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A new convergence proof of augmented Lagrangian-based method with full Jacobian decomposition for structured variational inequalities
1. | Higher Education Key Laboratory of Engineering and Scientific Computing, Taiyuan Normal University, Taiyuan 030012, Shanxi Province, China |
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S. Dafermos, Traffic Equilibrium and Variational Inequalities, Transportation Science, 14 (1980), 42-54.
doi: 10.1287/trsc.14.1.42. |
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doi: 10.1007/BF01581204. |
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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. |
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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. |
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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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
A. Nagurney, Network Economics: A Variational Inequality Approach, Kluwer Academic Publishers, Dordrecht, 1993.
doi: 10.1007/978-94-011-2178-1. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
A. Nagurney, Network Economics: A Variational Inequality Approach, Kluwer Academic Publishers, Dordrecht, 1993.
doi: 10.1007/978-94-011-2178-1. |
[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. |
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