Abstract
We consider combining the generalized alternating direction method of multipliers, proposed by Eckstein and Bertsekas, with the logarithmic–quadratic proximal method proposed by Auslender, Teboulle, and Ben-Tiba for solving a variational inequality with separable structures. For the derived algorithm, we prove its global convergence and establish its worst-case convergence rate measured by the iteration complexity in both the ergodic and nonergodic senses.
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Notes
Note that we follow the work [21] and many others to measure the worst-case convergence rate in term of the iteration complexity. That is, a worst-case \(O(1/t)\) convergence rate means the accuracy to a solution under certain criteria is of the order \(O(1/t)\) after \(t\) iterations of an iterative scheme; or equivalently, it requires at most \(O(1/\varepsilon )\) iterations to achieve an approximate solution with an accuracy of \(\varepsilon \).
References
Eckstein, J., Bertsekas, D.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)
Auslender, A., Teboulle, M., Ben-Tiba, S.: A logarithmic-quadratic proximal method for variational inequalities. Comput. Optim. Appl. 12, 31–40 (1999)
Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific, Belmont (1997)
Fukushima, M.: Application of the alternating directions method of multipliers to separable convex programming problems. Comput. Optim. Appl. 1(1), 93–111 (1992)
Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)
Glowinski, R., Tallec, P.L.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. Society for Industrial and Applied Mathematics, Philadelphia (1989)
Tseng, P.: Alternating projection-proximal methods for convex programming and variational inequalities. SIAM J. Optim. 7(4), 951–965 (1997)
Glowinski, R., Marrocco, A.: Sur l’approximation par éléments finis d’ordre un et la résolution par pénalisation-dualité d’une classe de problémes de Dirichlet non linéaires. Revue Fr. Autom. Inform. Rech. Opér. Anal. Numér. 2, 41–76 (1975)
Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Comput. Math. Appl. 2(1), 17–40 (1976)
Chan, T.F., Glowinski, R.: Finite element approximation and iterative solution of a class of mildly non-linear elliptic equations. Stanford University, Technical Report (1978)
He, B.S., Liao, L.-Z., Han, D., Yang, H.: A new inexact alternating directions method for monontone variational inequalities. Math. Program. 92, 103–118 (2002)
Kontogiorgis, S., Meyer, R.R.: A variable-penalty alternating directions method for convex optimization. Math. Program. 83, 29–53 (1998)
Tseng, P.: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Con. Optim. 29, 119–138 (1991)
Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3, 1–122 (2010)
Martinet, B.: Regularision d’inéquations variationnelles par approximations successive. Revue Fr. Autom. Inform. Rech. Opér. 126, 154–159 (1970)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)
Gol’shtein, E.G., Tret’yakov, N.V.: Modified Lagrangian in convex programming and their generalizations. Math. Program. Stud. 10, 86–97 (1979)
Cai, X.J., Gu, G.Y., He, B.S., Yuan, X.M.: A proximal point algorithm revisit on the alternating direction method of multipliers. Sci. China Math. 56(10), 2179–2186 (2013)
Auslender, A., Teboulle, M.: Entropic proximal decomposition method for convex programs and variational inequalities. Math. Program. 91(1), 33–47 (2001)
Yuan, X.M., Li, M.: An LQP-based decomposition method for solving a class of variational inequalities. SIAM J. Optim. 21(4), 1309–1318 (2011)
Nesterov, Y.E.: A method for solving the convex programming problem with convergence rate \(O(1/k^2)\). Dokl. Akad. Nauk SSSR 269, 543–547 (1983)
He, B.S., Yuan, X.M.: On the \(O(1/n)\) convergence rate of the Douglas-Rachford alternating direction method. SIAM J. Numer. Anal. 50(2), 700–709 (2012)
He, B.S., Yuan, X.M.: On non-ergodic convergence rate of Douglas-Rachford alternating direction method of multipliers. Submission, (2013)
Shefi, R., Teboulle, M.: Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization. SIAM J. Optim. 24(1), 269–297 (2014)
Tao, M., Yuan, X.M.: On the \(O(1/t)\) convergence rate of alternating direction method with logarithmic-quadratic proximal regularization. SIAM J. Optim. 22(4), 1431–1448 (2012)
Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementarity Problems, Vols. I and II. Springer, New York (2003)
Nesterov, Y.: Gradient methods for minimizing composite objective function. Math. Program. Ser. B 140, 125–161 (2013)
He, B.S., Liao, L.-Z., Yuan, X.M.: A LQP-based interior prediction-correction method for nonlinear complementarity problems. J. Comput. Math. 24(1), 33–44 (2006)
Ng, M.K., Wang, F., Yuan, X.M.: Inexact alternating direction methods for image recovery. SIAM J. Sci. Comput. 33(4), 1643–1668 (2011)
Acknowledgments
The first author was supported by the National Natural Science Foundation of China Grant 11001053, the Program for New Century Excellent Talents in University Grant NCET-12-0111, and the Natural Science Foundation of Jiangsu Province grant BK2012662. The third author was partially supported by the FRG Grant from Hong Kong Baptist University: FRG2/13-14/061 and the General Research Fund from Hong Kong Research Grants Council: 203613.
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Communicated by Masao Fukushima.
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Li, M., Li, X. & Yuan, X. Convergence Analysis of the Generalized Alternating Direction Method of Multipliers with Logarithmic–Quadratic Proximal Regularization. J Optim Theory Appl 164, 218–233 (2015). https://doi.org/10.1007/s10957-014-0567-x
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DOI: https://doi.org/10.1007/s10957-014-0567-x
Keywords
- Generalized alternating direction method of multipliers
- Logarithmic–quadratic proximal method
- Convergence rate
- Variational inequality