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Relaxed augmented Lagrangian-based proximal point algorithms for convex optimization with linear constraints
1. | School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing, 210023, China |
2. | School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China |
3. | Department of Mathematics, Nanjing University, Nanjing, 210093 |
References:
[1] |
M. V. Afonso, J. M. Bioucas-Dias and M. A. Figueiredo, An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems,, IEEE Trans. Imag. Process., 20 (2011), 681.
doi: 10.1109/TIP.2010.2076294. |
[2] |
D. P. Bertsekas, Multiplier methods: A survey,, Automatica, 12 (1976), 133.
doi: 10.1016/0005-1098(76)90077-7. |
[3] |
D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods,, Academic Press, (1982).
|
[4] |
S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers,, Found. Trends Mach. Learn., 3 (2010), 1.
doi: 10.1561/2200000016. |
[5] |
G. Chen and M. Teboulle, A proximal-based decomposition method for convex minimization problems,, Math. Program., 64 (1994), 81.
doi: 10.1007/BF01582566. |
[6] |
J. Eckstein, Some saddle-function splitting methods for convex programming,, Optim. Methods Softw., 4 (1994), 75.
doi: 10.1080/10556789408805578. |
[7] |
J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators,, Math. Program., 55 (1992), 293.
doi: 10.1007/BF01581204. |
[8] |
J. Eckstein and M. Fukushima, Some reformulation and applications of the alternating direction method of multipliers,, in Large Scale Optimization: State of the Art (Eds. W. W. Hager, (1994), 115.
|
[9] |
E. Esser, Applications of Lagrangian-Based Alternating Direction Methods and Connections to Split Bregman,, UCLA CAM Report, (2009), 9. Google Scholar |
[10] |
F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems,, Springer-Verlag, (2003). Google Scholar |
[11] |
X. L. Fu and B. S. He, Self-adaptive projection-based prediction-correction method for constrained variational inequalities,, Front. Math. China, 5 (2010), 3.
doi: 10.1007/s11464-009-0045-1. |
[12] |
M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems,, Comput. Optim. Appl., 1 (1992), 93.
doi: 10.1007/BF00247655. |
[13] |
D. Gabay, Applications of the method of multipliers to variational inequalities,, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems (eds. M. Fortin and R. Glowinski), (1983), 299.
doi: 10.1016/S0168-2024(08)70034-1. |
[14] |
R. Glowinski, Numerical Methods for Nonlinear Variational Problems,, Springer-Verlag, (1984).
|
[15] |
R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM, (1989).
doi: 10.1137/1.9781611970838. |
[16] |
T. Goldstein and S. Osher, The split bregman method for l1-regularized problems,, SIAM J. Imag. Sci., 2 (2009), 323.
doi: 10.1137/080725891. |
[17] |
G. H. Golub and C. F. Van Loan, Matrix Computations,, Johns Hopkins University Press, (1996).
|
[18] |
B. He, L. Liao, D. Han and H. Yang, A new inexact alternating directions method for monotone variational inequalities,, Math. Program., 92 (2002), 103.
doi: 10.1007/s101070100280. |
[19] |
B. He, L. Liao and X. Wang, Proximal-like contraction methods for monotone variational inequalities in a unified framework,, Comput. Optim. Appl., 51 (2012), 649.
doi: 10.1007/s10589-010-9372-0. |
[20] |
B. He and H. Yang, Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities,, Oper. Res. Lett., 23 (1998), 151.
doi: 10.1016/S0167-6377(98)00044-3. |
[21] |
B. He and X. Yuan, On the $\mathcalO(1/t)$ convergence rate of the alternating direction method,, SIAM J. Numer. Anal., 50 (2012), 700.
