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2011, 1(3): 495-508. doi: 10.3934/naco.2011.1.495

A unified nonlinear augmented Lagrangian approach for nonconvex vector optimization

Chunrong Chen 1,

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331

Received  April 2011 Revised  July 2011 Published  September 2011

In this paper, we propose a unified nonlinear augmented Lagrangian dual approach for a nonconvex vector optimization problem by applying a class of vector-valued nonlinear augmented Lagrangian penalty functions. We establish weak and strong duality results, necessary and sufficient conditions for uniformly exact penalization and exact penalization in the framework of nonlinear augmented Lagrangian.
Keywords: nonlinear augmented Lagrangian, set-valued maps., Vector optimization, duality, exact penalization.
Mathematics Subject Classification: Primary: 90C29; Secondary: 90C4.
Citation: Chunrong Chen. A unified nonlinear augmented Lagrangian approach for nonconvex vector optimization. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 495-508. doi: 10.3934/naco.2011.1.495
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.  Google Scholar

[2]

G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis," Springer, Berlin, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[8]

A. M. Rubinov and X. Q. Yang, "Lagrange-Type Functions in Constrained Non-Convex Optimization," Kluwer Academic Publishers, Dordrecht, 2003.  Google Scholar

[9]

Y. Sawaragi, H. Nakayama and T. Tanino, "Theory of Multiobjective Optimization," Academic Press, New York, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis," Springer, Berlin, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[8]

A. M. Rubinov and X. Q. Yang, "Lagrange-Type Functions in Constrained Non-Convex Optimization," Kluwer Academic Publishers, Dordrecht, 2003.  Google Scholar

[9]

Y. Sawaragi, H. Nakayama and T. Tanino, "Theory of Multiobjective Optimization," Academic Press, New York, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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