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Optimality conditions for approximate solutions of vector optimization problems
1. | Department of Mathematics, Chongqing Normal University, Chongqing 400047, China |
2. | Department of Mathematics and Statistics, Curtin University, G.P.O. Box U1987, Perth, WA 6845 |
References:
[1] |
E. M. Bednarczuk and M. J. Przybyla, The vector-valued variational principle in Banach spaces ordered by cones with nonempty interiors, SIAM J. Optim., 18 (2007), 907-913.
doi: 10.1137/060658989. |
[2] |
M. Beldiman, E. Panaitescu and L. Dogaru, Approximate quasi efficient solutions in multiobjective optimization, Bull. Math. Soc. Sci. Math. Roumanie Tome, 99 (2008), 109-121. |
[3] |
H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones, J. Math. Anal. Appl., 71 (1979), 232-241.
doi: 10.1016/0022-247X(79)90226-9. |
[4] |
S. Bolintinéanu, Vector variational principles: $\epsilon-$efficiency and scalar stationarity, J. Convex Anal., 8 (2001), 71-85. |
[5] |
J. Borwein, Proper efficient points for maximizations with respect to cones, SIAM J. Control Optim., 15 (1977), 57-63.
doi: 10.1137/0315004. |
[6] |
G. Y. Chen, X. X. Huang and X. M. Yang, "Vector Optimization. Set-Valued and Variational Analysis," Lecture Notes in Econom. and Math. Systems 541, Springer-Verlag, Berlin, 2005. |
[7] |
J. Dutta and V. Vetrivel, On approximate minima in vector optimization, Numer. Func. Anal. Optim., 22 (2001), 845-859.
doi: 10.1081/NFA-100108312. |
[8] |
M. Durea, J. Dutta and C. Tammer, Lagrange multipliers for $\epsilon$-Pareto solutions in vector optimization with nonsolid cones in Banach spaces, J. Optim. Theory Appl., 145 (2010), 196-211.
doi: 10.1007/s10957-009-9609-1. |
[9] |
J. B. G. Frenk and G. Kassay, On classes of generalized convex functions, Gordan-Farkas type theorems, and Lagrangian duality, J. Optim. Theory Appl., 102 (1999), 315-343.
doi: 10.1023/A:1021780423989. |
[10] |
A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces," Springer-Verlag, New York, 2003. |
[11] |
D. Gupta and A. Mehra, Two types of approximate saddle points, Numer. Func. Anal. Optim., 29 (2008), 532-550.
doi: 10.1080/01630560802099274. |
[12] |
C. Gutiérrez, B. Jiménez and V. Novo, A set-valued Ekeland's variational principle in vector optimization, SIAM J. Control Optim., 47 (2008), 883-903.
doi: 10.1137/060672868. |
[13] |
C. Gutiérrez, R. López and V. Novo, Generalized $\epsilon-$quasi-solutions in multiobjective optimization problems: Existence results and optimality conditions, Nonlinear Anal., 72 (2010), 4331-4346.
doi: 10.1016/j.na.2010.02.012. |
[14] |
C. Gutiérrez, B. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems, SIAM J. Optim., 17 (2006), 688-710.
doi: 10.1137/05062648X. |
[15] |
C. Gutiérrez, B. Jiménez and V. Novo, On approximate efficiency in multiobjective programming, Math. Methods Oper. Res., 64 (2006), 165-185.
doi: 10.1007/s00186-006-0078-0. |
[16] |
C. Gutiérrez, B. Jiménez and V. Novo, Optimality conditions via scalarization for a new $\epsilon$-efficiency concept in vector optimization problems, European J. Oper. Res., 201 (2010), 11-22.
doi: 10.1016/j.ejor.2009.02.007. |
[17] |
A. M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl., 22 (1968), 618-630.
doi: 10.1016/0022-247X(68)90201-1. |
[18] |
T. X. D. Ha, The Ekeland variational principle for Henig proper minimizers and super minimizers, J. Math. Anal. Appl., 364 (2010), 156-170.
