American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A primal-dual algorithm for nonlinear programming exploiting negative curvature directions
  • NACO Home
  • This Issue
  • Next Article
    A note on monotone approximations of minimum and maximum functions and multi-objective problems
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

[1]

Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35
[2]

Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673
[3]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[4]

Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327
[5]

Chunrong Chen, T. C. Edwin Cheng, Shengji Li, Xiaoqi Yang. Nonlinear augmented Lagrangian for nonconvex multiobjective optimization. Journal of Industrial & Management Optimization, 2011, 7 (1) : 157-174. doi: 10.3934/jimo.2011.7.157
[6]

Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022
[7]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519
[8]

Xing Wang, Nan-Jing Huang. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial & Management Optimization, 2013, 9 (1) : 57-74. doi: 10.3934/jimo.2013.9.57
[9]

Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019
[10]

Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051
[11]

Zhiang Zhou, Xinmin Yang, Kequan Zhao. $E$-super efficiency of set-valued optimization problems involving improvement sets. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1031-1039. doi: 10.3934/jimo.2016.12.1031
[12]

Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115
[13]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087
[14]

Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1
[15]

Sina Greenwood, Rolf Suabedissen. 2-manifolds and inverse limits of set-valued functions on intervals. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246
[16]

Mariusz Michta. Stochastic inclusions with non-continuous set-valued operators. Conference Publications, 2009, 2009 (Special) : 548-557. doi: 10.3934/proc.2009.2009.548
[17]

Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309
[18]

Zengjing Chen, Yuting Lan, Gaofeng Zong. Strong law of large numbers for upper set-valued and fuzzy-set valued probability. Mathematical Control & Related Fields, 2015, 5 (3) : 435-452. doi: 10.3934/mcrf.2015.5.435
[19]

Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020176
[20]

Xinmin Yang. On symmetric and self duality in vector optimization problem. Journal of Industrial & Management Optimization, 2011, 7 (3) : 523-529. doi: 10.3934/jimo.2011.7.523

 Impact Factor: 

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences