional.programming,, Optimization, 45 (1999), 1.
doi: 10.1080/02331939908844423. |
[6] |
S. Chandra and V. Kumar, Duality in fractional minimax programming,, J. Austral. Math. Soc. ser. A, 58 (1995), 376.
doi: 10.1017/S1446788700038362. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
R. G. Schroeder, Linear programming solutions to ratio games,, Operations Research, 18 (1970), 300.
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, (1990).
doi: 10.1007/978-3-642-46709-7_22. |
[14] |
I. M. Stancu-Minasian, Fractional Programming: Theory, Methods and Applications,, Kluwer, (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.
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.
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.
|
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. Google Scholar |
[2] |
T. Antczak, $(p, r)$-invex sets and functions,, J. Math. Anal. Appl., 263 (2001), 355.
doi: 10.1006/jmaa.2001.7574. |
[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. |
[4] |
T. Antczak, Lipschitz $r$-invex functions and nonsmooth programming,, Numer. Funct. Anal. Optim., 23 (2002), 265.
doi: 10.1081/NFA-120006693. |
[5] |
C. Bajona-Xandri and J. E. Martinez-Legaz, Lower subdifferentiability in minimax fractional.programming,, Optimization, 45 (1999), 1.
doi: 10.1080/02331939908844423. |
[6] |
S. Chandra and V. Kumar, Duality in fractional minimax programming,, J. Austral. Math. Soc. ser. A, 58 (1995), 376.
doi: 10.1017/S1446788700038362. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
R. G. Schroeder, Linear programming solutions to ratio games,, Operations Research, 18 (1970), 300.
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, (1990).
doi: 10.1007/978-3-642-46709-7_22. |
[14] |
I. M. Stancu-Minasian, Fractional Programming: Theory, Methods and Applications,, Kluwer, (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.
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.
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.
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