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Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps
1. | Department of Mathematics, Chongqing Normal University, Chongqing 400047 |
2. | Department of Mathematics, Chongqing Normal University, Chongqing, 400047 |
3. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China |
4. | Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hong Kong, China |
References:
[1] |
T. Amahroq and A. Taa, On Lagrange Kuhn-Tucker multipliers for multiobjective optimization problems,, Optimization, 41 (1997), 159.
doi: 10.1080/02331939708844332. |
[2] |
S. Bolintineanu, Vector variational principles: $\epsilon$-efficiency and scalar stationarity,, Journal of Convex Analysis, 8 (2001), 71.
|
[3] |
J. Borwein, Proper efficient points for maximizations with respect to cones,, SIAM Journal of Control and Optimization, 15 (1977), 57.
doi: 10.1137/0315004. |
[4] |
G. Y. Chen, X. X. Huang and S. H. Hou, General Ekeland's variational principle for set-valued mappings,, Journal of Optimization Theory and Applications, 106 (2000), 151.
doi: 10.1023/A:1004663208905. |
[5] |
G. Y. Chen and W. D. Rong, Characterization of the benson proper efficiency for nonconvex vector optimization,, Journal of Optimization Theory and Applications, 98 (1998), 365.
doi: 10.1023/A:1022689517921. |
[6] |
M. Ciligot-Travain, On Lagrange Kuhn-Tucker multipliers for Pareto optimization problems,, Numerical Functional Analysis and Optimization, 15 (1994), 689.
doi: 10.1080/01630569408816587. |
[7] |
M. Durea, J. Dutta and C. Tammer, Lagrange multipliers for $\epsilon$-pareto solutions in vector optimization with nonsolid cones in Banach spaces,, Journal of Optimization Theory and Applications, 145 (2010), 196.
doi: 10.1007/s10957-009-9609-1. |
[8] |
J. Dutta and V. Veterivel, On approximate minima in vector optimization,, Numerical Functional Analysis and Optimization, 22 (2001), 845.
doi: 10.1081/NFA-100108312. |
[9] |
S. Helbig, One New Concept for $\epsilon$-efficency,, talk at, (1992). Google Scholar |
[10] |
Y. Gao, S. H. Hou and X. M. Yang, Existence and Optimality Conditions for Approximate Solutions to Vector Optimization Problems,, Jornal of Optimization Theory and application, 152 (2012), 97.
doi: 10.1007/s10957-011-9891-6. |
[11] |
Y. Gao, X. M. Yang and K. L. Teo, Optimality conditions for approximate solutions of vector optimization problems,, Journal of Industrial and Management Optimization, 7 (2011), 483.
doi: 10.3934/jimo.2011.7.483. |
[12] |
D. Gupta and A. Mehra, Two types of approximate saddle points,, Numerical Functional Analysis and Optimization, 29 (2008), 532.
doi: 10.1080/01630560802099274. |
[13] |
C. Gutiérrez, L. Huerga and V. Novo, Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems,, Journal of Mathematical Analysis and Applications, 389 (2012), 1046.
doi: 10.1016/j.jmaa.2011.12.050. |
[14] |
C. Gutiérrez, B. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems,, SIAM Journal on Optimization, 17 (2006), 688. Google Scholar |
[15] |
C. Gutiérrez, B. Jiménez and V. Novo, On approximate efficiency in multiobjective programming,, Mathematical Methods of Operations Research, 64 (2006), 165. Google Scholar |
[16] |
C. Gutiérrez, B. Jiménez and V. Novo, A set-valued Ekeland's variational principle in vector optimization,, SIAM Journal on Control And Optimization, 47 (2008), 883. Google Scholar |
[17] |
C. Gutiérrez, B. Jiménez and V. Novo, Multiplier rules and saddle-point theorems for Helbig's approximate solutions in convex Pareto problems,, Journal of Global Optimization, 32 (2005), 367.
doi: 10.1007/s10898-004-5904-4. |
[18] |
C. Gutiérrez, R. Lopez and V. Novo, Generalized $\epsilon$-quasi-solutions in multiobjective optimization problems: Existence results and optimality conditions,, Nonlinear Analysis, 72 (2010), 4331.
doi: 10.1016/j.na.2010.02.012. |
[19] |
A. Hamel, An $\epsilon$-Lagrange multiplier rule for a mathematical programming problem on Banach spaces,, Optimization, 49 (2001), 137.
doi: 10.1080/02331930108844524. |
[20] |
X. X. Huang, Optimality conditions and approximate optimality conditions in locally Lipschitz vector optimization,, Optimization, 51 (2002), 309.
doi: 10.1080/02331930290019440. |
[21] |
Y. W. Huang, Optimality conditions for vector optimization with set-valued maps,, Bulletin of the Australian Mathematical Society 66 (2002), 66 (2002), 317.
