# American Institute of Mathematical Sciences

• Previous Article
Improved particle swarm optimization and neighborhood field optimization by introducing the re-sampling step of particle filter
• JIMO Home
• This Issue
• Next Article
Mechanism design in project procurement auctions with cost uncertainty and failure risk
January  2019, 15(1): 159-175. doi: 10.3934/jimo.2018037

## Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step

 School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing 210023, China

* Corresponding author: Yuan Shen

Received  May 2017 Revised  September 02, 2017 Published  April 2018

Fund Project: Research supported by National Natural Science Foundation of China under grant 11401295 and by National Natural Science Foundation Tianyuan Project under grant 11726618 and by Jiangsu Provincial Natural Science Foundation under grant BK20141007 and by Major Program of the National Social Science Foundation of China under Grant 12&ZD114 and by National Social Science Foundation of China under Grant 15BGL58 and by Jiangsu Provincial Social Science Foundation under Grant 14EUA001 and by Qinglan Project of Jiangsu Province.

In this paper, we propose a partial convolution model for image deblurring and denoising. We also devise a new linearized alternating direction method of multipliers (ADMM) with an extension step. As the computation of its subproblem is simple enough to have closed-form solutions, its per-iteration cost is low; however, the relaxed parameter condition together with the extra extension step inspired by Ye and Yuan's ADMM enables faster convergence than the original linearized ADMM. Preliminary experimental results show that our algorithm can produce better quality results than some existing efficient algorithms within a similar computation time. The performance advantage of our algorithm is particularly evident at high noise ratios.