doi: 10.1016/j.jmaa.2009.10.065. |
[19] |
S. Helbig, "On a new concept for $\epsilon$-efficency," A Talk at Optimization Days 1992, Montreal,1992. |
[20] |
M. I. Henig, Proper efficiency with respect to cones, J. Optim. Theory Appl., 36 (1982), 387-407.
doi: 10.1007/BF00934353. |
[21] |
J. B. Hiriart-Urruty, New concepts in nondifferentiable programming, Bull. Soc. Math. France Mém., 60 (1979), 57-85. |
[22] |
J. B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res., 4 (1979), 79-97.
doi: 10.1287/moor.4.1.79. |
[23] |
S. Kutateladze, Convex $\epsilon-$programming, Soviet Math. Dokl., 20 (1979), 391-393. |
[24] |
H. W. Kuhn and A. W. Tucker, Nonlinear programming, from "Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability," Berkeley, California, (1951), 481-492 |
[25] |
J. C. Liu, $\epsilon-$properly efficient solutions to nondifferentiable multiobjective programming problems, Appl. Math. Lett., 12 (1999), 109-113.
doi: 10.1016/S0893-9659(99)00087-7. |
[26] |
Z. Li and S. Wang, $\epsilon$-approximate solutions in multiobjective optimization, Optimization, 44 (1998), 161-174.
doi: 10.1080/02331939808844406. |
[27] |
C. G. Liu, K. F. Ng and W. H. Yang, Merit functions in vector optimization, Math. Program., Ser. A, 119 (2009), 215-237.
doi: 10.1016/j.colsurfa.2009.04.036. |
[28] |
A. B. Németh, A nonconvex vector minimization problem, Nonlinear Anal., 10 (1986), 669-678.
doi: 10.1016/0362-546X(86)90126-4. |
[29] |
W. D. Rong and Y. N. Wu, $\epsilon$-weak minimal solutions of vector optimization problems with set-valued maps, J. Optim. Theory Appl., 106 (2000), 569-579.
doi: 10.1023/A:1004657412928. |
[30] |
W. D. Rong, $\epsilon-$efficiency in vector optimization problems with cone subconvexlikeness, Acta Sci. Natur. Univ. NeiMongol., 28 (1997), 609-613. |
[31] |
T. Tanaka, A new approach to approximation of solutions in vector optimization problems, in "Proceedings of APORS, 1994" M. Fushimi and K. Tone (eds), World Scientific, Singapore, (1995), 497-504. |
[32] |
I. Vályi, Approximate saddle-point theorems in vector optimization, J. Optim. Theory Appl., 55 (1987), 435-448.
doi: 10.1007/BF00941179. |
[33] |
D. J. White, Epsilon efficiency, J. Optim. Theory Appl., 49 (1986), 319-337.
doi: 10.1007/BF00940762. |
[34] |
A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM J. Control Optim., 42 (2003), 1071-1086.
doi: 10.1137/S0363012902411532. |
show all references
References:
[1] |
E. M. Bednarczuk and M. J. Przybyla, The vector-valued variational principle in Banach spaces ordered by cones with nonempty interiors, SIAM J. Optim., 18 (2007), 907-913.
doi: 10.1137/060658989. |
[2] |
M. Beldiman, E. Panaitescu and L. Dogaru, Approximate quasi efficient solutions in multiobjective optimization, Bull. Math. Soc. Sci. Math. Roumanie Tome, 99 (2008), 109-121. |
[3] |
H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones, J. Math. Anal. Appl., 71 (1979), 232-241.
doi: 10.1016/0022-247X(79)90226-9. |
[4] |
S. Bolintinéanu, Vector variational principles: $\epsilon-$efficiency and scalar stationarity, J. Convex Anal., 8 (2001), 71-85. |
[5] |
J. Borwein, Proper efficient points for maximizations with respect to cones, SIAM J. Control Optim., 15 (1977), 57-63.
doi: 10.1137/0315004. |
[6] |
G. Y. Chen, X. X. Huang and X. M. Yang, "Vector Optimization. Set-Valued and Variational Analysis," Lecture Notes in Econom. and Math. Systems 541, Springer-Verlag, Berlin, 2005. |
[7] |
J. Dutta and V. Vetrivel, On approximate minima in vector optimization, Numer. Func. Anal. Optim., 22 (2001), 845-859.
doi: 10.1081/NFA-100108312. |
[8] |
M. Durea, J. Dutta and C. Tammer, Lagrange multipliers for $\epsilon$-Pareto solutions in vector optimization with nonsolid cones in Banach spaces, J. Optim. Theory Appl., 145 (2010), 196-211.
doi: 10.1007/s10957-009-9609-1. |
[9] |
J. B. G. Frenk and G. Kassay, On classes of generalized convex functions, Gordan-Farkas type theorems, and Lagrangian duality, J. Optim. Theory Appl., 102 (1999), 315-343.
