-
Previous Article
The inverse problem for electroseismic conversion: Stable recovery of the conductivity and the electrokinetic mobility parameter
- IPI Home
- This Issue
-
Next Article
On the stability of some imaging functionals
Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators
1. | University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria |
2. | Chemnitz University of Technology, Department of Mathematics, D-09107 Chemnitz, Germany |
References:
[1] |
H. Attouch, L. M. Briceño-Arias and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim., 48 (2010), 3246-3270.
doi: 10.1137/090754297. |
[2] |
H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York, 2011.
doi: 10.1007/978-1-4419-9467-7. |
[3] |
S. Becker and P. L. Combettes, An algorithm for splitting parallel sums of linearly composed monotone operators, with applications to signal recovery, J. Nonlinear Convex A., 15 (2014), 137-159. |
[4] |
R. I. Boţ, Conjugate Duality in Convex Optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 637, Springer, Berlin, 2010.
doi: 10.1007/978-3-642-04900-2. |
[5] |
R. I. Boţ, E. R. Csetnek and A. Heinrich, A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators, SIAM J. Optim., 23 (2013), 2011-2036.
doi: 10.1137/12088255X. |
[6] |
R. I. Boţ, E. R. Csetnek, A. Heinrich and C. Hendrich, On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems, Math. Program., 150 (2015), 251-279.
doi: 10.1007/s10107-014-0766-0. |
[7] |
R. I. Boţ, S. M. Grad and G. Wanka, Duality in Vector Optimization, Springer, Berlin, 2009.
doi: 10.1007/978-3-642-02886-1. |
[8] |
R. I. Boţ and C. Hendrich, A variable smoothing algorithm for solving convex optimization problems, TOP, 23 (2015), 124-150.
doi: 10.1007/s11750-014-0326-z. |
[9] |
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. |
[10] |
R. I. Boţ and C. Hendrich, On the acceleration of the double smoothing technique for unconstrained convex optimization problems, Optimization, 64 (2015), 265-288.
doi: 10.1080/02331934.2012.745530. |
[11] |
R. I. Boţ and C. Hendrich, A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems, Comput. Optim. Appl., 54 (2013), 239-262.
doi: 10.1007/s10589-012-9523-6. |
[12] |
R. I. Boţ and C. Hendrich, A Douglas-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. |
[13] |
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. |
[14] |
A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188.
doi: 10.1007/s002110050258. |
[15] |
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. |
[16] |
P. L. Combettes, Quasi-Fejérian analysis of some optimization algorithms, In: D. Butnariu, Y. Censor and S. Reich (Eds.), Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, Elsevier, New York, 8 (2001), 115-152.
doi: 10.1016/S1570-579X(01)80010-0. |
[17] |
P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504.
doi: 10.1080/02331930412331327157. |
[18] |
P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal., 16 (2009), 727-748. |
[19] |
P. L. Combettes, Systems of structured monotone inclusions: Duality, algorithms, and applications, SIAM J. Optim., 23 (2013), 2420-2447.
doi: 10.1137/130904160. |
[20] |
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. |
[21] |
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. |
[22] |
J. Douglas and H. H. Rachford, On the numerical solution of the heat conduction problem in two and three space variables, Trans. of the Amer. Math. Soc., 82 (1956), 421-439.
doi: 10.1090/S0002-9947-1956-0084194-4. |
[23] |
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), 293-318.
doi: 10.1007/BF01581204. |
[24] |
S. Harizanov, J.-C. Pesquet and G. Steidl, Epigraphical projection for solving least squares anscombe transformed constrained optimization problems, In: Scale Space and Variational Methods in Computer Vision, Springer, Berlin Heidelberg, 7893 (2013), 125-136.
doi: 10.1007/978-3-642-38267-3_11. |
[25] |
R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), 209-216.
doi: 10.2140/pjm.1970.33.209. |
[26] |
R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.
doi: 10.1137/0314056. |
[27] |
S. Setzer, G. Steidl and T. Teuber, Infimal convolution regularizations with discrete $l_1$-type functionals, Commun. Math. Sci., 9 (2011), 797-827.
