January  2011, 7(1): 157-174. doi: 10.3934/jimo.2011.7.157

Nonlinear augmented Lagrangian for nonconvex multiobjective optimization

1. 

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

2. 

Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

3. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

Received  October 2009 Revised  October 2010 Published  January 2011

In this paper, based on the ordering relations induced by a pointed, closed and convex cone with a nonempty interior, we propose a nonlinear augmented Lagrangian dual scheme for a nonconvex multiobjective optimization problem by applying a class of vector-valued nonlinear augmented Lagrangian penalty functions. We establish the weak and strong duality results, necessary and sufficient conditions for uniformly exact penalization and exact penalization in the framework of nonlinear augmented Lagrangian. Our results include several ones in the literature as special cases.
Citation: 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
References:
[1]

R. S. Burachik and A. Rubinov, Abstract convexity and augmented Lagrangians,, SIAM J. Optim., 18 (2007), 413.  doi: 10.1137/050647621.  Google Scholar

[2]

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

[3]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley and Sons, (1983).   Google Scholar

[4]

X. X. Huang and X. Q. Yang, A unified augmented Lagrangian approach to duality and exact penalization,, Math. Oper. Res., 28 (2003), 533.  doi: 10.1287/moor.28.3.533.16395.  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.  doi: 10.1137/S1052623401384850.  Google Scholar

[6]

X. X. Huang and X. Q. Yang, Duality for multiobjective optimization via nonlinear Lagrangian functions,, J. Optim. Theory Appl., 120 (2004), 111.  doi: 10.1023/B:JOTA.0000012735.86699.a1.  Google Scholar

[7]

X. X. Huang and X. Q. Yang, Duality and exact penalization for vector optimization via augmented Lagrangian,, J. Optim. Theory Appl., 111 (2001), 615.  doi: 10.1023/A:1012654128753.  Google Scholar

[8]

X. X. Huang, X. Q. Yang and K. L. Teo, Convergence analysis of a class of penalty methods for vector optimization problems with cone constraints,, J. Global Optim., 36 (2006), 637.  doi: 10.1007/s10898-004-1937-y.  Google Scholar

[9]

J. Jahn, "Vector Optimization-Theory, Applications and Extensions,", Springer, (2004).   Google Scholar

[10]

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.  doi: 10.1023/A:1009753224825.  Google Scholar

[11]

A. Nedić and A. Ozdaglar, Separation of nonconvex sets with general augmenting functions,, Math. Oper. Res., 33 (2008), 587.  doi: 10.1287/moor.1070.0296.  Google Scholar

[12]

A. Nedić and A. Ozdaglar, A geometric framework for nonconvex optimization duality using augmented Lagrangian functions,, J. Global Optim., 40 (2008), 545.  doi: 10.1007/s10898-006-9122-0.  Google Scholar

[13]

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

[14]

A. M. Rubinov, X. X. Huang and X. Q. Yang, The zero duality gap property and lower semicontinuity of the perturbation function,, Math. Oper. Res., 27 (2002), 775.  doi: 10.1287/moor.27.4.775.295.  Google Scholar

[15]

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

[16]

C. Singh, D. Bhatia and N. Rueda, Duality in nonlinear multiobjective programming using augmented Lagrangian functions,, J. Optim. Theory Appl., 88 (1996), 659.  doi: 10.1007/BF02192203.  Google Scholar

[17]

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)., (2009).   Google Scholar

[18]

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.  doi: 10.1007/s10957-007-9225-x.  Google Scholar

[19]

X. Q. Yang and X. X. Huang, A nonlinear Lagrangian approach to constrained optimization problems,, SIAM J. Optim., 11 (2001), 1119.  doi: 10.1137/S1052623400371806.  Google Scholar

[20]

Y. Y. Zhou and X. Q. Yang, Some results about duality and exact penalization,, J. Global Optim., 29 (2004), 497.  doi: 10.1023/B:JOGO.0000047916.73871.88.  Google Scholar

[21]

Y. Y. Zhou and X. Q. Yang, Augmented Lagrangian function, non-quadratic growth condition and exact penalization,, Oper. Res. Lett., 34 (2006), 127.  doi: 10.1016/j.orl.2005.03.008.  Google Scholar

