# American Institute of Mathematical Sciences

• Previous Article
Machine interference problem involving unsuccessful switchover for a group of repairable servers with vacations
• JIMO Home
• This Issue
• Next Article
Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities
doi: 10.3934/jimo.2020053

## The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems

 1 Department of Information and Computing Science, School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China 2 School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing 210023, China 3 LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University, Beijing 100191, China

* Corresponding author: Xingju Cai

Received  July 2019 Revised  October 2019 Published  March 2020

The symmetric alternating direction method of multipliers is an efficient algorithm, which updates the Lagrange multiplier twice at each iteration and the variables are treated in a symmetric manner. Considering that the convergence range of the parameters plays an important role in the implementation of the algorithm. In this paper, we analyze the convergence rate of the symmetric ADMM with a more relaxed parameter range for solving the two block nonconvex separable optimization problem under the assumption that the generated sequence is bounded. Two cases are considered. In the first case, both components of the objective function are nonconvex, we prove the convergence of the augmented Lagrangian function sequence, and establish the $O(1/\sqrt{k})$ worst-case complexity measured by the difference of two consecutive iterations. In the second case, one component of the objective function is convex and the error bound condition is assumed, then we can prove that the iterative sequence converges locally to a KKT point in a R-linear rate; and an auxiliary sequence converges in a Q-linear rate. Furthermore, a practical inexact symmetric ADMM with relative error criteria is proposed, and the associated convergence analysis is established under the same conditions.

Citation: Zehui Jia, Xue Gao, Xingju Cai, Deren Han. The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020053
##### References:
 [1] H. Attouch, J. Bolte, B. F. Svaiter and A. Soubeyran, Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods, Mathematical Programming, 137 (2013), 91-129.  doi: 10.1007/s10107-011-0484-9.  Google Scholar [2] L. E. Gibbons, D. W. Hearn, P. M. Pardalos and M. V. Ramana, Continuous characterizations of the maximum clique problem, Mathematics of Operations Research, 22 (1997), 754-768.  doi: 10.1287/moor.22.3.754.  Google Scholar [3] R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires, Revue Française D'Automatique, Informatique, Recherche Opérationnelle. Analyse Numérique, 9 (1975), 41–76. doi: 10.1051/m2an/197509R200411.  Google Scholar [4] R. Glowinski, T. W. Pan and X. C. Tai, Some facts about operator-splitting and alternating direcrion methods, Splitting Methods in Communication, Imaging, Science, and Engineering, (2016), 19–94.  Google Scholar [5] Y. Gu, B. Jiang and D. R. Han, A semi-proximal-based strictly contractive Peaceman-Rachford splitting method, arXiv: 1506.02221, 2015. Google Scholar [6] B. S. He, H. Liu, Z. R. Wang and X. M. Yuan, A strictly contractive Peaceman-Rachford splitting method for convex programming, SIAM Journal on Optimization, 24 (2014), 1011-1040.  