Skip to main content Accessibility help


  • L. LI (a1), G. Q. WANG (a1) and J. L. ZHANG (a1)


We focus on the convergence rate of the alternating direction method of multipliers (ADMM) in a complex domain. First, the complex form of variational inequality (VI) is established by using the Wirtinger calculus technique. Second, the $O(1/K)$ convergence rate of the ADMM in a complex domain is provided. Third, the ADMM in a complex domain is applied to the least absolute shrinkage and selectionator operator (LASSO). Finally, numerical simulations are provided to show that ADMM in a complex domain has the $O(1/K)$ convergence rate and that it has certain advantages compared with the ADMM in a real domain.


Corresponding author


Hide All
[1] Bai, Y. Q. and Shen, K. J., “Alternating direction method of multipliers for $\ell _{1}{-}\ell _{2}$ -regularized logistic regression model”, J. Oper. Res. Soc. China 4 (2016) 243253; doi:10.1007/s40305-015-0090-2.
[2] Boyd, S., Parikh, N., Chu, E., Peleato, B. and Eckstein, J., “Distributed optimization and statistical learning via the alternating direction method of multipliers”, Found. Trends Mach. Learn. 3 (2011) 1122; doi:10.1561/2200000016.
[3] Cai, X. J., Han, D. R. and Yuan, X. M., “On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function”, Comput. Optim. Appl. 66 (2017) 3973; doi:10.1007/s10589-016-9860-y.
[4] Candès, E. J., Romberg, J. and Tao, T., “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information”, IEEE Trans. Inform. Theory 52 (2006) 489509; doi:10.1109/TIT.2005.862083.
[5] Cottle, R. W., Giannessi, F. and Lions, J. L., Variational inequalities and complementarity problems: theory and applications (Wiley, Chichester, 1980); doi:10.1002/zamm.19810610316.
[6] Dafermosa, S. C., “An iterative scheme for variational inequalities”, Math. Program. 26 (1983) 4047; doi:10.1007/BF02591891.
[7] Donoho, D. L., “Compressed sensing”, IEEE Trans. Inform. Theory 52 (2006) 12891306; doi:10.1109/TIT.2006.871582.
[8] Eckstein, J. and Bertsekas, D. P., “On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators”, Math. Program. 55 (1992) 293318; doi:10.1007/BF01581204.
[9] Fichera, G., “Abstract unilateral problems”, in: Conference on the theory of ordinary and partial differential equations (2006) 5164; doi:10.1007/BFb0066918.
[10] Gabay, D., “Applications of the method of multipliers to variational inequalities”, Stud. Math. Appl. 15 (1983) 299331; doi:10.1016/S0168-2024(08)70034-1.
[11] Gabay, D. and Mercier, B., “A dual algorithm for the solution of nonlinear variational problems via finite element approximation”, Comput. Math. Appl. 2 (1976) 1740; doi:10.1016/0898-1221(76)90003-1.
[12] Glowinski, R., Lions, J. L. and Tremoliérès, R., Numerical analysis of variational inequalities (North-Holland, Amsterdam, 1981); ISBN: 9780080875293.
[13] Grant, M. and Boyd, S., “CVX: Matlab software for disciplined convex programming, version 2.1”, (2014);
[14] Hastie, T., Tibshirani, R. and Friedman, J., The elements of statistical learning: data mining, inference and prediction (Springer, New York, 2009); doi:10.1007/978-0-387-84858-7.
[15] He, B. S. and Yuan, X. M., “On the $O(1/n)$ convergence rate of the Douglas–Rachford alternating direction method”, SIAM J. Numer. Anal. 50 (2012) 700709; doi:10.1137/110836936.
[16] He, B. S. and Yuan, X. M., “On non-ergodic convergence rate of Douglas–Rachford alternating direction method of multipliers”, Numer. Math. 130 (2015) 567577; doi:10.1007/s00211-014-0673-6.
[17] Hestenes, M. R., “Multiplier and gradient methods”, J. Optim. Theory Appl. 4 (1969) 303320; doi:10.1007/BF00927673.
