Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications

Yixuan Yang; Yuchao Tang; Meng Wen; Tieyong Zeng

doi:10.3934/ipi.2021014

Previous Article
IPI Home
This Issue
Next Article
Synthetic-Aperture Radar image based positioning in GPS-denied environments using Deep Cosine Similarity Neural Networks

August 2021, 15(4): 787-825. doi: 10.3934/ipi.2021014

Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications

Yixuan Yang ^1, , Yuchao Tang ^1,, , Meng Wen ^2, and Tieyong Zeng ^3,

1.	Department of Mathematics, Nanchang University, Nanchang 330031, China
2.	School of Science, Xi'an Polytechnic University, Xi'an, 710048, China
3.	Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China

^* Corresponding author: Yuchao Tang

Received January 2020 Revised December 2020 Published August 2021 Early access February 2021

Fund Project: The second author is supported by the National Natural Science Foundation of China (12061045, 11661056), the China Postdoctoral Science Foundation (2015M571989) and the Jiangxi Province Postdoctoral Science Foundation (2015KY51). The third author is supported by the National Natural Science Foundation of China (12001416)

This paper is concerned with the monotone inclusion involving the sum of a finite number of maximally monotone operators and the parallel sum of two maximally monotone operators with bounded linear operators. To solve this monotone inclusion, we first transform it into the formulation of the sum of three maximally monotone operators in a proper product space. Then we derive two efficient iterative algorithms, which combine the partial inverse method with the preconditioned Douglas-Rachford splitting algorithm and the preconditioned proximal point algorithm. Furthermore, we develop an iterative algorithm, which relies on the preconditioned Douglas-Rachford splitting algorithm without using the partial inverse method. We carefully analyze the theoretical convergence of the proposed algorithms. Finally, in order to demonstrate the effectiveness and efficiency of these algorithms, we conduct numerical experiments on a novel image denoising model for salt-and-pepper noise removal. Numerical results show the good performance of the proposed algorithms.

Keywords: Maximally monotone operators, partial inverse, resolvent, Douglas-Rachford splitting algorithm, proximal point algorithm.

Mathematics Subject Classification: Primary: 90C25, 90C46; Secondary: 47A52.

Citation: Yixuan Yang, Yuchao Tang, Meng Wen, Tieyong Zeng. Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications. Inverse Problems & Imaging, 2021, 15 (4) : 787-825. doi: 10.3934/ipi.2021014

References:

