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Extragradient-projection method for solving constrained convex minimization problems
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 |
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. |
[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. |
[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. |
[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. |
[5] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley-Interscience, New York, 1983. |
[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. |
[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. |
[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. |
[9] |
V. Jeyakumar and B. Mond, On generalized convex mathematical programming, J. Austral. Math. Soc., 34 (1992), 43-53.
doi: 10.1017/S0334270000007372. |
[10] |
G. M. Lee, Nonsmooth invexity in multiobjective programming, J. Inform. Optim. Soc., 15 (1994), 127-136. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
T. W. Reiland, Nonsmooth invexity, Bull. Austral. Math. Soc., 42 (1990), 437-446.
doi: 10.1017/S0004972700028604. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley-Interscience, New York, 1983. |
[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. |
[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. |
[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. |
[9] |
V. Jeyakumar and B. Mond, On generalized convex mathematical programming, J. Austral. Math. Soc., 34 (1992), 43-53.
doi: 10.1017/S0334270000007372. |
[10] |
G. M. Lee, Nonsmooth invexity in multiobjective programming, J. Inform. Optim. Soc., 15 (1994), 127-136. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
T. W. Reiland, Nonsmooth invexity, Bull. Austral. Math. Soc., 42 (1990), 437-446.
doi: 10.1017/S0004972700028604. |
[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. |
[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. |
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