doi: 10.1137/13090849X.  Google Scholar [7] B. S. He, F. Ma and X. M. Yuan, Convergence study on the symmetric version of admm with larger step sizes, SIAM Journal on Imaging Sciences, 9 (2016), 1467-1501.  doi: 10.1137/15M1044448.  Google Scholar [8] D. R. Han, A new hybrid generalized proximal point algorithm for variational inequality problems, Journal of Global Optimization, 26 (2003), 125-140.  doi: 10.1023/A:1023087304476.  Google Scholar [9] D. R. Han, Inexact operator splitting methods with selfadaptive strategy for variational inequality problems, Journal of Optimization Theory and Applications, 132 (2007), 227-243.  doi: 10.1007/s10957-006-9060-5.  Google Scholar [10] Z. H. Jia, X. Gao, X. J. Cai and D. R. Han, Local Linear Convergence of the Alternating Direction Method of Multipliers for Nonconvex Separable Optimization Problems, Manuscript, 2018. Google Scholar [11] W. Kaplan, A test for copositive matrices, Linear Algebra and its Applications, 313 (2000), 203-206.  doi: 10.1016/S0024-3795(00)00138-5.  Google Scholar [12] J. Liu, J. H. Chen and J. P. Ye, Large-scale sparse logistic regression, Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, Paris, (2009), 547–556. doi: 10.1145/1557019.1557082.  Google Scholar [13] P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numercial Analusis, 16 (1979), 964-979.  doi: 10.1137/0716071.  Google Scholar [14] Z. Q. Luo and P. Tseng, Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem, SIAM Journal on Optimization, 2 (1992), 43-54.  doi: 10.1137/0802004.  Google Scholar [15] Z. Q. Luo and P. Tseng, Error bounds and convergence analysis of feasible descent methods: A general approach, Annals of Operations Research, 46 (1993), 157-178.  doi: 10.1007/BF02096261.  Google Scholar [16] D. H. Peaceman and H. H. Rachford, The numerical solution of parabolic and elliptic differential equations, Journal of the Society for Industrial and Applied Mathematics, 3 (1955), 28-41.  doi: 10.1137/0103003.  Google Scholar [17] N. Parikh and S. Boyd, Proximal algorithms, Foundations and Trends in Optimization, 1 (2014), 127-239.  doi: 10.1561/2400000003.  Google Scholar [18] H. V$\ddot{a}$liaho, Quadratic programming criteria for copositive matrices, Linear Algebra and its Applications, 119 (1989), 163-182.  doi: 10.1016/0024-3795(89)90076-1.  Google Scholar [19] Z. M. Wu, M. Li, David Z. W. Wang and D. R. Han, A symmetric alternating direction method of multipliers for separable nonconvex mimimization problems, Asia-Pacific Journal of Operational Research, 34 (2017), 1750030, 27pp. doi: 10.1142/S0217595917500300.  Google Scholar [20] B. Wen, X. J. Chen and T. K. Pong, Linear convergence of proximal gradient algorithm with extrapolation for a class of nonconvex nonsmooth minimization problems, SIAM Journal on Optimization, 27 (2017), 124-145.  doi: 10.1137/16M1055323.  Google Scholar [21] Z. B. Xu, X. Y. Chang, F. M. Xu and H. Zhang, $L_ {1/2}$ regularization: A thresholding representation theory and a fast solver, IEEE Transactions on Neural Networks and Learning Systems, 23 (2012), 1013-1027.   Google Scholar