[1]	M. A. Alghamdi, A. Alotaibi, P. L. Combettes and N. Shahzad, A primal-dual method of partial inverses for composite inclusions, Optim. Lett., 8 (2014), 2271-2284. doi: 10.1007/s11590-014-0734-x. Google Scholar
[2]	A. Alliney, A property of the minimum vectors of a regularization functional defined by means of the absolute norm, IEEE Trans. Signal Process., 45 (1997), 913-917. doi: 10.1109/78.564179. Google Scholar
[3]	A. Alotaibi, P. L. Combettes and N. Shahzad, Solving coupled composite monotone inclusions by successive fejér approximations of their Kukn-Tucker set, SIAM J. Optim., 24 (2014), 2076-2095. doi: 10.1137/130950616. Google Scholar
[4]	H. Attouch and M. Soueycatt, Augmented lagrangian and proximal alternating direction methods of multipliers in hilbert spaces, applications to games, pde's and control, Pacific. J. Optim., 5 (2008), 17-37. Google Scholar
[5]	H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2$^nd$ edition, Springer, Cham, 2017. doi: 10.1007/978-3-319-48311-5. Google Scholar
[6]	A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202. doi: 10.1137/080716542. Google Scholar
[7]	A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Trans. Image Process., 18 (2009), 2419-2434. doi: 10.1109/TIP.2009.2028250. Google Scholar
[8]	S. Becker, J. Bobin and E. J. Candès, NESTA: A fast and accurate first-order method for sparse recovery, SIAM J. Imaging Sci., 4 (2011), 1-39. doi: 10.1137/090756855. Google Scholar
[9]	S. R. Becker and P. L. Combettes, An algorithm for splitting parallel sums of linearly composed monotone operatos with applications to signal recovery, J. Nonlinear Convex Anal., 15 (2014), 137-159. Google Scholar
[10]	R. I. Boţ, E. R. Csetnek and A. Heinrich, A primal-dual splitting for finding zeros of sums of maximal monotone operators, SIAM J. Optim., 23 (2013), 2011-2036. doi: 10.1137/12088255X. Google Scholar
[11]	R. I. Boţ and C. Hendrich, Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators, Inverse Probl. Imaging, 10 (2016), 617-640. doi: 10.3934/ipi.2016014. Google Scholar
[12]	R. I. Boţ and C. Hendrich, Convergence analysis for a primal-dual monotone+skew splitting algorithm with applications to total variation minimization, J. Math. Imaging Vis., 49 (2014), 551-568. doi: 10.1007/s10851-013-0486-8. Google Scholar
[13]	R. I. Boţ and C. Hendrich, A Dougla-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel -sum type monotone operators, SIAM J. Optim., 23 (2013), 2541-2565. doi: 10.1137/120901106. Google Scholar
[14]	S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distrituted optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2010), 1-122. Google Scholar
[15]	L. M. Briceño-Arias and P. L. Combettes, A monotone+skew splitting model for composite monotone inclusions in duality, SIAM J. Optim., 21 (2011), 1230-1250. doi: 10.1137/10081602X. Google Scholar
[16]	L. M. Briceño-Arias, Forward-Douglas-Rachford splitting and forward-partial inverse method for solving monotone inclusions, Optim., 64 (2015), 1239-1261. doi: 10.1080/02331934.2013.855210. Google Scholar
[17]	L. M. Briceño-Arias, Foward-partial inverse forward splitting for solving monotone inclusions, J. Optim. Theory Appl., 166 (2015), 391-413. doi: 10.1007/s10957-015-0703-2. Google Scholar
[18]	A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Springer-Verlag, Berlin Heidelberg, 2012. Google Scholar
[19]	A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1. Google Scholar
[20]	A. Chambolle and T. Pock, An introduction to continuous optimization for imaging, Acta Numer., 25 (2016), 161-319. doi: 10.1017/S096249291600009X. Google Scholar
[21]	F. Chen, L. Shen, Y. Xu and X. Zeng, The moreau envelope approach for the l1/TV image denoising model, Inverse Probl. Imaging, 8 (2014), 53-77. doi: 10.3934/ipi.2014.8.53. Google Scholar
[22]	P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200. doi: 10.1137/050626090. Google Scholar
[23]	P. L. Combettes and J.-C. Pesquet, A Douglas-Rachford splitting approach to nonsmooth conve variational signal recovery, IEEE J. Sel. Top. Signal Process, 1 (2007), 564-574. doi: 10.1109/JSTSP.2007.910264. Google Scholar
[24]	P. L. Combettes and J.-C. Pesquet, Primal-dual splitting algorithm for solving inclusions with mixtures of composite, lipschitzian, and parallel-sum type monotone operators, Set-Valued Var. Anal., 20 (2012), 307-330. doi: 10.1007/s11228-011-0191-y. Google Scholar
[25]	P. L. Combettes and B. C. Vũ, Variable metric forward-backward splitting with applications to monotone inclusions in duality, Optim., 63 (2014), 1289-1318. doi: 10.1080/02331934.2012.733883. Google Scholar
[26]	P. L. Combettes and J. Eckstein, Asynchronous block-iterative primal-dual decomposition methods for monotone inclusions, Math. Program., 168 (2018), 645-672. doi: 10.1007/s10107-016-1044-0. Google Scholar
[27]	P. L. Combettes, Systems of structured monotone inclusions: Duality, algorithms, and applications, SIAM J. Optim., 23 (2013), 2420-2447. doi: 10.1137/130904160. Google Scholar
[28]	P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal., 16 (2009), 727-748. Google Scholar
[29]	L. Condat, A primal-dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl., 158 (2013), 460-479. doi: 10.1007/s10957-012-0245-9. Google Scholar
[30]	E. X. Fang, B. He, H. Liu and X. Yuan, Generalized alternating direction method of multipliers: new theoretical insights and applications, Math. Program. Comput., 7 (2015), 149-187. doi: 10.1007/s12532-015-0078-2. Google Scholar
[31]	M. L. N. Goncalves, M. M. Alves and J. G. Melo, Pointwise and ergodic convergence rates of variable metric proximal alternating direction method of multipliers, J. Optim. Theory Appl., 177 (2018), 448-478. doi: 10.1007/s10957-018-1232-6. Google Scholar
[32]	P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979. doi: 10.1137/0716071. Google Scholar
[33]	Y.-J. Liu, D. Sun and K.-C. Toh, An implementable proximal point algorithmic framework for nuclear norm minimization, Math. Program., 133 (2012), 399-436. doi: 10.1007/s10107-010-0437-8. Google Scholar
[34]	C. A. Micchelli, L. Shen, Y. Xu and X. Zeng, Proximity algorithms for the l1/TV image denoising model, Adv. Comput. Math., 38 (2013), 401-426. doi: 10.1007/s10444-011-9243-y. Google Scholar
[35]	C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denoising, Inverse Probl., 27 (2011), 045009, 30 pp. doi: 10.1088/0266-5611/27/4/045009. Google Scholar
[36]	M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vis., 20 (2004), 99-120. Google Scholar
[37]	H. Raguet, J. Fadili and G. Peyré, A generalized forward-backward splitting, SIAM J. Imaging Sci., 6 (2013), 1199-1226. doi: 10.1137/120872802. Google Scholar
[38]	H. Raguet and L. Landrieu, Preconditioning of a generalized forward-backward splitting and application to optimization on graphs, SIAM J. Imaging Sci., 8 (2015), 2706-2739. doi: 10.1137/15M1018253. Google Scholar
[39]	H. Raguet, A note on the forward-Douglas-Rachford splitting for monotone inclusion and convex optimization, Optim. Lett., 13 (2019), 717-740. doi: 10.1007/s11590-018-1272-8. Google Scholar
[40]	R. Shefi and M. Teboulle, Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization, SIAM J. Optim., 24 (2014), 269-297. doi: 10.1137/130910774. Google Scholar
[41]	E. Y. Sidky, J. H. Jørgensen and X. Pan, Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm, Phys. Med. Biol., 57 (2012), 3065-3091. doi: 10.1088/0031-9155/57/10/3065. Google Scholar
[42]	J. E. Spingarn, Partial inverse of a monotone operator, Appl. Math. Optim., 10 (1983), 247-265. doi: 10.1007/BF01448388. Google Scholar
[43]	J. E. Spingarn, Applications of the method of partial inverses to convex programming: Decomposition, Math. Programming, 32 (1985), 199-223. doi: 10.1007/BF01586091. Google Scholar
[44]	Y. C. Tang, G. R. Wu and C. X. Zhu, An inner-outer iteration method for solving convex optimization problems involving the sum of three convex functions (in chinese), Sci. Sic. Math., 49 (2019), 831-858. doi: 10.1360/SCM-2017-0313. Google Scholar
[45]	Y. C. Tang, C. X. Zhu, M. Wen and J. G. Peng, A splitting primal-dual proximity algorithm for solving composite optimization problems, Acta. Math. Sin.-English Ser., 33 (2017), 868-886. doi: 10.1007/s10114-016-5625-x. Google Scholar
[46]	P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446. doi: 10.1137/S0363012998338806. Google Scholar
[47]	B. C. Vũ, A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comput. Math., 38 (2013), 667-681. doi: 10.1007/s10444-011-9254-8. Google Scholar
[48]	Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861. Google Scholar
[49]	C. Zong, Y. Tang and Y. J. Cho, Convergence analysis of an inexact three-operator splitting algorithm, Symmetry, 10 (2018), 563. doi: 10.3390/sym10110563. Google Scholar

show all references

References:

