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A criterion for an approximation global optimal solution based on the filled functions
On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems
1. | Department of Mathematics and Physics, Shanghai Dianji University, Shanghai, 200240, China |
2. | School of Electric Engineering, Shanghai Dianji University, Shanghai, 200240, China |
3. | School of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai, 264005, China |
References:
[1] |
A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems,, SIAM Journal on Imaging Sciences, 2 (2009), 183.
doi: 10.1137/080716542. |
[2] |
A. Beck, M. Teboulle, A Fast Dual Proximal Gradient Algorithm for Convex Minimization and Applications,, Operations Research Letters, 42 (2014), 1.
doi: 10.1016/j.orl.2013.10.007. |
[3] |
D. P. Bertsekas, Constrained Optimization and Lagrangian Multipliers,, New York, (1982). Google Scholar |
[4] |
S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers,, Foundations and Trends in Machine Learning, 3 (2011), 1.
doi: 10.1561/2200000016. |
[5] |
P. L. Combettes and J. C. Presquet, Proximal splitting methods in signal processing,, in Fixed-Point Algorithms for Inverse Problems in Science and Engineering (eds. H. H. Bauschke, 49 (2011), 185.
doi: 10.1007/978-1-4419-9569-8_10. |
[6] |
D. Gabay, Applications of the method of multipliers to variational inequalities,, in Augmented Lagrangian Methods: Applications to the Solution of Boundary Value Problems, (1983), 299. Google Scholar |
[7] |
R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics,, volume 9, (1989).
doi: 10.1137/1.9781611970838. |
[8] |
P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators,, SIAM Journal on Numerical Analysis, 16 (1979), 964.
doi: 10.1137/0716071. |
[9] |
J. J. Moreau., Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273.
|
[10] |
Y. Nesterov, Gradient Methods for Minimizing Composite Objective Function,, CORE Discussion Papers, (2007). Google Scholar |
[11] |
G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space,, J. Math. Anal. Appl., 72 (1979), 383.
doi: 10.1016/0022-247X(79)90234-8. |
[12] |
R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM Journal on Control and Optimization, 14 (1976), 877.
doi: 10.1137/0314056. |
[13] |
R. T. Rockafellar, Convex Analysis,, Princeton NJ: Princeton Univ. Press, (1970).
|
[14] |
M. Schmidt, N. Le Roux, and F. Bach, Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization,, NIPS'11 - 25 th Annual Conference on Neural Information Processing Systems, (2011). Google Scholar |
[15] |
J. Yang and Y. Zhang, Alternating direction algorithms for l1-problems in compressive sensing,, SIAM journal on scientific computing, 33 (2011), 250.
doi: 10.1137/090777761. |
show all references
References:
[1] |
A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems,, SIAM Journal on Imaging Sciences, 2 (2009), 183.
doi: 10.1137/080716542. |
[2] |
A. Beck, M. Teboulle, A Fast Dual Proximal Gradient Algorithm for Convex Minimization and Applications,, Operations Research Letters, 42 (2014), 1.
doi: 10.1016/j.orl.2013.10.007. |
[3] |
D. P. Bertsekas, Constrained Optimization and Lagrangian Multipliers,, New York, (1982). Google Scholar |
[4] |
S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers,, Foundations and Trends in Machine Learning, 3 (2011), 1.
doi: 10.1561/2200000016. |
[5] |
P. L. Combettes and J. C. Presquet, Proximal splitting methods in signal processing,, in Fixed-Point Algorithms for Inverse Problems in Science and Engineering (eds. H. H. Bauschke, 49 (2011), 185.
doi: 10.1007/978-1-4419-9569-8_10. |
[6] |
D. Gabay, Applications of the method of multipliers to variational inequalities,, in Augmented Lagrangian Methods: Applications to the Solution of Boundary Value Problems, (1983), 299. Google Scholar |
[7] |
R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics,, volume 9, (1989).
doi: 10.1137/1.9781611970838. |
[8] |
P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators,, SIAM Journal on Numerical Analysis, 16 (1979), 964.
doi: 10.1137/0716071. |
[9] |
J. J. Moreau., Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273.
|
[10] |
Y. Nesterov, Gradient Methods for Minimizing Composite Objective Function,, CORE Discussion Papers, (2007). Google Scholar |
[11] |
G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space,, J. Math. Anal. Appl., 72 (1979), 383.
doi: 10.1016/0022-247X(79)90234-8. |
[12] |
R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM Journal on Control and Optimization, 14 (1976), 877.
doi: 10.1137/0314056. |
[13] |
R. T. Rockafellar, Convex Analysis,, Princeton NJ: Princeton Univ. Press, (1970).
|
[14] |
M. Schmidt, N. Le Roux, and F. Bach, Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization,, NIPS'11 - 25 th Annual Conference on Neural Information Processing Systems, (2011). Google Scholar |
[15] |
J. Yang and Y. Zhang, Alternating direction algorithms for l1-problems in compressive sensing,, SIAM journal on scientific computing, 33 (2011), 250.
doi: 10.1137/090777761. |
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