[22]

Y. Y. Zhou and X. Q. Yang, Duality and penalization in optimization via an augmented Lagrangian function with applications,, J. Optim. Theory Appl., 140 (2009), 171.  doi: 10.1007/s10957-008-9455-6.  Google Scholar

show all references

References:
[1]

R. S. Burachik and A. Rubinov, Abstract convexity and augmented Lagrangians,, SIAM J. Optim., 18 (2007), 413.  doi: 10.1137/050647621.  Google Scholar

[2]

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

[3]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley and Sons, (1983).   Google Scholar

[4]

X. X. Huang and X. Q. Yang, A unified augmented Lagrangian approach to duality and exact penalization,, Math. Oper. Res., 28 (2003), 533.  doi: 10.1287/moor.28.3.533.16395.  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.  doi: 10.1137/S1052623401384850.  Google Scholar

[6]

X. X. Huang and X. Q. Yang, Duality for multiobjective optimization via nonlinear Lagrangian functions,, J. Optim. Theory Appl., 120 (2004), 111.  doi: 10.1023/B:JOTA.0000012735.86699.a1.  Google Scholar

[7]

X. X. Huang and X. Q. Yang, Duality and exact penalization for vector optimization via augmented Lagrangian,, J. Optim. Theory Appl., 111 (2001), 615.  doi: 10.1023/A:1012654128753.  Google Scholar

[8]

X. X. Huang, X. Q. Yang and K. L. Teo, Convergence analysis of a class of penalty methods for vector optimization problems with cone constraints,, J. Global Optim., 36 (2006), 637.  doi: 10.1007/s10898-004-1937-y.  Google Scholar

[9]

J. Jahn, "Vector Optimization-Theory, Applications and Extensions,", Springer, (2004).   Google Scholar

[10]

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.  doi: 10.1023/A:1009753224825.  Google Scholar

[11]

A. Nedić and A. Ozdaglar, Separation of nonconvex sets with general augmenting functions,, Math. Oper. Res., 33 (2008), 587.  doi: 10.1287/moor.1070.0296.  Google Scholar

[12]

A. Nedić and A. Ozdaglar, A geometric framework for nonconvex optimization duality using augmented Lagrangian functions,, J. Global Optim., 40 (2008), 545.  doi: 10.1007/s10898-006-9122-0.  Google Scholar

[13]

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

[14]

A. M. Rubinov, X. X. Huang and X. Q. Yang, The zero duality gap property and lower semicontinuity of the perturbation function,, Math. Oper. Res., 27 (2002), 775.  doi: 10.1287/moor.27.4.775.295.  Google Scholar

[15]

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

[16]

C. Singh, D. Bhatia and N. Rueda, Duality in nonlinear multiobjective programming using augmented Lagrangian functions,, J. Optim. Theory Appl., 88 (1996), 659.  doi: 10.1007/BF02192203.  Google Scholar

[17]

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)., (2009).   Google Scholar

[18]

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.  doi: 10.1007/s10957-007-9225-x.  Google Scholar

[19]

X. Q. Yang and X. X. Huang, A nonlinear Lagrangian approach to constrained optimization problems,, SIAM J. Optim., 11 (2001), 1119.  doi: 10.1137/S1052623400371806.  Google Scholar

[20]

Y. Y. Zhou and X. Q. Yang, Some results about duality and exact penalization,, J. Global Optim., 29 (2004), 497.  doi: 10.1023/B:JOGO.0000047916.73871.88.  Google Scholar

[21]

Y. Y. Zhou and X. Q. Yang, Augmented Lagrangian function, non-quadratic growth condition and exact penalization,, Oper. Res. Lett., 34 (2006), 127.  doi: 10.1016/j.orl.2005.03.008.  Google Scholar

[22]

Y. Y. Zhou and X. Q. Yang, Duality and penalization in optimization via an augmented Lagrangian function with applications,, J. Optim. Theory Appl., 140 (2009), 171.  doi: 10.1007/s10957-008-9455-6.  Google Scholar

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