[1]	M. A. Alghamdi, A. Alotaibi, P. L. Combettes and N. Shahzad, A primal-dual method of partial inverses for composite inclusions, Optim. Lett., 8 (2014), 2271-2284. doi: 10.1007/s11590-014-0734-x. Google Scholar
[2]	A. Alliney, A property of the minimum vectors of a regularization functional defined by means of the absolute norm, IEEE Trans. Signal Process., 45 (1997), 913-917. doi: 10.1109/78.564179. Google Scholar
[3]	A. Alotaibi, P. L. Combettes and N. Shahzad, Solving coupled composite monotone inclusions by successive fejér approximations of their Kukn-Tucker set, SIAM J. Optim., 24 (2014), 2076-2095. doi: 10.1137/130950616. Google Scholar
[4]	H. Attouch and M. Soueycatt, Augmented lagrangian and proximal alternating direction methods of multipliers in hilbert spaces, applications to games, pde's and control, Pacific. J. Optim., 5 (2008), 17-37. Google Scholar
[5]	H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2$^nd$ edition, Springer, Cham, 2017. doi: 10.1007/978-3-319-48311-5. Google Scholar
[6]	A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202. doi: 10.1137/080716542. Google Scholar
[7]	A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Trans. Image Process., 18 (2009), 2419-2434. doi: 10.1109/TIP.2009.2028250. Google Scholar
[8]	S. Becker, J. Bobin and E. J. Candès, NESTA: A fast and accurate first-order method for sparse recovery, SIAM J. Imaging Sci., 4 (2011), 1-39. doi: 10.1137/090756855. Google Scholar
[9]	S. R. Becker and P. L. Combettes, An algorithm for splitting parallel sums of linearly composed monotone operatos with applications to signal recovery, J. Nonlinear Convex Anal., 15 (2014), 137-159. Google Scholar
[10]	R. I. Boţ, E. R. Csetnek and A. Heinrich, A primal-dual splitting for finding zeros of sums of maximal monotone operators, SIAM J. Optim., 23 (2013), 2011-2036. doi: 10.1137/12088255X. Google Scholar
[11]	R. I. Boţ and C. Hendrich, Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators, Inverse Probl. Imaging, 10 (2016), 617-640. doi: 10.3934/ipi.2016014. Google Scholar
[12]	R. I. Boţ and C. Hendrich, Convergence analysis for a primal-dual monotone+skew splitting algorithm with applications to total variation minimization, J. Math. Imaging Vis., 49 (2014), 551-568. doi: 10.1007/s10851-013-0486-8. Google Scholar
[13]	R. I. Boţ and C. Hendrich, A Dougla-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel -sum type monotone operators, SIAM J. Optim., 23 (2013), 2541-2565. doi: 10.1137/120901106. Google Scholar
[14]	S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distrituted optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2010), 1-122. Google Scholar
[15]	L. M. Briceño-Arias and P. L. Combettes, A monotone+skew splitting model for composite monotone inclusions in duality, SIAM J. Optim., 21 (2011), 1230-1250. doi: 10.1137/10081602X. Google Scholar
[16]	L. M. Briceño-Arias, Forward-Douglas-Rachford splitting and forward-partial inverse method for solving monotone inclusions, Optim., 64 (2015), 1239-1261. doi: 10.1080/02331934.2013.855210. Google Scholar
[17]	L. M. Briceño-Arias, Foward-partial inverse forward splitting for solving monotone inclusions, J. Optim. Theory Appl., 166 (2015), 391-413. doi: 10.1007/s10957-015-0703-2. Google Scholar
[18]	A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Springer-Verlag, Berlin Heidelberg, 2012. Google Scholar
[19]	A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1. Google Scholar
[20]	A. Chambolle and T. Pock, An introduction to continuous optimization for imaging, Acta Numer., 25 (2016), 161-319. doi: 10.1017/S096249291600009X. Google Scholar
[21]	F. Chen, L. Shen, Y. Xu and X. Zeng, The moreau envelope approach for the l1/TV image denoising model, Inverse Probl. Imaging, 8 (2014), 53-77. doi: 10.3934/ipi.2014.8.53. Google Scholar
[22]	P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200. doi: 10.1137/050626090. Google Scholar
[23]	P. L. Combettes and J.-C. Pesquet, A Douglas-Rachford splitting approach to nonsmooth conve variational signal recovery, IEEE J. Sel. Top. Signal Process, 1 (2007), 564-574. doi: 10.1109/JSTSP.2007.910264. Google Scholar
[24]	P. L. Combettes and J.-C. Pesquet, Primal-dual splitting algorithm for solving inclusions with mixtures of composite, lipschitzian, and parallel-sum type monotone operators, Set-Valued Var. Anal., 20 (2012), 307-330. doi: 10.1007/s11228-011-0191-y. Google Scholar
[25]	P. L. Combettes and B. C. Vũ, Variable metric forward-backward splitting with applications to monotone inclusions in duality, Optim., 63 (2014), 1289-1318. doi: 10.1080/02331934.2012.733883. Google Scholar
[26]	P. L. Combettes and J. Eckstein, Asynchronous block-iterative primal-dual decomposition methods for monotone inclusions, Math. Program., 168 (2018), 645-672. doi: 10.1007/s10107-016-1044-0. Google Scholar
[27]	P. L. Combettes, Systems of structured monotone inclusions: Duality, algorithms, and applications, SIAM J. Optim., 23 (2013), 2420-2447. doi: 10.1137/130904160. Google Scholar
[28]	P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal., 16 (2009), 727-748. Google Scholar
[29]	L. Condat, A primal-dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl., 158 (2013), 460-479. doi: 10.1007/s10957-012-0245-9. Google Scholar
[30]	E. X. Fang, B. He, H. Liu and X. Yuan, Generalized alternating direction method of multipliers: new theoretical insights and applications, Math. Program. Comput., 7 (2015), 149-187. doi: 10.1007/s12532-015-0078-2. Google Scholar
[31]	M. L. N. Goncalves, M. M. Alves and J. G. Melo, Pointwise and ergodic convergence rates of variable metric proximal alternating direction method of multipliers, J. Optim. Theory Appl., 177 (2018), 448-478. doi: 10.1007/s10957-018-1232-6. Google Scholar
[32]	P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979. doi: 10.1137/0716071. Google Scholar
[33]	Y.-J. Liu, D. Sun and K.-C. Toh, An implementable proximal point algorithmic framework for nuclear norm minimization, Math. Program., 133 (2012), 399-436. doi: 10.1007/s10107-010-0437-8. Google Scholar
[34]	C. A. Micchelli, L. Shen, Y. Xu and X. Zeng, Proximity algorithms for the l1/TV image denoising model, Adv. Comput. Math., 38 (2013), 401-426. doi: 10.1007/s10444-011-9243-y. Google Scholar
[35]	C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denoising, Inverse Probl., 27 (2011), 045009, 30 pp. doi: 10.1088/0266-5611/27/4/045009. Google Scholar
[36]	M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vis., 20 (2004), 99-120. Google Scholar
[37]	H. Raguet, J. Fadili and G. Peyré, A generalized forward-backward splitting, SIAM J. Imaging Sci., 6 (2013), 1199-1226. doi: 10.1137/120872802. Google Scholar
[38]	H. Raguet and L. Landrieu, Preconditioning of a generalized forward-backward splitting and application to optimization on graphs, SIAM J. Imaging Sci., 8 (2015), 2706-2739. doi: 10.1137/15M1018253. Google Scholar
[39]	H. Raguet, A note on the forward-Douglas-Rachford splitting for monotone inclusion and convex optimization, Optim. Lett., 13 (2019), 717-740. doi: 10.1007/s11590-018-1272-8. Google Scholar
[40]	R. Shefi and M. Teboulle, Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization, SIAM J. Optim., 24 (2014), 269-297. doi: 10.1137/130910774. Google Scholar
[41]	E. Y. Sidky, J. H. Jørgensen and X. Pan, Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm, Phys. Med. Biol., 57 (2012), 3065-3091. doi: 10.1088/0031-9155/57/10/3065. Google Scholar
[42]	J. E. Spingarn, Partial inverse of a monotone operator, Appl. Math. Optim., 10 (1983), 247-265. doi: 10.1007/BF01448388. Google Scholar
[43]	J. E. Spingarn, Applications of the method of partial inverses to convex programming: Decomposition, Math. Programming, 32 (1985), 199-223. doi: 10.1007/BF01586091. Google Scholar
[44]	Y. C. Tang, G. R. Wu and C. X. Zhu, An inner-outer iteration method for solving convex optimization problems involving the sum of three convex functions (in chinese), Sci. Sic. Math., 49 (2019), 831-858. doi: 10.1360/SCM-2017-0313. Google Scholar
[45]	Y. C. Tang, C. X. Zhu, M. Wen and J. G. Peng, A splitting primal-dual proximity algorithm for solving composite optimization problems, Acta. Math. Sin.-English Ser., 33 (2017), 868-886. doi: 10.1007/s10114-016-5625-x. Google Scholar
[46]	P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446. doi: 10.1137/S0363012998338806. Google Scholar
[47]	B. C. Vũ, A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comput. Math., 38 (2013), 667-681. doi: 10.1007/s10444-011-9254-8. Google Scholar
[48]	Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861. Google Scholar
[49]	C. Zong, Y. Tang and Y. J. Cho, Convergence analysis of an inexact three-operator splitting algorithm, Symmetry, 10 (2018), 563. doi: 10.3390/sym10110563. Google Scholar

