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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.
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.
|
[3] |
H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones,, J. Math. Anal. Appl., 71 (1979), 232.
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.
|
[5] |
J. Borwein, Proper efficient points for maximizations with respect to cones,, SIAM J. Control Optim., 15 (1977), 57.
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 \textbf{541}, 541 (2005).
|
[7] |
J. Dutta and V. Vetrivel, On approximate minima in vector optimization,, Numer. Func. Anal. Optim., 22 (2001), 845.
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.
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.
doi: 10.1023/A:1021780423989. |
[10] |
A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces,", Springer-Verlag, (2003).
|
[11] |
D. Gupta and A. Mehra, Two types of approximate saddle points,, Numer. Func. Anal. Optim., 29 (2008), 532.
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.
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.
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.
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.
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.
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.
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.
doi: 10.1016/j.jmaa.2009.10.065. |
[19] |
S. Helbig, "On a new concept for $\epsilon$-efficency,", A Talk at Optimization Days 1992, (1992). Google Scholar |
[20] |
M. I. Henig, Proper efficiency with respect to cones,, J. Optim. Theory Appl., 36 (1982), 387.
doi: 10.1007/BF00934353. |
[21] |
J. B. Hiriart-Urruty, New concepts in nondifferentiable programming,, Bull. Soc. Math. France M\'em., 60 (1979), 57.
|
[22] |
J. B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces,, Math. Oper. Res., 4 (1979), 79.
doi: 10.1287/moor.4.1.79. |
[23] |
S. Kutateladze, Convex $\epsilon-$programming,, Soviet Math. Dokl., 20 (1979), 391. Google Scholar |
[24] |
H. W. Kuhn and A. W. Tucker, Nonlinear programming,, from, (1951), 481.
|
[25] |
J. C. Liu, $\epsilon-$properly efficient solutions to nondifferentiable multiobjective programming problems,, Appl. Math. Lett., 12 (1999), 109.
doi: 10.1016/S0893-9659(99)00087-7. |
[26] |
Z. Li and S. Wang, $\epsilon$-approximate solutions in multiobjective optimization,, Optimization, 44 (1998), 161.
doi: 10.1080/02331939808844406. |
[27] |
C. G. Liu, K. F. Ng and W. H. Yang, Merit functions in vector optimization,, Math. Program., 119 (2009), 215.
doi: 10.1016/j.colsurfa.2009.04.036. |
[28] |
A. B. Németh, A nonconvex vector minimization problem,, Nonlinear Anal., 10 (1986), 669.
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.
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.
|
[31] |
T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, in, (1995), 497. Google Scholar |
[32] |
I. Vályi, Approximate saddle-point theorems in vector optimization,, J. Optim. Theory Appl., 55 (1987), 435.
doi: 10.1007/BF00941179. |
[33] |
D. J. White, Epsilon efficiency,, J. Optim. Theory Appl., 49 (1986), 319.
doi: 10.1007/BF00940762. |
[34] |
A. Zaffaroni, Degrees of efficiency and degrees of minimality,, SIAM J. Control Optim., 42 (2003), 1071.
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.
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.
|
[3] |
H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones,, J. Math. Anal. Appl., 71 (1979), 232.
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.
|
[5] |
J. Borwein, Proper efficient points for maximizations with respect to cones,, SIAM J. Control Optim., 15 (1977), 57.
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 \textbf{541}, 541 (2005).
|
[7] |
J. Dutta and V. Vetrivel, On approximate minima in vector optimization,, Numer. Func. Anal. Optim., 22 (2001), 845.
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.
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.
doi: 10.1023/A:1021780423989. |
[10] |
A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces,", Springer-Verlag, (2003).
|
[11] |
D. Gupta and A. Mehra, Two types of approximate saddle points,, Numer. Func. Anal. Optim., 29 (2008), 532.
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.
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.
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.
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.
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.
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.
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.
doi: 10.1016/j.jmaa.2009.10.065. |
[19] |
S. Helbig, "On a new concept for $\epsilon$-efficency,", A Talk at Optimization Days 1992, (1992). Google Scholar |
[20] |
M. I. Henig, Proper efficiency with respect to cones,, J. Optim. Theory Appl., 36 (1982), 387.
doi: 10.1007/BF00934353. |
[21] |
J. B. Hiriart-Urruty, New concepts in nondifferentiable programming,, Bull. Soc. Math. France M\'em., 60 (1979), 57.
|
[22] |
J. B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces,, Math. Oper. Res., 4 (1979), 79.
doi: 10.1287/moor.4.1.79. |
[23] |
S. Kutateladze, Convex $\epsilon-$programming,, Soviet Math. Dokl., 20 (1979), 391. Google Scholar |
[24] |
H. W. Kuhn and A. W. Tucker, Nonlinear programming,, from, (1951), 481.
|
[25] |
J. C. Liu, $\epsilon-$properly efficient solutions to nondifferentiable multiobjective programming problems,, Appl. Math. Lett., 12 (1999), 109.
doi: 10.1016/S0893-9659(99)00087-7. |
[26] |
Z. Li and S. Wang, $\epsilon$-approximate solutions in multiobjective optimization,, Optimization, 44 (1998), 161.
doi: 10.1080/02331939808844406. |
[27] |
C. G. Liu, K. F. Ng and W. H. Yang, Merit functions in vector optimization,, Math. Program., 119 (2009), 215.
doi: 10.1016/j.colsurfa.2009.04.036. |
[28] |
A. B. Németh, A nonconvex vector minimization problem,, Nonlinear Anal., 10 (1986), 669.
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.
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.
|
[31] |
T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, in, (1995), 497. Google Scholar |
[32] |
I. Vályi, Approximate saddle-point theorems in vector optimization,, J. Optim. Theory Appl., 55 (1987), 435.
doi: 10.1007/BF00941179. |
[33] |
D. J. White, Epsilon efficiency,, J. Optim. Theory Appl., 49 (1986), 319.
doi: 10.1007/BF00940762. |
[34] |
A. Zaffaroni, Degrees of efficiency and degrees of minimality,, SIAM J. Control Optim., 42 (2003), 1071.
doi: 10.1137/S0363012902411532. |
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