doi: 10.1137/110836936. |
[22] |
M. R. Hestenes, Multiplier and gradient methods,, J. Optim. Theory App., 4 (1969), 303.
doi: 10.1007/BF00927673. |
[23] |
Z. Lin, M. Chen, L. Wu and Y. Ma, The augmented lagrange multiplier method for exact recovery of a corrupted low-rank matrices,, preprint, (). Google Scholar |
[24] |
A. Nagurney, Network Economics, A Variational Inequality Approach,, Kluwer Academics Publishers, (1993).
doi: 10.1007/978-94-011-2178-1. |
[25] |
M. J. D. Powell, A method for nonlinear constraints in minimization problems,, in Optimization (ed. R. Fletcher), (1969), 283.
|
[26] |
S. Ramani and J. A. Fessler, Parallel mr image reconstruction using augmented lagrangian methods,, IEEE Trans. Medical Imaging, 30 (2011), 694.
doi: 10.1109/TMI.2010.2093536. |
[27] |
R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming,, Math. Oper. Res., 1 (1976), 97.
doi: 10.1287/moor.1.2.97. |
[28] |
Y. Shen, Partial convolution total variation problem by augmented Lagrangian-based proximal point descent algorithm,, submitted to J. Comput. Math., (2013). Google Scholar |
[29] |
Y. Shen and B. He, New augmented Lagrangian-based proximal point algorithms for constrained optimization,, Submitted to Front. Math. China, (). Google Scholar |
[30] |
Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction,, SIAM J. Imaging. Sci., 1 (2008), 248.
doi: 10.1137/080724265. |
[31] |
J. Yang and Y. Zhang, Alternating direction algorithms for $l_1$-problems in compressive sensing,, SIAM J. Sci. Comput., 33 (2011), 250.
doi: 10.1137/090777761. |
[32] |
J. Yang, Y. Zhang and W. Yin, An efficient tvl1 algorithm for deblurring multichannel images corrupted by impulsive noise,, SIAM J. Sci. Comput., 31 (2009), 2842.
doi: 10.1137/080732894. |
[33] |
W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for $l_1$-minimization with applications to compressed sensing,, SIAM J. Imaging Sci., 1 (2008), 143.
doi: 10.1137/070703983. |
show all references
References:
[1] |
M. V. Afonso, J. M. Bioucas-Dias and M. A. Figueiredo, An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems,, IEEE Trans. Imag. Process., 20 (2011), 681.
doi: 10.1109/TIP.2010.2076294. |
[2] |
D. P. Bertsekas, Multiplier methods: A survey,, Automatica, 12 (1976), 133.
doi: 10.1016/0005-1098(76)90077-7. |
[3] |
D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods,, Academic Press, (1982).
|
[4] |
S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers,, Found. Trends Mach. Learn., 3 (2010), 1.
doi: 10.1561/2200000016. |
[5] |
G. Chen and M. Teboulle, A proximal-based decomposition method for convex minimization problems,, Math. Program., 64 (1994), 81.
doi: 10.1007/BF01582566. |
[6] |
J. Eckstein, Some saddle-function splitting methods for convex programming,, Optim. Methods Softw., 4 (1994), 75.
doi: 10.1080/10556789408805578. |
[7] |
J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators,, Math. Program., 55 (1992), 293.
doi: 10.1007/BF01581204. |
[8] |
J. Eckstein and M. Fukushima, Some reformulation and applications of the alternating direction method of multipliers,, in Large Scale Optimization: State of the Art (Eds. W. W. Hager, (1994), 115.
|
[9] |
E. Esser, Applications of Lagrangian-Based Alternating Direction Methods and Connections to Split Bregman,, UCLA CAM Report, (2009), 9. Google Scholar |
[10] |
F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems,, Springer-Verlag, (2003). Google Scholar |
[11] |
X. L. Fu and B. S. He, Self-adaptive projection-based prediction-correction method for constrained variational inequalities,, Front. Math. China, 5 (2010), 3.
doi: 10.1007/s11464-009-0045-1. |
[12] |
M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems,, Comput. Optim. Appl., 1 (1992), 93.