Figure 1. Test image. (a) The original image; (b) The noise image

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Original images. (a) "Barbara"; (b) "Building"

Figure Options

Download full-size image

Download as PowerPoint slide

$ 10 \% $ to $ 90 \% $; The second column is the results of the PADMM; The third column is the results of the PDCP; The forth column is the results of the proposed Algorithm 2">Figure 3. Noisy and restored "Barbara" images. The first column is the noise images of "Barbara" with the noise level from $ 10 \% $ to $ 90 \% $; The second column is the results of the PADMM; The third column is the results of the PDCP; The forth column is the results of the proposed Algorithm 2

Figure Options

Download full-size image

Download as PowerPoint slide

$ 10 \% $ to $ 90 \% $; The second column is the results of the PADMM; The third column is the results of the PDCP; The forth column is the results of the proposed Algorithm 2">Figure 4. Noisy and restored "Building" images. The first column is the noise images of "Building" with the noise level from $ 10 \% $ to $ 90 \% $; The second column is the results of the PADMM; The third column is the results of the PDCP; The forth column is the results of the proposed Algorithm 2

Figure Options

Download full-size image

Download as PowerPoint slide

Table 1. Numerical results of Algorithm 1, Algorithm 2 and Algorithm 3 for different selection of regularization parameters in terms of SNR, SSIM and the number of iterations (Iter)