doi: 10.1017/S0004972700040168. |
[22] |
V. Jeyakumar, Convexlike alternative theorems and mathematical programming,, Optimization, 16 (1985), 643.
doi: 10.1080/02331938508843061. |
[23] |
S. S. Kutateladze, Convex $\epsilon-$programming,, Dokl. Akad. Nauk SSSR, 245 (1979), 1048.
|
[24] |
Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps,, Journal of Optimization Theory and Application, 100 (1999), 365.
doi: 10.1023/A:1021786303883. |
[25] |
Z. F. Li, Benson proper efficiency in the vector optimization of set-valued maps,, Journal of Optimization Theory and Applications, 98 (1998), 623.
doi: 10.1023/A:1022676013609. |
[26] |
Z. F. Li and S. Y. Wang, Lagrange multipliers and saddle points in multiobjective programming,, Journal of Optimization Theory and Applications, 83 (1994), 63.
doi: 10.1007/BF02191762. |
[27] |
J. H. Qiu, Dual characterization and scalarization for Benson properly efficiency,, SIAM Journal on Optimization, 19 (2008), 144.
doi: 10.1137/060676465. |
[28] |
J. H. Qiu, A generalized Ekeland vector variational principle and its applications in optimization,, Nonlinear Analysis, 71 (2009), 4705.
doi: 10.1016/j.na.2009.03.034. |
[29] |
J. H. Qiu, Ekeland's variational principle in locally convex spaces and the density of extremal points,, Journal of Mathematical Analysis and Applications, 360 (2009), 317.
doi: 10.1016/j.jmaa.2009.06.054. |
[30] |
P. H. Sach, Nearly subconvexlike set-valued maps and vector optimization problems,, Journal of Optimization Theory and Applications, 119 (2003), 335.
doi: 10.1023/B:JOTA.0000005449.20614.41. |
[31] |
T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, In Fushimi, (1994), 497. Google Scholar |
[32] |
D. J. White, Epsilon efficiency,, Journal of Optimization Theory and Applications, 49 (1986), 319.
doi: 10.1007/BF00940762. |
[33] |
X. M. Yang, Alternative theorems and optimality conditions with weakened convexity,, Opsearch, 29 (1992), 125. Google Scholar |
[34] |
X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions,, Journal of Optimization Theory and Applications, 110 (2001), 413.
doi: 10.1023/A:1017535631418. |
[35] |
X. M. Yang, X. Q. Yang and G. Y. Chen, Theorems of the alternative and optimization with set-valued maps,, Journal of Optimization Theory and Applications, 107 (2000), 627.
doi: 10.1023/A:1026407517917. |
show all references
References:
[1] |
T. Amahroq and A. Taa, On Lagrange Kuhn-Tucker multipliers for multiobjective optimization problems,, Optimization, 41 (1997), 159.
doi: 10.1080/02331939708844332. |
[2] |
S. Bolintineanu, Vector variational principles: $\epsilon$-efficiency and scalar stationarity,, Journal of Convex Analysis, 8 (2001), 71.
|
[3] |
J. Borwein, Proper efficient points for maximizations with respect to cones,, SIAM Journal of Control and Optimization, 15 (1977), 57.
doi: 10.1137/0315004. |
[4] |
G. Y. Chen, X. X. Huang and S. H. Hou, General Ekeland's variational principle for set-valued mappings,, Journal of Optimization Theory and Applications, 106 (2000), 151.
doi: 10.1023/A:1004663208905. |
[5] |
G. Y. Chen and W. D. Rong, Characterization of the benson proper efficiency for nonconvex vector optimization,, Journal of Optimization Theory and Applications, 98 (1998), 365.
doi: 10.1023/A:1022689517921. |
[6] |
M. Ciligot-Travain, On Lagrange Kuhn-Tucker multipliers for Pareto optimization problems,, Numerical Functional Analysis and Optimization, 15 (1994), 689.
doi: 10.1080/01630569408816587. |
[7] |
M. Durea, J. Dutta and C. Tammer, Lagrange multipliers for $\epsilon$-pareto solutions in vector optimization with nonsolid cones in Banach spaces,, Journal of Optimization Theory and Applications, 145 (2010), 196.
doi: 10.1007/s10957-009-9609-1. |
[8] |
J. Dutta and V. Veterivel, On approximate minima in vector optimization,, Numerical Functional Analysis and Optimization, 22 (2001), 845.
doi: 10.1081/NFA-100108312. |
[9] |
S. Helbig, One New Concept for $\epsilon$-efficency,, talk at, (1992). Google Scholar |
[10] |
Y. Gao, S. H. Hou and X. M. Yang, Existence and Optimality Conditions for Approximate Solutions to Vector Optimization Problems,, Jornal of Optimization Theory and application, 152 (2012), 97.
doi: 10.1007/s10957-011-9891-6. |
[11] |
Y. Gao, X. M. Yang and K. L. Teo, Optimality conditions for approximate solutions of vector optimization problems,, Journal of Industrial and Management Optimization, 7 (2011), 483.