Citation: Yuan Shen, Lei Ji. Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step. Journal of Industrial & Management Optimization, 2019, 15 (1) : 159-175. doi: 10.3934/jimo.2018037
##### References:
 [1] R. Acar and C. R. Vogel, Analysis of total variation penalty methods, Inverse Probl., 10 (1994), 1217-1229.  doi: 10.1088/0266-5611/10/6/003.  Google Scholar [2] S. Alliney, Digital filters as absolute norm regularizers, IEEE Trans. Signal Proces., 40 (1992), 1548-1562.   Google Scholar [3] J. Barzilai and J.M. Borwein, Two-point step size gradient methods, IMA J. Numer. Anal., 8 (1988), 141-148.  doi: 10.1093/imanum/8.1.141.  Google Scholar [4] 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 [5] S. Becker, J. Bobin and E. Candes, 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 [6] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.  Google Scholar [7] J. Bioucas-Dias and M. Figueiredo, A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration, IEEE Trans. Image Proces., 16 (2007), 2992-3004.  doi: 10.1109/TIP.2007.909319.  Google Scholar [8] 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 (2011), 1-122.   Google Scholar [9] J. Cai, R.H. Chan and M. Nikolova, Two phase methods for deblurring images corrupted by impulse plus gaussian noise, Inverse Probl. Imag., 2 (2008), 187-204.  doi: 10.3934/ipi.2008.2.187.  Google Scholar [10] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar [11] R.H. Chan, C.W. Ho and M. Nikolova, Salt-and-pepper noise removal by median-type noise detector and detail-preserving regularization, IEEE Trans. Imag. Process., 14 (2005), 1479-1485.   Google Scholar [12] R.H. Chan, M. Tao and X. Yuan, Linearized alternating direction method of multipliers for constrained linear least-squares problem, E. Asian J. Appl. Math., 2 (2012), 326-341.  doi: 10.4208/eajam.270812.161112a.  Google Scholar [13] T.F. Chan and S. Esedoglu, Aspects of total variation regularized l1 function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837.  doi: 10.1137/040604297.  Google Scholar [14] T.F. Chan, G. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.  doi: 10.1137/S1064827596299767.  Google Scholar [15] C. Chen, B. He, Y. Ye and X. Yuan, The direct extension of admm for multi-block convex minimization problems is not necessarily convergent, Math. Program. Series A, 155 (2016), 57-79.  doi: 10.1007/s10107-014-0826-5.  Google Scholar [16] C. Chen, Y. Shen and Y. You, On the convergence analysis of the alternating direction method of multipliers with three blocks, Abstr. Appl. Anal., 2013 (2013), Art. ID 183961, 7 pp.  Google Scholar [17] S. W.J. Cho and S. Lee, Handling outliers in non-blind image deconvolution, International Conference on Computer Vision, (2011), 495-502.   Google Scholar [18] I. Daubechies, M. Defrise and C. DeMol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042.  Google Scholar [19] J. Douglas and H.H. Rachford, On the numerical solution of the heat conduction problem in 2 and 3 space variables, Trans. Amer. Math. Soc., 82 (1956), 421-439.  doi: 10.1090/S0002-9947-1956-0084194-4.  Google Scholar [20] E. Esser, Applications of lagrangian-based alternating direction methods and connections to split bregman, Manuscript, (2009), ftp://ftp.math.ucla.edu/pub/camreport/cam09-31.pdf. Google Scholar [21] M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, North-Holland, Amsterdam, 1983.  Google Scholar [22] D. Gabay, Applications of the method of multipliers to variational inequalities, In Augmented Lagrangian Methods: Applications to the Solution of Boundary Value Problems, M. Fortin and R. Glowinski, Eds. Horth-Hollan, Amsterdam, 1983.  Google Scholar [23] D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Comput. Math. Appl., (1976), 17-40.   Google Scholar [24] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984.   Google Scholar [25] R. Glowinski and A. Marrocco, Sur lapproximation par elements finis dordre un, et la resolution par penalisation-dualite dune classe de problemes de dirichlet nonlineaires, Rev. Francaise dAut. Inf. Rech. Oper., 9 (1975), 41-76.   Google Scholar [26] B. He, M. Tao and X. Yuan, Alternating direction method with gaussian back substitution for separable convex programming, SIAM J. Optim., 22 (2012), 313-340.  doi: 10.1137/110822347.  Google Scholar [27] B. He and X. Yuan, On the convergence rate of douglas-rachford operator splitting method, Math. Program., 153 (2015), 715-722.  doi: 10.1007/s10107-014-0805-x.  Google Scholar [28] M. Hong and Z. Luo, On the linear convergence of the alternating direction method of multipliers, arXiv preprint, arXiv: 1208.3922, (2013). Google Scholar [29] Y. Huang, M. K. Ng and Y.-W. Wen, A fast total variation minimization method for image restoration, Multiscale Model. Sim., 7 (2008), 774-795.  doi: 10.1137/070703533.  Google Scholar [30] Y. Jiao, Q. Jin, X. Lu and W. Wang, Alternating direction method of multipliers for linear inverse problems, SIAM J. Numer. Anal., 54 (2016), 2114-2137.  doi: 10.1137/15M1029308.  Google Scholar [31] Y. Jiao, Q. Jin, X. Lu and W. Wang, Preconditioned alternating direction method of multipliers for inverse problems with constraints, nverse Probl., 33 (2017), 025004(34pp).   Google Scholar [32] C. Li, W. Yin and Y. Zhang, Tv Minimization by Augmented Lagrangian and Alternating Direction Algorithms, Tech. rep., Department of CAAM, Rice University, Houston, Texas, 77005, 2009. http://www.caam.rice.edu/~optimization/L1/TVAL3/. Google Scholar [33] 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 [34] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685.  doi: 10.1002/cpa.3160420503.  Google Scholar [35] Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program., 103 (2005), 127-152.  doi: 10.1007/s10107-004-0552-5.  Google Scholar [36] Y. Ouyang, Y. Chen, G. Lan and E.J. Pasiliao, An accelerated linearized alternating direction method of multipliers, SIAM J Imaging Sci., 8 (2015), 644-681.  doi: 10.1137/14095697X.  Google Scholar [37] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [38] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston, New York, 1977.   Google Scholar [39] C. Vogel and M. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Trans. Image Proces., 7 (1998), 813-824.  doi: 10.1109/83.679423.  Google Scholar [40] X. Wang and X. Yuan, The linearized alternating direction method of multipliers for dantzig selector, SIAM J. Sci. Comput., 34 (2012), 2792-2811.  doi: 10.1137/110833543.  Google Scholar [41] Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Image Sci., 1 (2008), 248-272.  doi: 10.1137/080724265.  Google Scholar [42] T. Wu, Variable splitting based method for image restoration with impulse plus gaussian noise, Math. Probl. Eng., 2016 (2016), Article ID 3151303, 16 pages. Google Scholar [43] J. Yang and Y. Zhang, Alternating direction algorithms for $\ell_1$-problems in compressive sensing, SIAM J. Sci. Comput., 33 (2011), 250-278.  doi: 10.1137/090777761.  Google Scholar [44] 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), 2842-2865.  doi: 10.1137/080732894.  Google Scholar [45] C. Ye and X. Yuan, A descent method for structured monotone variational inequalities, Optim. Methods Softw., 22 (2007), 329-338.  doi: 10.1080/10556780600552693.  Google Scholar [46] W. Yin, D. Goldfarb and S. Osher, The total variation regularized $L^1$ multiscale decomposition, Multiscale Model. Sim., 6 (2007), 190-211.  doi: 10.1137/060663027.  Google Scholar