Methods	$ \lambda_1 $	$ \lambda_2 $	$ \epsilon = 10^{-2} $			$ \epsilon = 10^{-4} $
Methods	$ \lambda_1 $	$ \lambda_2 $	SNR (dB)	SSIM	Iter	SNR (dB)	SSIM	Iter
Algorithm 1	$ 0 $	0.5	$ 4.4173 $	$ 0.0159 $	$ 149 $	$ 4.4648 $	$ 0.0162 $	$ - $
		$ 1 $	$ 4.4182 $	$ 0.0159 $	$ 147 $	$ 4.4749 $	$ 0.0162 $	$ - $
		$ 20 $	$ 53.0111 $	$ 0.9988 $	$ 738 $	$ 62.4224 $	$ 0.9999 $	$ - $
	$ 0.5 $	$ 0 $	$ 23.8749 $	$ 0.8813 $	$ 29 $	$ 24.3263 $	$ 0.9195 $	$ 110 $
	$ 1 $		$ 30.3788 $	$ 0.9340 $	$ 30 $	$ 34.3648 $	$ 0.9905 $	$ 151 $
	$ 2 $		$ 31.2330 $	$ 0.9331 $	$ 30 $	$ 52.4987 $	$ 0.9996 $	$ 276 $
	$ 0.5 $	$ 0.5 $	$ 27.2958 $	$ 0.9269 $	$ 24 $	$ 30.7593 $	$ 0.9837 $	$ 139 $
	$ 1 $	$ 1 $	$ 29.2181 $	$ 0.9434 $	$ 24 $	$ 47.8240 $	$ 0.9988 $	$ 135 $
	$ 2 $	$ 2 $	$ 29.3063 $	$ 0.9391 $	$ 25 $	$ 79.2541 $	$ 1 $	$ 210 $
Algorithm 2	$ 0 $	0.5	$ 4.5444 $	$ 0.0164 $	$ 24 $	$ 4.4263 $	$ 0.0159 $	$ 45 $
		$ 1 $	$ 4.5044 $	$ 0.0164 $	$ 25 $	$ 4.4260 $	$ 0.0159 $	$ 46 $
		$ 20 $	$ 32.8316 $	$ 0.9507 $	$ 33 $	$ 71.4020 $	$ 1 $	$ 89 $
	$ 0.5 $	$ 0 $	$ 21.2139 $	$ 0.0.7467 $	$ 42 $	$ 22.8320 $	$ 0.9143 $	$ 223 $
	$ 1 $		$ 27.0660 $	$ 0.7962 $	$ 49 $	$ 34.2508 $	$ 0.9903 $	$ 236 $
	$ 2 $		$ 26.9064 $	$ 0.7831 $	$ 50 $	$ 53.0912 $	$ 0.9996 $	$ 272 $
	$ 0.5 $	$ 0.5 $	$ 23.7828 $	$ 0.8072 $	$ 36 $	$ 30.5978 $	$ 0.9808 $	$ 168 $
	$ 1 $	$ 1 $	$ 28.3761 $	$ 0.8454 $	$ 35 $	$ 48.3065 $	$ 0.9990 $	$ 111 $
	$ 2 $	$ 2 $	$ 26.4981 $	$ 0.9092 $	$ 33 $	$ 76.5604 $	$ 1 $	$ 195 $
Algorithm 3	$ 0 $	0.5	$ 4.4169 $	$ 0.0159 $	$ 373 $	$ 4.4219 $	$ 0.0159 $	$ - $
		$ 1 $	$ 4.4246 $	$ 0.0159 $	$ 299 $	$ 4.4235 $	$ 0.0159 $	$ - $
		$ 20 $	$ 39.9637 $	$ 0.9762 $	$ - $	$ 39.9637 $	$ 0.9762 $	$ - $
	$ 0.5 $	$ 0 $	$ 23.1899 $	$ 0.9011 $	$ 61 $	$ 23.3077 $	$ 0.9151 $	$ 448 $
	$ 1 $		$ 32.4599 $	$ 0.9695 $	$ 49 $	$ 34.7962 $	$ 0.9913 $	$ 308 $
	$ 2 $		$ 35.2484 $	$ 0.9751 $	$ 53 $	$ 55.4857 $	$ 0.9998 $	$ 389 $
	$ 0.5 $	$ 0.5 $	$ 29.7816 $	$ 0.9611 $	$ 54 $	$ 30.7667 $	$ 0.9838 $	$ 512 $
	$ 1 $	$ 1 $	$ 35.3009 $	$ 0.9731 $	$ 43 $	$ 48.0544 $	$ 0.9989 $	$ 321 $
	$ 2 $	$ 2 $	$ 35.5435 $	$ 0.9767 $	$ 41 $	$ 85.6388 $	$ 1 $	$ 279 $

Table Options

Download as excel

Methods	$ \lambda_1 $	$ \lambda_2 $	$ \epsilon = 10^{-2} $			$ \epsilon = 10^{-4} $
Methods	$ \lambda_1 $	$ \lambda_2 $	SNR (dB)	SSIM	Iter	SNR (dB)	SSIM	Iter
Algorithm 1	$ 0 $	0.5	$ 4.4173 $	$ 0.0159 $	$ 149 $	$ 4.4648 $	$ 0.0162 $	$ - $
		$ 1 $	$ 4.4182 $	$ 0.0159 $	$ 147 $	$ 4.4749 $	$ 0.0162 $	$ - $
		$ 20 $	$ 53.0111 $	$ 0.9988 $	$ 738 $	$ 62.4224 $	$ 0.9999 $	$ - $
	$ 0.5 $	$ 0 $	$ 23.8749 $	$ 0.8813 $	$ 29 $	$ 24.3263 $	$ 0.9195 $	$ 110 $
	$ 1 $		$ 30.3788 $	$ 0.9340 $	$ 30 $	$ 34.3648 $	$ 0.9905 $	$ 151 $
	$ 2 $		$ 31.2330 $	$ 0.9331 $	$ 30 $	$ 52.4987 $	$ 0.9996 $	$ 276 $
	$ 0.5 $	$ 0.5 $	$ 27.2958 $	$ 0.9269 $	$ 24 $	$ 30.7593 $	$ 0.9837 $	$ 139 $
	$ 1 $	$ 1 $	$ 29.2181 $	$ 0.9434 $	$ 24 $	$ 47.8240 $	$ 0.9988 $	$ 135 $
	$ 2 $	$ 2 $	$ 29.3063 $	$ 0.9391 $	$ 25 $	$ 79.2541 $	$ 1 $	$ 210 $
Algorithm 2	$ 0 $	0.5	$ 4.5444 $	$ 0.0164 $	$ 24 $	$ 4.4263 $	$ 0.0159 $	$ 45 $
		$ 1 $	$ 4.5044 $	$ 0.0164 $	$ 25 $	$ 4.4260 $	$ 0.0159 $	$ 46 $
		$ 20 $	$ 32.8316 $	$ 0.9507 $	$ 33 $	$ 71.4020 $	$ 1 $	$ 89 $
	$ 0.5 $	$ 0 $	$ 21.2139 $	$ 0.0.7467 $	$ 42 $	$ 22.8320 $	$ 0.9143 $	$ 223 $
	$ 1 $		$ 27.0660 $	$ 0.7962 $	$ 49 $	$ 34.2508 $	$ 0.9903 $	$ 236 $
	$ 2 $		$ 26.9064 $	$ 0.7831 $	$ 50 $	$ 53.0912 $	$ 0.9996 $	$ 272 $
	$ 0.5 $	$ 0.5 $	$ 23.7828 $	$ 0.8072 $	$ 36 $	$ 30.5978 $	$ 0.9808 $	$ 168 $
	$ 1 $	$ 1 $	$ 28.3761 $	$ 0.8454 $	$ 35 $	$ 48.3065 $	$ 0.9990 $	$ 111 $
	$ 2 $	$ 2 $	$ 26.4981 $	$ 0.9092 $	$ 33 $	$ 76.5604 $	$ 1 $	$ 195 $
Algorithm 3	$ 0 $	0.5	$ 4.4169 $	$ 0.0159 $	$ 373 $	$ 4.4219 $	$ 0.0159 $	$ - $
		$ 1 $	$ 4.4246 $	$ 0.0159 $	$ 299 $	$ 4.4235 $	$ 0.0159 $	$ - $
		$ 20 $	$ 39.9637 $	$ 0.9762 $	$ - $	$ 39.9637 $	$ 0.9762 $	$ - $
	$ 0.5 $	$ 0 $	$ 23.1899 $	$ 0.9011 $	$ 61 $	$ 23.3077 $	$ 0.9151 $	$ 448 $
	$ 1 $		$ 32.4599 $	$ 0.9695 $	$ 49 $	$ 34.7962 $	$ 0.9913 $	$ 308 $
	$ 2 $		$ 35.2484 $	$ 0.9751 $	$ 53 $	$ 55.4857 $	$ 0.9998 $	$ 389 $
	$ 0.5 $	$ 0.5 $	$ 29.7816 $	$ 0.9611 $	$ 54 $	$ 30.7667 $	$ 0.9838 $	$ 512 $
	$ 1 $	$ 1 $	$ 35.3009 $	$ 0.9731 $	$ 43 $	$ 48.0544 $	$ 0.9989 $	$ 321 $
	$ 2 $	$ 2 $	$ 35.5435 $	$ 0.9767 $	$ 41 $	$ 85.6388 $	$ 1 $	$ 279 $