doi: 10.4310/CMS.2011.v9.n3.a7. |
[28] |
P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.
doi: 10.1137/S0363012998338806. |
[29] |
B. C. Vũ, A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comp. Math., 38 (2013), 667-681.
doi: 10.1007/s10444-011-9254-8. |
[30] |
C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002.
doi: 10.1142/9789812777096. |
show all references
References:
[1] |
H. Attouch, L. M. Briceño-Arias and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim., 48 (2010), 3246-3270.
doi: 10.1137/090754297. |
[2] |
H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York, 2011.
doi: 10.1007/978-1-4419-9467-7. |
[3] |
S. Becker and P. L. Combettes, An algorithm for splitting parallel sums of linearly composed monotone operators, with applications to signal recovery, J. Nonlinear Convex A., 15 (2014), 137-159. |
[4] |
R. I. Boţ, Conjugate Duality in Convex Optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 637, Springer, Berlin, 2010.
doi: 10.1007/978-3-642-04900-2. |
[5] |
R. I. Boţ, E. R. Csetnek and A. Heinrich, A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators, SIAM J. Optim., 23 (2013), 2011-2036.
doi: 10.1137/12088255X. |
[6] |
R. I. Boţ, E. R. Csetnek, A. Heinrich and C. Hendrich, On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems, Math. Program., 150 (2015), 251-279.
doi: 10.1007/s10107-014-0766-0. |
[7] |
R. I. Boţ, S. M. Grad and G. Wanka, Duality in Vector Optimization, Springer, Berlin, 2009.
doi: 10.1007/978-3-642-02886-1. |
[8] |
R. I. Boţ and C. Hendrich, A variable smoothing algorithm for solving convex optimization problems, TOP, 23 (2015), 124-150.
doi: 10.1007/s11750-014-0326-z. |
[9] |
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. |
[10] |
R. I. Boţ and C. Hendrich, On the acceleration of the double smoothing technique for unconstrained convex optimization problems, Optimization, 64 (2015), 265-288.
doi: 10.1080/02331934.2012.745530. |
[11] |
R. I. Boţ and C. Hendrich, A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems, Comput. Optim. Appl., 54 (2013), 239-262.
doi: 10.1007/s10589-012-9523-6. |
[12] |
R. I. Boţ and C. Hendrich, A Douglas-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. |
[13] |
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. |
[14] |
A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188.
doi: 10.1007/s002110050258. |
[15] |
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. |
[16] |
P. L. Combettes, Quasi-Fejérian analysis of some optimization algorithms, In: D. Butnariu, Y. Censor and S. Reich (Eds.), Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, Elsevier, New York, 8 (2001), 115-152.
doi: 10.1016/S1570-579X(01)80010-0. |
[17] |
P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504.
doi: 10.1080/02331930412331327157. |
[18] |
P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal., 16 (2009), 727-748. |
[19] |
P. L. Combettes, Systems of structured monotone inclusions: Duality, algorithms, and applications, SIAM J. Optim., 23 (2013), 2420-2447.
doi: 10.1137/130904160. |
[20] |
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. |
[21] |
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. |
[22] |
J. Douglas and H. H. Rachford, On the numerical solution of the heat conduction problem in two and three space variables, Trans. of the Amer. Math. Soc., 82 (1956), 421-439.
doi: 10.1090/S0002-9947-1956-0084194-4. |
[23] |
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), 293-318.
doi: 10.1007/BF01581204. |
[24] |
S. Harizanov, J.-C. Pesquet and G. Steidl, Epigraphical projection for solving least squares anscombe transformed constrained optimization problems, In: Scale Space and Variational Methods in Computer Vision, Springer, Berlin Heidelberg, 7893 (2013), 125-136.
doi: 10.1007/978-3-642-38267-3_11. |
[25] |
R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), 209-216.
doi: 10.2140/pjm.1970.33.209. |
[26] |
R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.
doi: 10.1137/0314056. |
[27] |
S. Setzer, G. Steidl and T. Teuber, Infimal convolution regularizations with discrete $l_1$-type functionals, Commun. Math. Sci., 9 (2011), 797-827.
