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A note on monotone approximations of minimum and maximum functions and multi-objective problems
A unified nonlinear augmented Lagrangian approach for nonconvex vector optimization
1. | College of Mathematics and Statistics, Chongqing University, Chongqing, 401331 |
References:
[1] |
C. R. Chen, T. C. E. Cheng, S. J. Li and X. Q. Yang, Nonlinear augmented Lagrangian for nonconvex multiobjective optimization, J. Ind. Manag. Optim., 7 (2011), 157-174.
doi: 10.3934/jimo.2011.7.157. |
[2] |
G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis," Springer, Berlin, 2005. |
[3] |
X. X. Huang and X. Q. Yang, A unified augmented Lagrangian approach to duality and exact penalization, Math. Oper. Res., 28 (2003), 533-552.
doi: 10.1287/moor.28.3.533.16395. |
[4] |
X. X. Huang and X. Q. Yang, Duality and exact penalization for vector optimization via augmented Lagrangian, J. Optim. Theory Appl., 111 (2001), 615-640.
doi: 10.1023/A:1012654128753. |
[5] |
X. X. Huang and X. Q. Yang, Nonlinear Lagrangian for multiobjective optimization and applications to duality and exact penalization, SIAM J. Optim., 13 (2002), 675-692.
doi: 10.1137/S1052623401384850. |
[6] |
P. Q. Khanh, T. H. Nuong and M. Thera, On duality in nonconvex vector optimization in Banach spaces using augmented Lagrangians, Positivity, 3 (1999), 49-64.
doi: 10.1023/A:1009753224825. |
[7] |
R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-642-02431-3. |
[8] |
A. M. Rubinov and X. Q. Yang, "Lagrange-Type Functions in Constrained Non-Convex Optimization," Kluwer Academic Publishers, Dordrecht, 2003. |
[9] |
Y. Sawaragi, H. Nakayama and T. Tanino, "Theory of Multiobjective Optimization," Academic Press, New York, 1985. |
[10] |
C. Singh, D. Bhatia and N. Rueda, Duality in nonlinear multiobjective programming using augmented Lagrangian functions, J. Optim. Theory Appl., 88 (1996), 659-670.
doi: 10.1007/BF02192203. |
[11] |
C. Y. Wang, X. Q. Yang and X. M. Yang, Unified nonlinear Lagrangian approach to duality and optimal paths, J. Optim. Theory Appl., 135 (2007), 85-100.
doi: 10.1007/s10957-007-9225-x. |
[12] |
C. Y. Wang, X. Q. Yang and X. M. Yang, Zero duality gap and convergence of sub-optimal paths for optimization problems via a nonlinear augmented Lagrangian, (2009) (preprint). |
[13] |
X. Q. Yang and X. X. Huang, A nonlinear Lagrangian approach to constrained optimization problems, SIAM J. Optim., 11 (2001), 1119-1144.
doi: 10.1137/S1052623400371806. |
[14] |
Y. Y. Zhou and X. Q. Yang, Augmented Lagrangian function, non-quadratic growth condition and exact penalization, Oper. Res. Lett., 34 (2006), 127-134.
doi: 10.1016/j.orl.2005.03.008. |
[15] |
Y. Y. Zhou and X. Q. Yang, Some results about duality and exact penalization, J. Global Optim., 29 (2004), 497-509.
doi: 10.1023/B:JOGO.0000047916.73871.88. |
show all references
References:
[1] |
C. R. Chen, T. C. E. Cheng, S. J. Li and X. Q. Yang, Nonlinear augmented Lagrangian for nonconvex multiobjective optimization, J. Ind. Manag. Optim., 7 (2011), 157-174.
doi: 10.3934/jimo.2011.7.157. |
[2] |
G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis," Springer, Berlin, 2005. |
[3] |
X. X. Huang and X. Q. Yang, A unified augmented Lagrangian approach to duality and exact penalization, Math. Oper. Res., 28 (2003), 533-552.
doi: 10.1287/moor.28.3.533.16395. |
[4] |
X. X. Huang and X. Q. Yang, Duality and exact penalization for vector optimization via augmented Lagrangian, J. Optim. Theory Appl., 111 (2001), 615-640.
doi: 10.1023/A:1012654128753. |
[5] |
X. X. Huang and X. Q. Yang, Nonlinear Lagrangian for multiobjective optimization and applications to duality and exact penalization, SIAM J. Optim., 13 (2002), 675-692.
doi: 10.1137/S1052623401384850. |
[6] |
P. Q. Khanh, T. H. Nuong and M. Thera, On duality in nonconvex vector optimization in Banach spaces using augmented Lagrangians, Positivity, 3 (1999), 49-64.
doi: 10.1023/A:1009753224825. |
[7] |
R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-642-02431-3. |
[8] |
A. M. Rubinov and X. Q. Yang, "Lagrange-Type Functions in Constrained Non-Convex Optimization," Kluwer Academic Publishers, Dordrecht, 2003. |
[9] |
Y. Sawaragi, H. Nakayama and T. Tanino, "Theory of Multiobjective Optimization," Academic Press, New York, 1985. |
[10] |
C. Singh, D. Bhatia and N. Rueda, Duality in nonlinear multiobjective programming using augmented Lagrangian functions, J. Optim. Theory Appl., 88 (1996), 659-670.
doi: 10.1007/BF02192203. |
[11] |
C. Y. Wang, X. Q. Yang and X. M. Yang, Unified nonlinear Lagrangian approach to duality and optimal paths, J. Optim. Theory Appl., 135 (2007), 85-100.
doi: 10.1007/s10957-007-9225-x. |
[12] |
C. Y. Wang, X. Q. Yang and X. M. Yang, Zero duality gap and convergence of sub-optimal paths for optimization problems via a nonlinear augmented Lagrangian, (2009) (preprint). |
[13] |
X. Q. Yang and X. X. Huang, A nonlinear Lagrangian approach to constrained optimization problems, SIAM J. Optim., 11 (2001), 1119-1144.
doi: 10.1137/S1052623400371806. |
[14] |
Y. Y. Zhou and X. Q. Yang, Augmented Lagrangian function, non-quadratic growth condition and exact penalization, Oper. Res. Lett., 34 (2006), 127-134.
doi: 10.1016/j.orl.2005.03.008. |
[15] |
Y. Y. Zhou and X. Q. Yang, Some results about duality and exact penalization, J. Global Optim., 29 (2004), 497-509.
doi: 10.1023/B:JOGO.0000047916.73871.88. |
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