Table 2. Regularization parameters $ \lambda_1 $ and $ \lambda_2 $ for the PADMM (76) and Algorithm 2 under different noise levels

Images	10$ \% $		30$ \% $		50$ \% $		70$ \% $		90$ \% $
Images	$ \lambda_{1} $	$ \lambda_{2} $	$ \lambda_{1} $	$ \lambda_{2} $	$ \lambda_{1} $	$ \lambda_{2} $	$ \lambda_{1} $	$ \lambda_{2} $	$ \lambda_{1} $	$ \lambda_{2} $
Barbara	$ 0.3 $	$ 3.4 $	$ 0.4 $	$ 3.3 $	$ 0.5 $	$ 4.3 $	$ 0.6 $	$ 4.6 $	$ 0.7 $	$ 4.0 $
Building	$ 0.3 $	$ 5.8 $	$ 0.4 $	$ 5.6 $	$ 0.4 $	$ 8.4 $	$ 0.4 $	$ 10.2 $	$ 0.4 $	$ 10.5 $

Table Options

Download as excel

Images	10$ \% $		30$ \% $		50$ \% $		70$ \% $		90$ \% $
Images	$ \lambda_{1} $	$ \lambda_{2} $	$ \lambda_{1} $	$ \lambda_{2} $	$ \lambda_{1} $	$ \lambda_{2} $	$ \lambda_{1} $	$ \lambda_{2} $	$ \lambda_{1} $	$ \lambda_{2} $
Barbara	$ 0.3 $	$ 3.4 $	$ 0.4 $	$ 3.3 $	$ 0.5 $	$ 4.3 $	$ 0.6 $	$ 4.6 $	$ 0.7 $	$ 4.0 $
Building	$ 0.3 $	$ 5.8 $	$ 0.4 $	$ 5.6 $	$ 0.4 $	$ 8.4 $	$ 0.4 $	$ 10.2 $	$ 0.4 $	$ 10.5 $

Table 3. Regularization parameters $ \lambda_1 $ for the PDCP (77) under different noise levels

Images	10$ \% $	30$ \% $	50$ \% $	70$ \% $	90$ \% $
Images	$ \lambda_{1} $	$ \lambda_{1} $	$ \lambda_{1} $	$ \lambda_{1} $	$ \lambda_{1} $
Barbara	$ 0.4 $	$ 0.55 $	$ 0.75 $	$ 0.9 $	$ 1.2 $
Building	$ 0.4 $	$ 0.55 $	$ 0.7 $	$ 0.85 $	$ 1.15 $

Table Options

Download as excel

Images	10$ \% $	30$ \% $	50$ \% $	70$ \% $	90$ \% $
Images	$ \lambda_{1} $	$ \lambda_{1} $	$ \lambda_{1} $	$ \lambda_{1} $	$ \lambda_{1} $
Barbara	$ 0.4 $	$ 0.55 $	$ 0.75 $	$ 0.9 $	$ 1.2 $
Building	$ 0.4 $	$ 0.55 $	$ 0.7 $	$ 0.85 $	$ 1.15 $

Table 4. Numerical results of the PADMM (76), the PDCP (77) and the proposed Algorithm 2 for different noise levels in terms of SNR (dB), PSNR (dB), SSIM, number of iterations (Iter) and CPU time (in seconds)