show all references

##### References:
 [1] R. Acar and C. R. Vogel, Analysis of total variation penalty methods, Inverse Probl., 10 (1994), 1217-1229.  doi: 10.1088/0266-5611/10/6/003.  Google Scholar [2] S. Alliney, Digital filters as absolute norm regularizers, IEEE Trans. Signal Proces., 40 (1992), 1548-1562.   Google Scholar [3] J. Barzilai and J.M. Borwein, Two-point step size gradient methods, IMA J. Numer. Anal., 8 (1988), 141-148.  doi: 10.1093/imanum/8.1.141.  Google Scholar [4] 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 [5] S. Becker, J. Bobin and E. Candes, 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 [6] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.  Google Scholar [7] J. Bioucas-Dias and M. Figueiredo, A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration, IEEE Trans. Image Proces., 16 (2007), 2992-3004.  doi: 10.1109/TIP.2007.909319.  Google Scholar [8] 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 (2011), 1-122.   Google Scholar [9] J. Cai, R.H. Chan and M. Nikolova, Two phase methods for deblurring images corrupted by impulse plus gaussian noise, Inverse Probl. Imag., 2 (2008), 187-204.  doi: 10.3934/ipi.2008.2.187.  Google Scholar [10] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar [11] R.H. Chan, C.W. Ho and M. Nikolova, Salt-and-pepper noise removal by median-type noise detector and detail-preserving regularization, IEEE Trans. Imag. Process., 14 (2005), 1479-1485.   Google Scholar [12] R.H. Chan, M. Tao and X. Yuan, Linearized alternating direction method of multipliers for constrained linear least-squares problem, E. Asian J. Appl. Math., 2 (2012), 326-341.  doi: 10.4208/eajam.270812.161112a.  Google Scholar [13] T.F. Chan and S. Esedoglu, Aspects of total variation regularized l1 function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837.  doi: 10.1137/040604297.  Google Scholar [14] T.F. Chan, G. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.  doi: 10.1137/S1064827596299767.  Google Scholar [15] C. Chen, B. He, Y. Ye and X. Yuan, The direct extension of admm for multi-block convex minimization problems is not necessarily convergent, Math. Program. Series A, 155 (2016), 57-79.  doi: 10.1007/s10107-014-0826-5.  Google Scholar [16] C. Chen, Y. Shen and Y. You, On the convergence analysis of the alternating direction method of multipliers with three blocks, Abstr. Appl. Anal., 2013 (2013), Art. ID 183961, 7 pp.  Google Scholar [17] S. W.J. Cho and S. Lee, Handling outliers in non-blind image deconvolution, International Conference on Computer Vision, (2011), 495-502.   Google Scholar [18] I. Daubechies, M. Defrise and C. DeMol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042.  Google Scholar [19] J. Douglas and H.H. Rachford, On the numerical solution of the heat conduction problem in 2 and 3 space variables, Trans. Amer. Math. Soc., 82 (1956), 421-439.  doi: 10.1090/S0002-9947-1956-0084194-4.  Google Scholar [20] E. Esser, Applications of lagrangian-based alternating direction methods and connections to split bregman, Manuscript, (2009), ftp://ftp.math.ucla.edu/pub/camreport/cam09-31.pdf. Google Scholar [21] M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, North-Holland, Amsterdam, 1983.  Google Scholar [22] D. Gabay, Applications of the method of multipliers to variational inequalities, In Augmented Lagrangian Methods: Applications to the Solution of Boundary Value Problems, M. Fortin and R. Glowinski, Eds. Horth-Hollan, Amsterdam, 1983.  Google Scholar [23] D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Comput. Math. Appl., (1976), 17-40.   Google Scholar [24] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984.   Google Scholar [25] R. Glowinski and A. Marrocco, Sur lapproximation par elements finis dordre un, et la resolution par penalisation-dualite dune classe de problemes de dirichlet nonlineaires, Rev. Francaise dAut. Inf. Rech. Oper., 9 (1975), 41-76.   