doi: 10.1007/BF00247655. |
[13] |
D. Gabay, Applications of the method of multipliers to variational inequalities,, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems (eds. M. Fortin and R. Glowinski), (1983), 299.
doi: 10.1016/S0168-2024(08)70034-1. |
[14] |
R. Glowinski, Numerical Methods for Nonlinear Variational Problems,, Springer-Verlag, (1984).
|
[15] |
R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM, (1989).
doi: 10.1137/1.9781611970838. |
[16] |
T. Goldstein and S. Osher, The split bregman method for l1-regularized problems,, SIAM J. Imag. Sci., 2 (2009), 323.
doi: 10.1137/080725891. |
[17] |
G. H. Golub and C. F. Van Loan, Matrix Computations,, Johns Hopkins University Press, (1996).
|
[18] |
B. He, L. Liao, D. Han and H. Yang, A new inexact alternating directions method for monotone variational inequalities,, Math. Program., 92 (2002), 103.
doi: 10.1007/s101070100280. |
[19] |
B. He, L. Liao and X. Wang, Proximal-like contraction methods for monotone variational inequalities in a unified framework,, Comput. Optim. Appl., 51 (2012), 649.
doi: 10.1007/s10589-010-9372-0. |
[20] |
B. He and H. Yang, Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities,, Oper. Res. Lett., 23 (1998), 151.
doi: 10.1016/S0167-6377(98)00044-3. |
[21] |
B. He and X. Yuan, On the $\mathcalO(1/t)$ convergence rate of the alternating direction method,, SIAM J. Numer. Anal., 50 (2012), 700.
doi: 10.1137/110836936. |
[22] |
M. R. Hestenes, Multiplier and gradient methods,, J. Optim. Theory App., 4 (1969), 303.
doi: 10.1007/BF00927673. |
[23] |
Z. Lin, M. Chen, L. Wu and Y. Ma, The augmented lagrange multiplier method for exact recovery of a corrupted low-rank matrices,, preprint, (). Google Scholar |
[24] |
A. Nagurney, Network Economics, A Variational Inequality Approach,, Kluwer Academics Publishers, (1993).
doi: 10.1007/978-94-011-2178-1. |
[25] |
M. J. D. Powell, A method for nonlinear constraints in minimization problems,, in Optimization (ed. R. Fletcher), (1969), 283.
|
[26] |
S. Ramani and J. A. Fessler, Parallel mr image reconstruction using augmented lagrangian methods,, IEEE Trans. Medical Imaging, 30 (2011), 694.
doi: 10.1109/TMI.2010.2093536. |
[27] |
R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming,, Math. Oper. Res., 1 (1976), 97.
doi: 10.1287/moor.1.2.97. |
[28] |
Y. Shen, Partial convolution total variation problem by augmented Lagrangian-based proximal point descent algorithm,, submitted to J. Comput. Math., (2013). Google Scholar |
[29] |
Y. Shen and B. He, New augmented Lagrangian-based proximal point algorithms for constrained optimization,, Submitted to Front. Math. China, (). Google Scholar |
[30] |
Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction,, SIAM J. Imaging. Sci., 1 (2008), 248.
doi: 10.1137/080724265. |
[31] |
J. Yang and Y. Zhang, Alternating direction algorithms for $l_1$-problems in compressive sensing,, SIAM J. Sci. Comput., 33 (2011), 250.
doi: 10.1137/090777761. |
[32] |
J. Yang, Y. Zhang and W. Yin, An efficient tvl1 algorithm for deblurring multichannel images corrupted by impulsive noise,, SIAM J. Sci. Comput., 31 (2009), 2842.
doi: 10.1137/080732894. |
[33] |
W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for $l_1$-minimization with applications to compressed sensing,, SIAM J. Imaging Sci., 1 (2008), 143.
doi: 10.1137/070703983. |
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