Level	Methods	Barbara					Building
Level	Methods	SNR	PSNR	SSIM	Iter	CPU	SNR	PSNR	SSIM	Iter	CPU
10$ \% $	PADMM	$ 23.1733 $	$ 29.5804 $	$ 0.9210 $	$ 890 $	$ 186.60 $	$ 25.7280 $	$ 31.5560 $	$ 0.9418 $	$ 990 $	$ 200.47 $
	PDCP	$ 21.7728 $	$ 28.1799 $	$ 0.9015 $	$ 297 $	$ 21.47 $	$ 21.9205 $	$ 27.7484 $	$ 0.8984 $	$ 262 $	$ 18.86 $
	Algorithm 2	$ 23.1722 $	$ 29.5792 $	$ 0.9210 $	$ 122 $	$ 25.61 $	$ 25.7268 $	$ 31.5547 $	0.9418	$ 110 $	$ 23.91 $
30$ \% $	PADMM	18.9947	$ 25.4018 $	$ 0.7976 $	$ 1232 $	$ 241.68 $	$ 20.0999 $	$ 25.9278 $	$ 0.7996 $	$ 1356 $	$ 260.67 $
	PDCP	$ 18.0216 $	$ 24.4287 $	$ 0.7553 $	$ 686 $	$ 48.35 $	$ 18.1008 $	$ 23.9288 $	$ 0.7109 $	$ 688 $	$ 52.04 $
	Algorithm 2	$ 18.9937 $	$ 25.4008 $	$ 0.7981 $	$ 180 $	$ 37.26 $	$ 20.0902 $	$ 25.9181 $	$ 0.7988 $	$ 159 $	$ 33.12 $
50$ \% $	PADMM	$ 17.2073 $	$ 23.6144 $	$ 0.6889 $	$ 2804 $	$ 557.60 $	$ 17.6625 $	$ 23.4905 $	$ 0.6647 $	$ 2001 $	$ 397.35 $
	PDCP	$ 17.0137 $	$ 23.4205 $	$ 0.6700 $	$ 904 $	$ 64.98 $	$ 15.9925 $	$ 21.8204 $	$ 0.5546 $	$ 861 $	$ 59.20 $
	Algorithm 2	$ 17.2135 $	$ 23.6206 $	$ 0.6905 $	$ 210 $	$ 43.40 $	$ 17.6619 $	$ 23.4898 $	$ 0.6646 $	$ 169 $	$ 34.49 $
70$ \% $	PADMM	$ 15.8733 $	$ 22.2804 $	$ 0.5972 $	$ 2001 $	$ 789.97 $	$ 15.6335 $	$ 21.4615 $	$ 0.5067 $	$ 2001 $	$ 529.72 $
70$ \% $	PDCP	$ 15.6142 $	$ 22.0213 $	$ 0.5852 $	$ 1322 $	$ 93.27 $	$ 13.8911 $	$ 19.7190 $	$ 0.3736 $	$ 1360 $	$ 94.47 $
	Algorithm 2	$ 15.8856 $	$ 22.2927 $	$ 0.6000 $	$ 234 $	$ 47.93 $	$ 15.6387 $	$ 21.4666 $	$ 0.5070 $	$ 196 $	$ 41.08 $
90$ \% $	PADMM	$ 12.4587 $	$ 18.8658 $	$ 0.4401 $	$ 2001 $	$ 856.01 $	$ 12.4100 $	$ 18.2379 $	$ 0.2730 $	$ 2001 $	$ 569.67 $
	PDCP	$ 12.4733 $	$ 18.8804 $	$ 0.4617 $	$ 2278 $	$ 172.28 $	$ 11.7455 $	$ 17.5735 $	$ 0.2278 $	$ 1987 $	$ 134.76 $
	Algorithm 2	$ 12.4789 $	$ 18.8860 $	$ 0.4441 $	$ 362 $	$ 79.30 $	$ 12.4224 $	$ 18.2503 $	$ 0.2735 $	$ 332 $	$ 72.85 $

Table Options

Download as excel

Level	Methods	Barbara					Building
Level	Methods	SNR	PSNR	SSIM	Iter	CPU	SNR	PSNR	SSIM	Iter	CPU
10$ \% $	PADMM	$ 23.1733 $	$ 29.5804 $	$ 0.9210 $	$ 890 $	$ 186.60 $	$ 25.7280 $	$ 31.5560 $	$ 0.9418 $	$ 990 $	$ 200.47 $
	PDCP	$ 21.7728 $	$ 28.1799 $	$ 0.9015 $	$ 297 $	$ 21.47 $	$ 21.9205 $	$ 27.7484 $	$ 0.8984 $	$ 262 $	$ 18.86 $
	Algorithm 2	$ 23.1722 $	$ 29.5792 $	$ 0.9210 $	$ 122 $	$ 25.61 $	$ 25.7268 $	$ 31.5547 $	0.9418	$ 110 $	$ 23.91 $
30$ \% $	PADMM	18.9947	$ 25.4018 $	$ 0.7976 $	$ 1232 $	$ 241.68 $	$ 20.0999 $	$ 25.9278 $	$ 0.7996 $	$ 1356 $	$ 260.67 $
	PDCP	$ 18.0216 $	$ 24.4287 $	$ 0.7553 $	$ 686 $	$ 48.35 $	$ 18.1008 $	$ 23.9288 $	$ 0.7109 $	$ 688 $	$ 52.04 $
	Algorithm 2	$ 18.9937 $	$ 25.4008 $	$ 0.7981 $	$ 180 $	$ 37.26 $	$ 20.0902 $	$ 25.9181 $	$ 0.7988 $	$ 159 $	$ 33.12 $
50$ \% $	PADMM	$ 17.2073 $	$ 23.6144 $	$ 0.6889 $	$ 2804 $	$ 557.60 $	$ 17.6625 $	$ 23.4905 $	$ 0.6647 $	$ 2001 $	$ 397.35 $
	PDCP	$ 17.0137 $	$ 23.4205 $	$ 0.6700 $	$ 904 $	$ 64.98 $	$ 15.9925 $	$ 21.8204 $	$ 0.5546 $	$ 861 $	$ 59.20 $
	Algorithm 2	$ 17.2135 $	$ 23.6206 $	$ 0.6905 $	$ 210 $	$ 43.40 $	$ 17.6619 $	$ 23.4898 $	$ 0.6646 $	$ 169 $	$ 34.49 $
70$ \% $	PADMM	$ 15.8733 $	$ 22.2804 $	$ 0.5972 $	$ 2001 $	$ 789.97 $	$ 15.6335 $	$ 21.4615 $	$ 0.5067 $	$ 2001 $	$ 529.72 $
70$ \% $	PDCP	$ 15.6142 $	$ 22.0213 $	$ 0.5852 $	$ 1322 $	$ 93.27 $	$ 13.8911 $	$ 19.7190 $	$ 0.3736 $	$ 1360 $	$ 94.47 $
	Algorithm 2	$ 15.8856 $	$ 22.2927 $	$ 0.6000 $	$ 234 $	$ 47.93 $	$ 15.6387 $	$ 21.4666 $	$ 0.5070 $	$ 196 $	$ 41.08 $
90$ \% $	PADMM	$ 12.4587 $	$ 18.8658 $	$ 0.4401 $	$ 2001 $	$ 856.01 $	$ 12.4100 $	$ 18.2379 $	$ 0.2730 $	$ 2001 $	$ 569.67 $
	PDCP	$ 12.4733 $	$ 18.8804 $	$ 0.4617 $	$ 2278 $	$ 172.28 $	$ 11.7455 $	$ 17.5735 $	$ 0.2278 $	$ 1987 $	$ 134.76 $
	Algorithm 2	$ 12.4789 $	$ 18.8860 $	$ 0.4441 $	$ 362 $	$ 79.30 $	$ 12.4224 $	$ 18.2503 $	$ 0.2735 $	$ 332 $	$ 72.85 $

