American Institute of Mathematical Sciences

Advanced
2016, 6(1): 45-54. doi: 10.3934/naco.2016.6.45

A new convergence proof of augmented Lagrangian-based method with full Jacobian decomposition for structured variational inequalities

Xi-Hong Yan 1,

1. 

Higher Education Key Laboratory of Engineering and Scientific Computing, Taiyuan Normal University, Taiyuan 030012, Shanxi Province, China

Received  June 2015 Revised  January 2016 Published  January 2016

In the work, we present a new proof for global convergence of a classical method, augmented Lagrangian-based method with full Jacobian decomposition, for a special class of variational inequality problems with a separable structure. This work can be regarded as an improvement to work [14]. The convergence result of the work is established under more general conditions and proven in a new way.
Keywords: Jacobian decomposition, augmented Lagrangian-based method., convergence, Variational inequality, separable structure.
Mathematics Subject Classification: Primary: 49M27, 65K15, 93B4.
Citation: Xi-Hong Yan. A new convergence proof of augmented Lagrangian-based method with full Jacobian decomposition for structured variational inequalities. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 45-54. doi: 10.3934/naco.2016.6.45
References:
[1]

S. Dafermos, Traffic Equilibrium and Variational Inequalities, Transportation Science, 14 (1980), 42-54. doi: 10.1287/trsc.14.1.42.  Google Scholar

[2]

J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318. doi: 10.1007/BF01581204.  Google Scholar

[3]

M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems, Computational Optimization and Applications, 1 (1992), 93-111. doi: 10.1007/BF00247655.  Google Scholar

[4]

D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Computers and Mathematics with Applications, 2 (1976), 17-40. Google Scholar

[5]

D. Han, H. He and L. Xu, A proximal parallel splitting method for minimizing sum of convex functions with linear constraints, Journal of Computational and Applied Mathematics, 256 (2014), 36-51. doi: 10.1016/j.cam.2013.07.010.  Google Scholar

[6]

D. Han and X. Yuan, A Note on the Alternating Direction Method of Multipliers, Journal of Optimization Theory and Applications, 155 (2012), 227-238. doi: 10.1007/s10957-012-0003-z.  Google Scholar

[7]

D. Han, X. Yuan and W. Zhang, An augmented Lagrangian based parallel splitting method for separable convex minimization with applications to image processing, Mathematics of Computation, 83 (2014), 2263-2291. doi: 10.1090/S0025-5718-2014-02829-9.  Google Scholar

[8]

P. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical Programming, 48 (1990), 161-220. doi: 10.1007/BF01582255.  Google Scholar

[9]

B. He, L. Hou and X. Yuan, On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming, Preprint, 2013. doi: 10.1137/130922793.  Google Scholar

[10]

B. He, M. Tao, M. Xu and X. Yuan, An alternating direction-based contraction method for linearly constrained separable convex programming problems, Optimization, 62 (2013), 573-596. doi: 10.1080/02331934.2011.611885.  Google Scholar

[11]

B. S. He, M. Tao and X.M. Yuan, Alternating direction method with Gaussian-back substitution for separable convex programming, SIAM Journal on Optimization, 22 (2012), 313-340. doi: 10.1137/110822347.  Google Scholar

[12]

B. He, M. Tao and X. Yuan, A splitting method for separable convex programming, IMA Journal of Numerical Analysis, 35 (2015), 394-426. doi: 10.1093/imanum/drt060.  Google Scholar

[13]

A. Nagurney, Network Economics: A Variational Inequality Approach, Kluwer Academic Publishers, Dordrecht, 1993. doi: 10.1007/978-94-011-2178-1.  Google Scholar

[14]

K. Wang, J. Desai and H. He, A note on augmented Lagrangian-based parallel splitting method, Optimization Letters, (2014), 1-14. doi: 10.1007/s11590-014-0825-8.  Google Scholar

show all references

References:
[1]

S. Dafermos, Traffic Equilibrium and Variational Inequalities, Transportation Science, 14 (1980), 42-54. doi: 10.1287/trsc.14.1.42.  Google Scholar

[2]

J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318. doi: 10.1007/BF01581204.  Google Scholar

[3]

M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems, Computational Optimization and Applications, 1 (1992), 93-111. doi: 10.1007/BF00247655.  Google Scholar

[4]

D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Computers and Mathematics with Applications, 2 (1976), 17-40. Google Scholar

[5]

D. Han, H. He and L. Xu, A proximal parallel splitting method for minimizing sum of convex functions with linear constraints, Journal of Computational and Applied Mathematics, 256 (2014), 36-51. doi: 10.1016/j.cam.2013.07.010.  Google Scholar

[6]

D. Han and X. Yuan, A Note on the Alternating Direction Method of Multipliers, Journal of Optimization Theory and Applications, 155 (2012), 227-238. doi: 10.1007/s10957-012-0003-z.  Google Scholar

[7]

D. Han, X. Yuan and W. Zhang, An augmented Lagrangian based parallel splitting method for separable convex minimization with applications to image processing, Mathematics of Computation, 83 (2014), 2263-2291. doi: 10.1090/S0025-5718-2014-02829-9.  Google Scholar

[8]

P. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical Programming, 48 (1990), 161-220. doi: 10.1007/BF01582255.  Google Scholar

[9]

B. He, L. Hou and X. Yuan, On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming, Preprint, 2013. doi: 10.1137/130922793.  Google Scholar

[10]

B. He, M. Tao, M. Xu and X. Yuan, An alternating direction-based contraction method for linearly constrained separable convex programming problems, Optimization, 62 (2013), 573-596. doi: 10.1080/02331934.2011.611885.  Google Scholar

[11]

B. S. He, M. Tao and X.M. Yuan, Alternating direction method with Gaussian-back substitution for separable convex programming, SIAM Journal on Optimization, 22 (2012), 313-340. doi: 10.1137/110822347.  Google Scholar

[12]

B. He, M. Tao and X. Yuan, A splitting method for separable convex programming, IMA Journal of Numerical Analysis, 35 (2015), 394-426. doi: 10.1093/imanum/drt060.  Google Scholar

[13]

A. Nagurney, Network Economics: A Variational Inequality Approach, Kluwer Academic Publishers, Dordrecht, 1993. doi: 10.1007/978-94-011-2178-1.  Google Scholar

[14]

K. Wang, J. Desai and H. He, A note on augmented Lagrangian-based parallel splitting method, Optimization Letters, (2014), 1-14. doi: 10.1007/s11590-014-0825-8.  Google Scholar

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