doi: 10.1023/A:1021780423989. |
[10] |
A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces," Springer-Verlag, New York, 2003. |
[11] |
D. Gupta and A. Mehra, Two types of approximate saddle points, Numer. Func. Anal. Optim., 29 (2008), 532-550.
doi: 10.1080/01630560802099274. |
[12] |
C. Gutiérrez, B. Jiménez and V. Novo, A set-valued Ekeland's variational principle in vector optimization, SIAM J. Control Optim., 47 (2008), 883-903.
doi: 10.1137/060672868. |
[13] |
C. Gutiérrez, R. López and V. Novo, Generalized $\epsilon-$quasi-solutions in multiobjective optimization problems: Existence results and optimality conditions, Nonlinear Anal., 72 (2010), 4331-4346.
doi: 10.1016/j.na.2010.02.012. |
[14] |
C. Gutiérrez, B. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems, SIAM J. Optim., 17 (2006), 688-710.
doi: 10.1137/05062648X. |
[15] |
C. Gutiérrez, B. Jiménez and V. Novo, On approximate efficiency in multiobjective programming, Math. Methods Oper. Res., 64 (2006), 165-185.
doi: 10.1007/s00186-006-0078-0. |
[16] |
C. Gutiérrez, B. Jiménez and V. Novo, Optimality conditions via scalarization for a new $\epsilon$-efficiency concept in vector optimization problems, European J. Oper. Res., 201 (2010), 11-22.
doi: 10.1016/j.ejor.2009.02.007. |
[17] |
A. M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl., 22 (1968), 618-630.
doi: 10.1016/0022-247X(68)90201-1. |
[18] |
T. X. D. Ha, The Ekeland variational principle for Henig proper minimizers and super minimizers, J. Math. Anal. Appl., 364 (2010), 156-170.
doi: 10.1016/j.jmaa.2009.10.065. |
[19] |
S. Helbig, "On a new concept for $\epsilon$-efficency," A Talk at Optimization Days 1992, Montreal,1992. |
[20] |
M. I. Henig, Proper efficiency with respect to cones, J. Optim. Theory Appl., 36 (1982), 387-407.
doi: 10.1007/BF00934353. |
[21] |
J. B. Hiriart-Urruty, New concepts in nondifferentiable programming, Bull. Soc. Math. France Mém., 60 (1979), 57-85. |
[22] |
J. B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res., 4 (1979), 79-97.
doi: 10.1287/moor.4.1.79. |
[23] |
S. Kutateladze, Convex $\epsilon-$programming, Soviet Math. Dokl., 20 (1979), 391-393. |
[24] |
H. W. Kuhn and A. W. Tucker, Nonlinear programming, from "Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability," Berkeley, California, (1951), 481-492 |
[25] |
J. C. Liu, $\epsilon-$properly efficient solutions to nondifferentiable multiobjective programming problems, Appl. Math. Lett., 12 (1999), 109-113.
doi: 10.1016/S0893-9659(99)00087-7. |
[26] |
Z. Li and S. Wang, $\epsilon$-approximate solutions in multiobjective optimization, Optimization, 44 (1998), 161-174.
doi: 10.1080/02331939808844406. |
[27] |
C. G. Liu, K. F. Ng and W. H. Yang, Merit functions in vector optimization, Math. Program., Ser. A, 119 (2009), 215-237.
doi: 10.1016/j.colsurfa.2009.04.036. |
[28] |
A. B. Németh, A nonconvex vector minimization problem, Nonlinear Anal., 10 (1986), 669-678.
doi: 10.1016/0362-546X(86)90126-4. |
[29] |
W. D. Rong and Y. N. Wu, $\epsilon$-weak minimal solutions of vector optimization problems with set-valued maps, J. Optim. Theory Appl., 106 (2000), 569-579.
doi: 10.1023/A:1004657412928. |
[30] |
W. D. Rong, $\epsilon-$efficiency in vector optimization problems with cone subconvexlikeness, Acta Sci. Natur. Univ. NeiMongol., 28 (1997), 609-613. |
[31] |
T. Tanaka, A new approach to approximation of solutions in vector optimization problems, in "Proceedings of APORS, 1994" M. Fushimi and K. Tone (eds), World Scientific, Singapore, (1995), 497-504. |
[32] |
I. Vályi, Approximate saddle-point theorems in vector optimization, J. Optim. Theory Appl., 55 (1987), 435-448.
doi: 10.1007/BF00941179. |
[33] |
D. J. White, Epsilon efficiency, J. Optim. Theory Appl., 49 (1986), 319-337.
doi: 10.1007/BF00940762. |
[34] |
A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM J. Control Optim., 42 (2003), 1071-1086.
doi: 10.1137/S0363012902411532. |
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