Google Scholar [26] B. He, M. Tao and X. Yuan, Alternating direction method with gaussian back substitution for separable convex programming, SIAM J. Optim., 22 (2012), 313-340.  doi: 10.1137/110822347.  Google Scholar [27] B. He and X. Yuan, On the convergence rate of douglas-rachford operator splitting method, Math. Program., 153 (2015), 715-722.  doi: 10.1007/s10107-014-0805-x.  Google Scholar [28] M. Hong and Z. Luo, On the linear convergence of the alternating direction method of multipliers, arXiv preprint, arXiv: 1208.3922, (2013). Google Scholar [29] Y. Huang, M. K. Ng and Y.-W. Wen, A fast total variation minimization method for image restoration, Multiscale Model. Sim., 7 (2008), 774-795.  doi: 10.1137/070703533.  Google Scholar [30] Y. Jiao, Q. Jin, X. Lu and W. Wang, Alternating direction method of multipliers for linear inverse problems, SIAM J. Numer. Anal., 54 (2016), 2114-2137.  doi: 10.1137/15M1029308.  Google Scholar [31] Y. Jiao, Q. Jin, X. Lu and W. Wang, Preconditioned alternating direction method of multipliers for inverse problems with constraints, nverse Probl., 33 (2017), 025004(34pp).   Google Scholar [32] C. Li, W. Yin and Y. Zhang, Tv Minimization by Augmented Lagrangian and Alternating Direction Algorithms, Tech. rep., Department of CAAM, Rice University, Houston, Texas, 77005, 2009. http://www.caam.rice.edu/~optimization/L1/TVAL3/. Google Scholar [33] 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 [34] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685.  doi: 10.1002/cpa.3160420503.  Google Scholar [35] Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program., 103 (2005), 127-152.  doi: 10.1007/s10107-004-0552-5.  Google Scholar [36] Y. Ouyang, Y. Chen, G. Lan and E.J. Pasiliao, An accelerated linearized alternating direction method of multipliers, SIAM J Imaging Sci., 8 (2015), 644-681.  doi: 10.1137/14095697X.  Google Scholar [37] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [38] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston, New York, 1977.   Google Scholar [39] C. Vogel and M. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Trans. Image Proces., 7 (1998), 813-824.  doi: 10.1109/83.679423.  Google Scholar [40] X. Wang and X. Yuan, The linearized alternating direction method of multipliers for dantzig selector, SIAM J. Sci. Comput., 34 (2012), 2792-2811.  doi: 10.1137/110833543.  Google Scholar [41] Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Image Sci., 1 (2008), 248-272.  doi: 10.1137/080724265.  Google Scholar [42] T. Wu, Variable splitting based method for image restoration with impulse plus gaussian noise, Math. Probl. Eng., 2016 (2016), Article ID 3151303, 16 pages. Google Scholar [43] J. Yang and Y. Zhang, Alternating direction algorithms for $\ell_1$-problems in compressive sensing, SIAM J. Sci. Comput., 33 (2011), 250-278.  doi: 10.1137/090777761.  Google Scholar [44] 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), 2842-2865.  doi: 10.1137/080732894.  Google Scholar [45] C. Ye and X. Yuan, A descent method for structured monotone variational inequalities, Optim. Methods Softw., 22 (2007), 329-338.  doi: 10.1080/10556780600552693.  Google Scholar [46] W. Yin, D. Goldfarb and S. Osher, The total variation regularized $L^1$ multiscale decomposition, Multiscale Model. Sim., 6 (2007), 190-211.  doi: 10.1137/060663027.  Google Scholar
From left to right are the results of FTVD, New algorithm, TVAL3, and SNR history plot. First row: $NR$ = 60%, $airplane$; second row: $NR$ = 80%, $lena$
Performance v.s. noise ratio. Left: time; right: SNR. First row: $RC = 0.1$, $coins$; second row: $RC = 0$, $westconcordorthophoto$
Performance v.s. indices matrix quality. Left: time, right: SNR. First row: $moon$, $NR = 60%$; second row: $tire$, $NR = 80%$
From left to right to below are the results of FTVD, New algorithm, and TVAL3, and SNR history plot. First row: "average" kernel with dirichlet boundary condition; second row: "Gaussian" kernel; third row: "Disk" kernel; fourth row: "motion" kernel
