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Iteration complexity on the generalized Peaceman-Rachford splitting method for separable convex programming. (English) Zbl 1431.90114
Summary: Recently, a generalized version of Peaceman-Rachford splitting method (GPRSM) for solving a convex minimization model with a general separable structure has been proposed and its global convergence has been proved by M. Sun and J. Liu [J. Appl. Math. Comput. 50, No. 1–2, 349–363 (2016; Zbl 1330.90079)]. In this paper, we establish the worst-case \(\mathcal{O}(1/t)\) convergence rate for the GPRSM in both the ergodic and a nonergodic senses, then we give some numerical results to demonstrate the convergence rate of the GPRSM.
90C25 Convex programming
90C60 Abstract computational complexity for mathematical programming problems
Full Text: DOI
[1] Glowinski, R.; Marrocco, A., Approximation par elements finis dordre un, et la resolution, par penalisation-dualite, dune classe de problmes de Dirichlet non lineaires, RAIRO Anal Numer, 9, 41-76, (1975) · Zbl 0368.65053
[2] 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) · Zbl 0352.65034
[3] 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) · Zbl 0765.90073
[4] Han, D.; Yuan, X., Local linear convergence of the alternating direction method of multipliers for quadratic programs, SIAM J Numer Anal, 51, 6, 3446-3457, (2013) · Zbl 1285.90033
[5] Peaceman, Dh; Rachford, Hh., The numerical solution of parabolic elliptic differential equations, SIAM J Appl Math, 3, 28-41, (1955) · Zbl 0067.35801
[6] Lions, Pl; Mercier, B., Splitting algorithms for the sum of two nonlinear operators, SIAM J Numer Anal, 16, 964-979, (1979) · Zbl 0426.65050
[7] He, B.; Liu, H.; Wang, Z., A strictly contractive Peaceman-Rachford splitting method for convex programming, SIAM J Optim, 24, 3, 1011-1040, (2014) · Zbl 1327.90210
[8] Bertsekas, Dp., Constrained optimization and Lagrange multiplier methods, (1982), New York (NY): Academic Press, New York (NY)
[9] Sun, M.; Liu, J., A proximal Peaceman-Rachford splitting method for compressive sensing, J Appl Math Comput, 50, 349-363, (2016) · Zbl 1330.90079
[10] Applications of the method of multipliers to variational inequalities. In: Fortin M, Glowinski R, editors. Augmented Lagrange methods: applications to the solution of boundary-valued problems. Amsterdam: North-Holland; 1983. p. 299-331
[11] Glowinski, R.; Le Tallec, P., Augmented Lagrangian and operator splitting methods in nonlinear mechanics, (1989), Philadelphia (PA): SIAM, Philadelphia (PA) · Zbl 0698.73001
[12] Sun, M.; Liu, J., Generalized Peaceman-Rachford splitting method for separable convex programming with applications to image processing, J Appl Math Comput, 51, 605-622, (2016) · Zbl 1338.90307
[13] Sun, M.; Wang, Y.; Liu, J., Generalized Peaceman-Rachford splitting method for multiple-block separable convex programming with applications to robust PCA, Calcolo, 54, 77-94, (2017) · Zbl 1368.90129
[14] Li, M.; Yuan, X., A strictly contractive Peaceman-Rachford splitting method with logarithmic-quadratic proximal regularization for convex programming, Math Oper Res, 40, 4, 842-858, (2015) · Zbl 1329.90106
[15] He, B.; Yang, H., Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities, Oper Res Lett, 23, 151-161, (1998) · Zbl 0963.49006
[16] He, B.; Yuan, X., On the O(1/n) convergence rate of Douglas-Rachford alternating direction method, SIAM J Numer Anal, 50, 700-709, (2012) · Zbl 1245.90084
[17] Rockafellar, Rt., Convex analysis, (1970), Princeton (NJ): Princeton University Press, Princeton (NJ)
[18] Boyd, S.; Parikh, N.; Chu, E., Distributed optimization and statistical learning via the alternating direction method of multipliers, Found Trends Mach Learn, 3, 1-122, (2011) · Zbl 1229.90122
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