show all references

##### References:
 [1] H. Attouch, J. Bolte, B. F. Svaiter and A. Soubeyran, Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods, Mathematical Programming, 137 (2013), 91-129.  doi: 10.1007/s10107-011-0484-9.  Google Scholar [2] L. E. Gibbons, D. W. Hearn, P. M. Pardalos and M. V. Ramana, Continuous characterizations of the maximum clique problem, Mathematics of Operations Research, 22 (1997), 754-768.  doi: 10.1287/moor.22.3.754.  Google Scholar [3] R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires, Revue Française D'Automatique, Informatique, Recherche Opérationnelle. Analyse Numérique, 9 (1975), 41–76. doi: 10.1051/m2an/197509R200411.  Google Scholar [4] R. Glowinski, T. W. Pan and X. C. Tai, Some facts about operator-splitting and alternating direcrion methods, Splitting Methods in Communication, Imaging, Science, and Engineering, (2016), 19–94.  Google Scholar [5] Y. Gu, B. Jiang and D. R. Han, A semi-proximal-based strictly contractive Peaceman-Rachford splitting method, arXiv: 1506.02221, 2015. Google Scholar [6] B. S. He, H. Liu, Z. R. Wang and X. M. Yuan, A strictly contractive Peaceman-Rachford splitting method for convex programming, SIAM Journal on Optimization, 24 (2014), 1011-1040.  doi: 10.1137/13090849X.  Google Scholar [7] B. S. He, F. Ma and X. M. Yuan, Convergence study on the symmetric version of admm with larger step sizes, SIAM Journal on Imaging Sciences, 9 (2016), 1467-1501.  doi: 10.1137/15M1044448.  Google Scholar [8] D. R. Han, A new hybrid generalized proximal point algorithm for variational inequality problems, Journal of Global Optimization, 26 (2003), 125-140.  doi: 10.1023/A:1023087304476.  Google Scholar [9] D. R. Han, Inexact operator splitting methods with selfadaptive strategy for variational inequality problems, Journal of Optimization Theory and Applications, 132 (2007), 227-243.  doi: 10.1007/s10957-006-9060-5.  Google Scholar [10] Z. H. Jia, X. Gao, X. J. Cai and D. R. Han, Local Linear Convergence of the Alternating Direction Method of Multipliers for Nonconvex Separable Optimization Problems, Manuscript, 2018. Google Scholar [11] W. Kaplan, A test for copositive matrices, Linear Algebra and its Applications, 313 (2000), 203-206.  doi: 10.1016/S0024-3795(00)00138-5.  Google Scholar [12] J. Liu, J. H. Chen and J. P. Ye, Large-scale sparse logistic regression, Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, Paris, (2009), 547–556. doi: 10.1145/1557019.1557082.  Google Scholar [13] P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numercial Analusis, 16 (1979), 964-979.  doi: 10.1137/0716071.  Google Scholar [14] Z. Q. Luo and P. Tseng, Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem, SIAM Journal on Optimization, 2 (1992), 43-54.  doi: 10.1137/0802004.  Google Scholar [15] Z. Q. Luo and P. Tseng, Error bounds and convergence analysis of feasible descent methods: A general approach, Annals of Operations Research, 46 (1993), 157-178.  doi: 10.1007/BF02096261.  Google Scholar [16] D. H. Peaceman and H. H. Rachford, The numerical solution of parabolic and elliptic differential equations, Journal of the Society for Industrial and Applied Mathematics, 3 (1955), 28-41.  doi: 10.1137/0103003.  Google Scholar [17] N. Parikh and S. Boyd, Proximal algorithms, Foundations and Trends in Optimization, 1 (2014), 127-239.  doi: 10.1561/2400000003.  Google Scholar [18] H. V$\ddot{a}$liaho, Quadratic programming criteria for copositive matrices, Linear Algebra and its Applications, 119 (1989), 163-182.  doi: 10.1016/0024-3795(89)90076-1.  Google Scholar [19] Z. M. Wu, M. Li, David Z. W. Wang and D. R. Han, A symmetric alternating direction method of multipliers for separable nonconvex mimimization problems, Asia-Pacific Journal of Operational Research, 34 (2017), 1750030, 27pp. doi: 10.1142/S0217595917500300.  Google Scholar [20] B. Wen, X. J. Chen and T. K. Pong, Linear convergence of proximal gradient algorithm with extrapolation for a class of nonconvex nonsmooth minimization problems, SIAM Journal on Optimization, 27 (2017), 124-145.  doi: 10.1137/16M1055323.  Google Scholar [21] Z. B. Xu, X. Y. Chang, F. M. Xu and H. Zhang, $L_ {1/2}$ regularization: A thresholding representation theory and a fast solver, IEEE Transactions on Neural Networks and Learning Systems, 23 (2012), 1013-1027.   Google Scholar
The relationship between $\alpha$ and $\theta$
 [1] Hui Gao, Jian Lv, Xiaoliang Wang, Liping Pang. An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 805-825. doi: 10.3934/jimo.2019135 [2] Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 [3] Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105 [4] George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003 [5] Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1 [6] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [7] Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465 [8] Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020367 [9] Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29 [10] Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347 [11] Tengfei Yan, Qunying Liu, Bowen Dou, Qing Li, Bowen Li. An adaptive dynamic programming method for torque ripple minimization of PMSM. Journal of Industrial & Management Optimization, 2021, 17 (2) : 827-839. doi: 10.3934/jimo.2019136 [12] Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018 [13] Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129 [14] Xiuli Xu, Xueke Pu. Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 987-1010. doi: 10.3934/dcdsb.2020150 [15] Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388 [16] Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 [17] Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 [18] Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345 [19] Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 [20] Gheorghe Craciun, Jiaxin Jin, Casian Pantea, Adrian Tudorascu. Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1305-1335. doi: 10.3934/dcdsb.2020164

2019 Impact Factor: 1.366