[18] Hinamoto, T., Doi, A. and Lu, W. S., “Realization of 3-D separable-denominator digital filters with low $l_{2}$ -sensitivity”, IEEE Trans. Signal Process. 60 (2012) 62826293; doi:10.1109/TSP.2012.2215027.
[19] Li, C. J., Liu, C., Deng, K., Yu, X. H. and Huang, T. W., “Data-driven charging strategy of PEVs under transformer aging risk”, IEEE Trans. Control Syst. Technol. Vol PP (2017) 114; doi:10.1109/TCST.2017.2713321.
[20] Li, J. Y., Li, C. J., Wu, Z. Y., Wang, X. Y., Teo, K. L. and Wu, C. Z., “Sparsity-promoting distributed charging control for plug-in electric vehicles over distribution networks”, Appl. Math. Model. 58 (2018) 111127; doi:10.1016/j.apm.2017.10.034.
[21] Li, J. Y., Li, C. J., Xu, Y., Dong, Z., Wong, K. and Huang, T. W., “Noncooperative game-based distributed charging control for plug-in electric vehicles in distribution networks”, IEEE Trans. Ind. Inform. 14 (2018) 301310; doi:10.1109/TII.2016.2632761.
[22] Li, L., Wang, X. Y. and Wang, G. Q., “Alternating direction method of multipliers for separable convex optimization of real functions in complex variables”, Math. Probl. Eng. 2015 (2015) 114; doi:10.1155/2015/104531.
[23] Lin, T. Y., Ma, S. Q. and Zhang, S. Z., “On the sublinear convergence rate of multi-block ADMM”, J. Oper. Res. Soc. China 3 (2015) 251274; doi:10.1007/s40305-015-0092-0.
[24] Lofberg, J., “YALMIP: a toolbox for modeling and optimization in MATLAB”, Optim. 2004 (2004) 284289; doi:10.1109/CACSD.2004.1393890.
[25] Lu, W. S. and Hinamoto, T., “Two-dimensional digital filters with sparse coefficients”, Multidimens. Syst. Signal Process. 22 (2011) 173189; doi:10.1007/s11045-010-0129-9.
[26] Monteiro, R. D. C. and Svaiter, B. F., “Iteration-complexity of block-decomposition algorithms and the alternating direction method of multipliers”, SIAM J. Optim. 23 (2010) 475507; doi:10.1137/110849468.
[27] Ng, M., Weiss, P. and Yuan, X. M., “Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods”, SIAM J. Sci. Comput. 32 (2010) 27102736; doi:10.1137/090774823.
[28] Rudin, L. I., Osher, S. and Fatemi, E., “Nonlinear total variation based noise removal algorithms”, in: Experimental mathematics, computational issues in nonlinear science: Proc. of the Eleventh Annual Int. Conf. of the Center for Nonlinear Studies, Los Alamos, NM, USA, 20–24 May 1991. Volume 60 (Elsevier Science Publishers B.V., Amsterdam, 1992) 259268; doi:10.1016/0167-2789(92)90242-F.
[29] Scheinberg, K., Ma, S. Q. and Goldfarb, D., “Sparse inverse covariance selection via alternating linearization methods”, in: Twenty-fourth annual conference on neural information processing systems (NIPS) Vancouver, Canada, 6–9 December, 2010, Volume 23 Advances in Neural Information Processing Systems (Curran Associates Inc., Red Hook, New York, 2010) 21012109; ISBN: 978-1-61782-380-0.
[30] Sorber, L., Barel, M. V. and Lathauwer, L. D., “Unconstrained optimization of real functions in complex variables”, SIAM J. Optim. 22 (2012) 87918898; doi:10.1137/110832124.
[31] Sturm, J. F., “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cons”, Optim. Methods Softw. 11 (1999) 625653; doi:10.1080/10556789908805766.
[32] Tibshirani, R., “Regression shrinkage selection via the LASSO”, J. R. Stat. Soc. Ser. B Stat. Methodol. 58 (1996) 267288; URL:
[33] Toh, K. C., Todd, M. J. and Tütüncü, R. H., “SDPT3 – a Matlab software package for semidefinite programming, Version 1.3”, Optim. Methods Softw. 11 (1999) 545581; doi:10.1080/10556789908805762.
[34] Zhao, L. and Dafermos, S., “General economic equilibrium and variational inequalities”, Oper. Res. Lett. 10 (1991) 369376; doi:10.1016/0167-6377(91)90037-P.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed