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

[1]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164

[2]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[3]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[4]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020031

[5]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029

[6]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[7]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072

[8]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[9]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[10]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[11]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[12]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[13]

M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014

[14]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[15]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[16]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[17]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[18]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[19]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[20]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (6)

[Back to Top]