Numerical results of basic test
 FTVD New LADMM TVAL3 Noise Ratio time SNR time SNR time SNR 60% 3.806 20.894 20.155 23.594 14.212 11.806 80% 8.174 8.751 19.687 15.226 26.442 7.106
 FTVD New LADMM TVAL3 Noise Ratio time SNR time SNR time SNR 60% 3.806 20.894 20.155 23.594 14.212 11.806 80% 8.174 8.751 19.687 15.226 26.442 7.106
 [1] Yuan Shen, Wenxing Zhang, Bingsheng He. Relaxed augmented Lagrangian-based proximal point algorithms for convex optimization with linear constraints. Journal of Industrial & Management Optimization, 2014, 10 (3) : 743-759. doi: 10.3934/jimo.2014.10.743 [2] Chunlin Wu, Juyong Zhang, Xue-Cheng Tai. Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Problems & Imaging, 2011, 5 (1) : 237-261. doi: 10.3934/ipi.2011.5.237 [3] Igor Griva, Roman A. Polyak. Proximal point nonlinear rescaling method for convex optimization. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 283-299. doi: 10.3934/naco.2011.1.283 [4] 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 [5] Chunrong Chen, T. C. Edwin Cheng, Shengji Li, Xiaoqi Yang. Nonlinear augmented Lagrangian for nonconvex multiobjective optimization. Journal of Industrial & Management Optimization, 2011, 7 (1) : 157-174. doi: 10.3934/jimo.2011.7.157 [6] 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 [7] Yan Gu, Nobuo Yamashita. A proximal ADMM with the Broyden family for convex optimization problems. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020091 [8] Yanqin Bai, Lipu Zhang. A full-Newton step interior-point algorithm for symmetric cone convex quadratic optimization. Journal of Industrial & Management Optimization, 2011, 7 (4) : 891-906. doi: 10.3934/jimo.2011.7.891 [9] Chunrong Chen. A unified nonlinear augmented Lagrangian approach for nonconvex vector optimization. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 495-508. doi: 10.3934/naco.2011.1.495 [10] Xueyong Wang, Yiju Wang, Gang Wang. An accelerated augmented Lagrangian method for multi-criteria optimization problem. Journal of Industrial & Management Optimization, 2020, 16 (1) : 1-9. doi: 10.3934/jimo.2018136 [11] Chunming Tang, Jinbao Jian, Guoyin Li. A proximal-projection partial bundle method for convex constrained minimax problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 757-774. doi: 10.3934/jimo.2018069 [12] Tim Hoheisel, Maxime Laborde, Adam Oberman. A regularization interpretation of the proximal point method for weakly convex functions. Journal of Dynamics & Games, 2020, 7 (1) : 79-96. doi: 10.3934/jdg.2020005 [13] 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 [14] Foxiang Liu, Lingling Xu, Yuehong Sun, Deren Han. A proximal alternating direction method for multi-block coupled convex optimization. Journal of Industrial & Management Optimization, 2019, 15 (2) : 723-737. doi: 10.3934/jimo.2018067 [15] Fan Jiang, Zhongming Wu, Xingju Cai. Generalized ADMM with optimal indefinite proximal term for linearly constrained convex optimization. Journal of Industrial & Management Optimization, 2020, 16 (2) : 835-856. doi: 10.3934/jimo.2018181 [16] Egil Bae, Xue-Cheng Tai, Wei Zhu. Augmented Lagrangian method for an Euler's elastica based segmentation model that promotes convex contours. Inverse Problems & Imaging, 2017, 11 (1) : 1-23. doi: 10.3934/ipi.2017001 [17] Adil Bagirov, Sona Taheri, Soodabeh Asadi. A difference of convex optimization algorithm for piecewise linear regression. Journal of Industrial & Management Optimization, 2019, 15 (2) : 909-932. doi: 10.3934/jimo.2018077 [18] Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial & Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134 [19] Xiaojing Ye, Haomin Zhou. Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm. Inverse Problems & Imaging, 2013, 7 (3) : 1031-1050. doi: 10.3934/ipi.2013.7.1031 [20] 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

2018 Impact Factor: 1.025

## Tools

Article outline

Figures and Tables