Abstract
The generalized alternating direction method of multipliers (GADMM), which expands the dual step length to (0,2), is a benchmark for solving the two-block separable convex programming. Recently, there are many ADMM-based improved algorithms with indefinite term, that is, the second subproblem is linearized by a specialized indefinite matrix. In this paper, we propose a generalized proximal Peaceman–Rachford splitting method (abbreviated as GPRSM-S) with substitution step and indefinite term. We will find out the relationship between linearized parameter, dual step length and substitution factor. The global convergence and the worst-case convergence rate in nonergodic sense are established theoretically by variational inequality. Finally, some numerical results on LASSO and total variation based denoising problems are presented to verify the feasibility of the introduced method.
Similar content being viewed by others
References
Adona, V.A., Goncalves, M.L.N., Melo, J.G.: Iteration-complexity analysis of a generalized alternating direction method of multipliers. J. Glob. Optim. 1, 1–18 (2018)
Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9(1–2), 3–11 (2001)
Bai, J., Li, J., Xu, F., Zhang, H.: Generalized symmetric admm for separable convex optimization. Comput. Optim. Appl. 70(1), 129–170 (2018)
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)
Boyd, S.P., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. Arch. 3(1), 1–122 (2011)
Chang, X., Liu, S., Zhao, P., Li, X.: Convergent prediction-correction-based admm for multi-block separable convex programming. J. Comput. Appl. Math. 335, 270–288 (2017)
Chen, C.H., Chan, R.H., Ma, S.Q., Yang, J.F.: Inertial proximal admm for linearly constrained separable convex optimization. SIAM J. Imaging Sci. 8, 2239–2267 (2015)
Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Modeling Simul. 4(4), 1168–1200 (2006)
Corman, E., Yuan, X.: A generalized proximal point algorithm and its convergence rate. SIAM J. Optim. 24(4), 1614–1638 (2014)
Donoho, D.L., Tsaig, Y.: Fast solution of \(l_1\)-norm minimization problems when the solution may be sparse. IEEE Trans. Inf. Theory 54(11), 4789–4812 (2008)
Dou, M.Y., Li, H., Liu, X.W.: An inertial proximal Peaceman–Rachford splitting method. Scientia Sinica (2017)
Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)
Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(3), 293–318 (1992)
Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. 2, No. 1. Springer, New York (2003)
Fu, X., He, B., Wang, X., Yuan, X.: Block-wise alternating direction method of multipliers with gaussian back substitution for multiple-block convex programming (2014)
Gabay, D.: Chapter ix applications of the method of multipliers to variational inequalities. Stud. Math. Its Appl. 15, 299–331 (1983)
Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl. 2(1), 17–40 (1976)
Gao, B., Ma, F.: Symmetric alternating direction method with indefinite proximal regularization for linearly constrained convex optimization. J. Optim. Theory Appl. 176(1), 178–204 (2018)
Glowinski, R., Marrocco, A.: Sur l’approximation, par elements finis d’ordre un, et la resolution, par penalisation-dualit’e, d’une classe de problems de dirichlet non lineares. Ann. Stat. 9, 41–76 (1975)
Gu, Y., Jiang, B., Han, D.: A semi-proximal-based strictly contractive Peaceman–Rachford splitting method. Mathematics (2015)
He, B., Ma, F., Yuan, X.: Optimal linearized alternating direction method of multipliers for convex programming. http://www.optimization-online.org (2017)
He, B., Tao, M., Yuan, X.: Alternating direction method with Gaussian back substitution for separable convex programming. SIAM J. Optim. 22(2), 313–340 (2012)
He, B., Yuan, X.: On non-ergodic convergence rate of Douglas–Rachford alternating direction method of multipliers. Numer. Math. 130(3), 567–577 (2015)
He, B.S., Ma, F., Yuan, X.M.: Convergence study on the symmetric version of admm with larger step sizes. SIAM J. Imaging Sci. 9(3), 1467–1501 (2016)
He, Y., Li, H., Liu, X.: Relaxed inertial proximal Peaceman–Rachford splitting method for separable convex programming. Front. Math. China 13(3), 1–24 (2018)
Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4(5), 303–320 (1969)
Jiang, F., Wu, Z., Cai, X.: Generalized admm with optimal indefinite proximal term for linearly constrained convex optimization. J. Ind. Manag. Optim. 13(5), 1–22 (2017)
Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)
Monteiro, R.D.C., Svaiter, B.F.: Iteration-complexity of block-decomposition algorithms and the alternating direction method of multipliers. SIAM J. Optim. 23(1), 475–507 (2013)
Peaceman, D.W., Rachford, H.H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955)
Powell, M.J.D.: A method for nonlinear constraints in minimization problems. Optimization 5(6), 283–298 (1969)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. In: Eleventh International Conference of the Center for Nonlinear Studies on Experimental Mathematics: Computational Issues in Nonlinear Science: Computational Issues in Nonlinear Science, pp. 259–268 (1992)
Sun, H., Tian, M., Sun, M.: The symmetric admm with indefinite proximal regularization and its application. J. Inequal. Appl. 2017(1), 172 (2017)
Sun, M., Liu, J.: Generalized Peaceman–Rachford splitting method for separable convex programming with applications to image processing. J. Appl. Math. Comput. 51(1–2), 605–622 (2016)
Tao, M., Yuan, X.: The generalized proximal point algorithm with step size 2 is not necessarily convergent. Comput. Optim. Appl. 70(3), 1–13 (2018)
Tibshirani, R.: Regression shrinkage and selection via the lasso: a retrospective. J. R. Stat. Soc. Ser. B Stat. Methodol. 73(3), 273–282 (2011)
Wang, J.J., Song, W.: An algorithm twisted from generalized admm for multi-block separable convex minimization models. J. Comput. Appl. Math. 309, 342–358 (2017)
Acknowledgements
This work was supported by National Natural Science Foundation of China (No. 61877046), Natural Science Basic Research Program of Shanxi (No. 2017JM1001), Fundamental Research Funds for the Central Universities of China (No. JB150716).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Deng, Z., Liu, S. Generalized Peaceman–Rachford splitting method with substitution for convex programming. Optim Lett 14, 1781–1802 (2020). https://doi.org/10.1007/s11590-019-01473-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-019-01473-2