doi: 10.4310/CMS.2011.v9.n3.a7. |
[28] |
P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.
doi: 10.1137/S0363012998338806. |
[29] |
B. C. Vũ, A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comp. Math., 38 (2013), 667-681.
doi: 10.1007/s10444-011-9254-8. |
[30] |
C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002.
doi: 10.1142/9789812777096. |
[1] |
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 and Imaging, 2021, 15 (4) : 787-825. doi: 10.3934/ipi.2021014 |
[2] |
Yanqin Bai, Xuerui Gao, Guoqiang Wang. Primal-dual interior-point algorithms for convex quadratic circular cone optimization. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 211-231. doi: 10.3934/naco.2015.5.211 |
[3] |
Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations and Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 |
[4] |
Gianni Di Pillo, Giampaolo Liuzzi, Stefano Lucidi. A primal-dual algorithm for nonlinear programming exploiting negative curvature directions. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 509-528. doi: 10.3934/naco.2011.1.509 |
[5] |
Kai Wang, Deren Han. On the linear convergence of the general first order primal-dual algorithm. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021134 |
[6] |
Xiaojing Ye, Haomin Zhou. Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm. Inverse Problems and Imaging, 2013, 7 (3) : 1031-1050. doi: 10.3934/ipi.2013.7.1031 |
[7] |
Yu-Hong Dai, Zhouhong Wang, Fengmin Xu. A Primal-dual algorithm for unfolding neutron energy spectrum from multiple activation foils. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2367-2387. doi: 10.3934/jimo.2020073 |
[8] |
Siqi Li, Weiyi Qian. Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization. Numerical Algebra, Control and Optimization, 2015, 5 (1) : 37-46. doi: 10.3934/naco.2015.5.37 |
[9] |
Guoqiang Wang, Zhongchen Wu, Zhongtuan Zheng, Xinzhong Cai. Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 101-113. doi: 10.3934/naco.2015.5.101 |
[10] |
Yuying Zhou, Gang Li. The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions. Numerical Algebra, Control and Optimization, 2014, 4 (1) : 9-23. doi: 10.3934/naco.2014.4.9 |
[11] |
Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial and Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723 |
[12] |
Fengmin Wang, Dachuan Xu, Donglei Du, Chenchen Wu. Primal-dual approximation algorithms for submodular cost set cover problems with linear/submodular penalties. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 91-100. doi: 10.3934/naco.2015.5.91 |
[13] |
Yu-Hong Dai, Xin-Wei Liu, Jie Sun. A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs. Journal of Industrial and Management Optimization, 2020, 16 (2) : 1009-1035. doi: 10.3934/jimo.2018190 |
[14] |
Xiayang Zhang, Yuqian Kong, Shanshan Liu, Yuan Shen. A relaxed parameter condition for the primal-dual hybrid gradient method for saddle-point problem. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022008 |
[15] |
Anulekha Dhara, Aparna Mehra. Conjugate duality for generalized convex optimization problems. Journal of Industrial and Management Optimization, 2007, 3 (3) : 415-427. doi: 10.3934/jimo.2007.3.415 |
[16] |
Yazheng Dang, Fanwen Meng, Jie Sun. Convergence analysis of a parallel projection algorithm for solving convex feasibility problems. Numerical Algebra, Control and Optimization, 2016, 6 (4) : 505-519. doi: 10.3934/naco.2016023 |
[17] |
Lican Kang, Yuan Luo, Jerry Zhijian Yang, Chang Zhu. A primal and dual active set algorithm for truncated $L_1$ regularized logistic regression. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022050 |
[18] |
Adil Bagirov, Sona Taheri, Soodabeh Asadi. A difference of convex optimization algorithm for piecewise linear regression. Journal of Industrial and Management Optimization, 2019, 15 (2) : 909-932. doi: 10.3934/jimo.2018077 |
[19] |
Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial and Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134 |
[20] |
Nadia Hazzam, Zakia Kebbiche. A primal-dual interior point method for $ P_{\ast }\left( \kappa \right) $-HLCP based on a class of parametric kernel functions. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 513-531. doi: 10.3934/naco.2020053 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]