[1]	Hadi Khatibzadeh, Vahid Mohebbi, Mohammad Hossein Alizadeh. On the cyclic pseudomonotonicity and the proximal point algorithm. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 441-449. doi: 10.3934/naco.2018027
[2]	Jiulong Liu, Nanguang Chen, Hui Ji. Learnable Douglas-Rachford iteration and its applications in DOT imaging. Inverse Problems & Imaging, 2020, 14 (4) : 683-700. doi: 10.3934/ipi.2020031
[3]	Leyu Hu, Xingju Cai. Convergence of a randomized Douglas-Rachford method for linear system. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 463-474. doi: 10.3934/naco.2020045
[4]	Ram U. Verma. On the generalized proximal point algorithm with applications to inclusion problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 381-390. doi: 10.3934/jimo.2009.5.381
[5]	Yu-Lin Chang, Jein-Shan Chen, Jia Wu. Proximal point algorithm for nonlinear complementarity problem based on the generalized Fischer-Burmeister merit function. Journal of Industrial & Management Optimization, 2013, 9 (1) : 153-169. doi: 10.3934/jimo.2013.9.153
[6]	Radu Ioan Boţ, Christopher Hendrich. Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators. Inverse Problems & Imaging, 2016, 10 (3) : 617-640. doi: 10.3934/ipi.2016014
[7]	Qin Sheng, David A. Voss, Q. M. Khaliq. An adaptive splitting algorithm for the sine-Gordon equation. Conference Publications, 2005, 2005 (Special) : 792-797. doi: 10.3934/proc.2005.2005.792
[8]	Sanming Liu, Zhijie Wang, Chongyang Liu. Proximal iterative Gaussian smoothing algorithm for a class of nonsmooth convex minimization problems. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 79-89. doi: 10.3934/naco.2015.5.79
[9]	Kangkang Deng, Zheng Peng, Jianli Chen. Sparse probabilistic Boolean network problems: A partial proximal-type operator splitting method. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1881-1896. doi: 10.3934/jimo.2018127
[10]	Pilar Bayer, Dionís Remón. A reduction point algorithm for cocompact Fuchsian groups and applications. Advances in Mathematics of Communications, 2014, 8 (2) : 223-239. doi: 10.3934/amc.2014.8.223
[11]	Leyu Hu, Wenxing Zhang, Xingju Cai, Deren Han. A parallel operator splitting algorithm for solving constrained total-variation retinex. Inverse Problems & Imaging, 2020, 14 (6) : 1135-1156. doi: 10.3934/ipi.2020058
[12]	Lacramioara Grecu, Constantin Popa. Constrained SART algorithm for inverse problems in image reconstruction. Inverse Problems & Imaging, 2013, 7 (1) : 199-216. doi: 10.3934/ipi.2013.7.199
[13]	Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73
[14]	Lu Han, Min Li, Dachuan Xu, Dongmei Zhang. Stochastic-Lazier-Greedy Algorithm for monotone non-submodular maximization. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2607-2614. doi: 10.3934/jimo.2020085
[15]	Pierre Frankel. Alternating proximal algorithm with costs-to-move, dual description and application to PDE's. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 545-557. doi: 10.3934/dcdss.2012.5.545
[16]	Laetitia Paoli. A proximal-like algorithm for vibro-impact problems with a non-smooth set of constraints. Conference Publications, 2011, 2011 (Special) : 1186-1195. doi: 10.3934/proc.2011.2011.1186
[17]	Jie Shen, Jian Lv, Fang-Fang Guo, Ya-Li Gao, Rui Zhao. A new proximal chebychev center cutting plane algorithm for nonsmooth optimization and its convergence. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1143-1155. doi: 10.3934/jimo.2018003
[18]	Zheng-Hai Huang, Shang-Wen Xu. Convergence properties of a non-interior-point smoothing algorithm for the PNCP. Journal of Industrial & Management Optimization*, 2007, 3 (3) : 569-584. doi: 10.3934/jimo.2007.3.569
[19]	Behrouz Kheirfam, Morteza Moslemi. On the extension of an arc-search interior-point algorithm for semidefinite optimization. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 261-275. doi: 10.3934/naco.2018015
[20]	Barbara Kaltenbacher, Jonas Offtermatt. A refinement and coarsening indicator algorithm for finding sparse solutions of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 391-406. doi: 10.3934/ipi.2011.5.391

2020 Impact Factor: 1.639

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.639

Tools

Figures and Tables

2020 Impact Factor: 1.639

American Institute of Mathematical Sciences

Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors