Abstract
Due to its significant efficiency, the alternating direction method (ADM) has attracted a lot of attention in solving linearly constrained structured convex optimization. In this paper, in order to make implementation of ADM relatively easy, some linearized proximal ADMs are proposed and the associated convergence results of the proposed linearized proximal ADMs are given. Additionally, theoretical analysis shows that the relaxation factor for the linearized proximal ADMs can have the same restriction region as that for the general ADM.
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Peaceman, D.W., Rachford, H.H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3, 28–41 (1955)
Douglas, J., Rachford, H.H.: On the numerical solution of the heat conduction problem in 2 and 3 space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)
Douglas, J., Gunn, J.E.: A general formulation of alternating direction methods, Part I. Parabolic and hyperbolic problems. Numer. Math. 6, 428–453 (1964)
Glowinski, R., Marroco, A.: Sur L’Approximation, par Elements Finis d’Ordre Un, et la Resolution, par Penalisation-Dualité, d’une Classe de Problemes de Dirichlet non Lineares. Revne Francaise d’Automatique, Informatique et Recherche Opérationelle 9(R-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, 17–40 (1976)
Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrange Methods: Applications to the Solution of Boundary-valued Problems, pp. 299–331. North Holland, Amsterdam (1983)
Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)
Fukushima, M.: Application of the alternating direction method of multipliers to separable convex programming problems. Comput. Optim. Appl. 2, 93–111 (1992)
Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)
Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Prentice-Hall, Englewood Cliffs (1989)
Chen, G., Teboulle, M.: A proximal-based decomposition method for convex minimization problems. Math. Program. 64, 81–101 (1994)
Eckstein, J., Fukushima, M.: Some reformulation and applications of the alternating directions method of multipliers. In: Hager, W.W. et al. (eds.) Large Scale Optimization: State of the Art, pp. 115–134. Kluwer Academic, Norwell (1994)
Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM Studies in Applied Mathematics. Philadelphia (1989)
He, B.S., Liao, L.Z., Han, D.R., Yang, H.: A new inexact alternating directions method for monotone variational inequalities. Math. Program. 92, 103–118 (2002)
He, B.S., Yang, H.: Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities. Oper. Res. Lett. 23, 151–161 (1998)
Kontogiorgis, S., Meyer, R.R.: A variable-penalty alternating directions method for convex optimization. Math. Program. 83, 29–53 (1998)
He, B.S., Xu, M.H., Yuan, X.M.: Solving large-scale least squares semidefinite programming by alternating direction methods. SIAM J. Matrix Anal. Appl. 32(1), 136–152 (2011)
Sun, J., Zhang, S.: A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs. Eur. J. Oper. Res. 207(3), 1210–1220 (2010)
Wen, Z.W., Yin, W.T., Goldfarb, D.: Alternating direction augmented lagrangian methods for semidefinite programming. Math. Program. Comput. 2(3–4), 203–230 (2010)
Yang, J.F., Zhang, Y., Yin, W.T.: A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data. IEEE J. Sel. Top. Signal Process. 4(2), 288–297 (2010)
He, B.S., Wang, S.L., Yang, H.: A modified variable-penalty alternating directions method for monotone variational inequalities. J. Comput. Math. 21, 495–504 (2003)
Xu, M.H.: Proximal alternating directions method for structured variational inequalities. J. Optim. Theory Appl. 134, 107–117 (2007)
Pardalos, P.M., Resende, M.G.C.: Handbook of Applied Optimization. Oxford University Press, London (2002)
Pardalos, P.M., Rassias, T.M., Khan, A.A.: Nonlinear Analysis and Variational Problems: In Honor of George Isac, 1st edn. Springer Optimization and Its Applications, vol. 35. Springer, Berlin (2010)
Floudas, C.A., Pardalos, P.M.: Encyclopedia of Optimization, 2nd edn. Springer, Berlin (2008)
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Operations Research. Springer, Berlin (2003)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1972)
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Xu, M.H., Wu, T. A Class of Linearized Proximal Alternating Direction Methods. J Optim Theory Appl 151, 321–337 (2011). https://doi.org/10.1007/s10957-011-9876-5
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DOI: https://doi.org/10.1007/s10957-011-9876-5