doi: 10.3934/jimo.2011.7.483. |
[12] |
D. Gupta and A. Mehra, Two types of approximate saddle points,, Numerical Functional Analysis and Optimization, 29 (2008), 532.
doi: 10.1080/01630560802099274. |
[13] |
C. Gutiérrez, L. Huerga and V. Novo, Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems,, Journal of Mathematical Analysis and Applications, 389 (2012), 1046.
doi: 10.1016/j.jmaa.2011.12.050. |
[14] |
C. Gutiérrez, B. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems,, SIAM Journal on Optimization, 17 (2006), 688. Google Scholar |
[15] |
C. Gutiérrez, B. Jiménez and V. Novo, On approximate efficiency in multiobjective programming,, Mathematical Methods of Operations Research, 64 (2006), 165. Google Scholar |
[16] |
C. Gutiérrez, B. Jiménez and V. Novo, A set-valued Ekeland's variational principle in vector optimization,, SIAM Journal on Control And Optimization, 47 (2008), 883. Google Scholar |
[17] |
C. Gutiérrez, B. Jiménez and V. Novo, Multiplier rules and saddle-point theorems for Helbig's approximate solutions in convex Pareto problems,, Journal of Global Optimization, 32 (2005), 367.
doi: 10.1007/s10898-004-5904-4. |
[18] |
C. Gutiérrez, R. Lopez and V. Novo, Generalized $\epsilon$-quasi-solutions in multiobjective optimization problems: Existence results and optimality conditions,, Nonlinear Analysis, 72 (2010), 4331.
doi: 10.1016/j.na.2010.02.012. |
[19] |
A. Hamel, An $\epsilon$-Lagrange multiplier rule for a mathematical programming problem on Banach spaces,, Optimization, 49 (2001), 137.
doi: 10.1080/02331930108844524. |
[20] |
X. X. Huang, Optimality conditions and approximate optimality conditions in locally Lipschitz vector optimization,, Optimization, 51 (2002), 309.
doi: 10.1080/02331930290019440. |
[21] |
Y. W. Huang, Optimality conditions for vector optimization with set-valued maps,, Bulletin of the Australian Mathematical Society 66 (2002), 66 (2002), 317.
doi: 10.1017/S0004972700040168. |
[22] |
V. Jeyakumar, Convexlike alternative theorems and mathematical programming,, Optimization, 16 (1985), 643.
doi: 10.1080/02331938508843061. |
[23] |
S. S. Kutateladze, Convex $\epsilon-$programming,, Dokl. Akad. Nauk SSSR, 245 (1979), 1048.
|
[24] |
Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps,, Journal of Optimization Theory and Application, 100 (1999), 365.
doi: 10.1023/A:1021786303883. |
[25] |
Z. F. Li, Benson proper efficiency in the vector optimization of set-valued maps,, Journal of Optimization Theory and Applications, 98 (1998), 623.
doi: 10.1023/A:1022676013609. |
[26] |
Z. F. Li and S. Y. Wang, Lagrange multipliers and saddle points in multiobjective programming,, Journal of Optimization Theory and Applications, 83 (1994), 63.
doi: 10.1007/BF02191762. |
[27] |
J. H. Qiu, Dual characterization and scalarization for Benson properly efficiency,, SIAM Journal on Optimization, 19 (2008), 144.
doi: 10.1137/060676465. |
[28] |
J. H. Qiu, A generalized Ekeland vector variational principle and its applications in optimization,, Nonlinear Analysis, 71 (2009), 4705.
doi: 10.1016/j.na.2009.03.034. |
[29] |
J. H. Qiu, Ekeland's variational principle in locally convex spaces and the density of extremal points,, Journal of Mathematical Analysis and Applications, 360 (2009), 317.
doi: 10.1016/j.jmaa.2009.06.054. |
[30] |
P. H. Sach, Nearly subconvexlike set-valued maps and vector optimization problems,, Journal of Optimization Theory and Applications, 119 (2003), 335.
doi: 10.1023/B:JOTA.0000005449.20614.41. |
[31] |
T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, In Fushimi, (1994), 497. Google Scholar |
[32] |
D. J. White, Epsilon efficiency,, Journal of Optimization Theory and Applications, 49 (1986), 319.
doi: 10.1007/BF00940762. |
[33] |
X. M. Yang, Alternative theorems and optimality conditions with weakened convexity,, Opsearch, 29 (1992), 125. Google Scholar |
[34] |
X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions,, Journal of Optimization Theory and Applications, 110 (2001), 413.
doi: 10.1023/A:1017535631418. |
[35] |
X. M. Yang, X. Q. Yang and G. Y. Chen, Theorems of the alternative and optimization with set-valued maps,, Journal of Optimization Theory and Applications, 107 (2000), 627.
doi: 10.1023/A:1026407517917. |
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