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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 and 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.

[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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