Inertial-relaxed splitting for composite monotone inclusions
Open Journal of Mathematical Optimization, Volume 4 (2023), article no. 3, 20 p.

In a similar spirit of the extension of the proximal point method developed by Alves et al. [2], we propose in this work an Inertial-Relaxed primal-dual splitting method to address the problem of decomposing the minimization of the sum of three convex functions, one of them being smooth, and considering a general coupling subspace. A unified setting is formalized and applied to different average maps whose corresponding fixed points are related to the solutions of the inclusion problem associated with our extended model. An interesting feature of the resulting algorithms we have designed is that they present two distinct versions with a Gauss–Seidel or a Jacobi flavor, extending in that sense former proximal ADMM methods, both including inertial and relaxation parameters. Finally we show computational experiments on a class of the fused LASSO instances of medium size.

Published online:
DOI: 10.5802/ojmo.22
Keywords: Operator splitting methods, Convex composite optimization
Ernesto Oré 1; Philippe Mahey 2; Eladio Ocaña 1

1 IMCA, Instituto de Matemática y Ciencias Afines, Universidad Nacional de Ingeniería, Lima, Perú
2 LIMOS, CNRS, Université Clermont Auvergne, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Ernesto Oré; Philippe Mahey; Eladio Ocaña. Inertial-relaxed splitting for composite monotone inclusions. Open Journal of Mathematical Optimization, Volume 4 (2023), article  no. 3, 20 p. doi : 10.5802/ojmo.22.

[1] Felipe Alvarez; Hedy Attouch An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., Volume 9 (2001), pp. 3-11 | DOI | Zbl

[2] M. Marques Alves; Jonathan Eckstein; Marina Geremia; Jefferson G. Melo Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms, Comput. Optim. Appl., Volume 75 (2020), pp. 389-422 | DOI | Zbl

[3] M. Marques Alves; Marina Geremia Iteration complexity of an inexact Douglas-Rachford and a Douglas-Rachford-Tseng Forward-Backward four-operator splitting method for solving monotone inclusions, Numer. Algorithms, Volume 82 (2019) no. 1, pp. 263-295 | DOI | Zbl

[4] Heinz H. Bauschke; Patrick L. Combettes Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, 2011 | DOI

[5] Stephen Boyd; Neal Parikh; Eric Chu; Borja Peleato; Jonathan Eckstein Distributed optimization and statistical learning with the alternating direction method of multipliers, Found. Trends Mach. Learn., Volume 3 (2011) no. 1, pp. 1-122 | DOI | Zbl

[6] Luis Briceño-Arias; Julio Deride; Sergio López-Rivera; Francisco J. Silva Primal-Dual Partial Inverse Splitting for Constrained Monotone Inclusions (2020) (

[7] Luis Briceño-Arias; Fernando Roldán Primal-dual splittings as fixed-point iterations in the range of linear operators (2019) (

[8] Antonin Chambolle; Thomas Pock A first-order primal-dual algorithm for convex programs with applications to imaging, J. Math. Imaging Vis., Volume 40 (2011), pp. 1-26 | Zbl

[9] Laurent Condat A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl., Volume 158 (2013), pp. 460-479 | DOI | Zbl

[10] Laurent Condat; Daichi Kitahara; Andrés Contreras; Akira Hirabayashi Proximal splitting algorithms : a tour of recent advances with new twists (2020) (

[11] Damek Davis; Wotao Yin Convergence rate analysis of several splitting schemes, Splitting Methods in Communication, Imaging, Science and Engineering (R. Glowinski; S. J. Osher; W. Yin, eds.) (Scientific Computation), Springer, 2016, pp. 115-163 | DOI | Zbl

[12] Damek Davis; Wotao Yin A three-operator splitting scheme and its optimization applications, Set-Valued Var. Anal., Volume 25 (2017) no. 4, pp. 829-858 | DOI | Zbl

[13] Jonathan Eckstein; Dimitri P. Bertsekas On the Douglas-Rachford Splitting Method and the Proximal Point Algorithm for Maximal Monotone Operators, Math. Program., Volume 55 (1992) no. 3, pp. 293-318 | DOI | Zbl

[14] Daniel Gabay; Bertrand Mercier A dual algorithm for the solution of nonlinear variational problems via finite-element approximations, Comput. Math. Appl., Volume 2 (1976), pp. 17-40 | DOI | Zbl

[15] Osman Güler New proximal point algorithms for convex optimization, SIAM J. Optim., Volume 2 (1992), pp. 649-664 | DOI | Zbl

[16] Bingsheng He; Feng Ma; Xiaoming Yuan Optimally linearizing the alternating direction method of multipliers for convex programming, Comput. Optim. Appl., Volume 75 (2020) no. 2, pp. 361-388 | Zbl

[17] Min Li; Defeng Sun; Kim-Chuan Toh A majorized ADMM with indefinite proximal terms for linearly constrained convex composite optimization, SIAM J. Optim., Volume 26 (2016) no. 2, pp. 922-950 | Zbl

[18] Yurii Nesterov A method for unconstrained convex minimization problem with the rate of convergence O(1/k 2 ), Dokl. Akad. Nauk SSSR, Volume 269 (1983), pp. 543-547

[19] Ernesto Oré-Albornoz; Philippe Mahey; Eladio Ocana-Anaya A unified splitting algorithm for composite monotone inclusions, J. Convex Anal., Volume 27 (2020) no. 3, pp. 893-922 | Zbl

[20] Fabian Pedregosa; Gauthier Gidel Adaptive three-operator splitting, Proceedings of 35th Int. Conf. on Machine Learning (2018), pp. 4085-4094

[21] R. Tyrell Rockafellar Augmented lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., Volume 1 (1976), pp. 97-116 | DOI | Zbl

[22] Ron Shefi; Marc Teboulle Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization, SIAM J. Optim., Volume 24 (2014) no. 1, pp. 269-297 | DOI | Zbl

[23] Băng Công Vũ A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comput. Math., Volume 38 (2013) no. 3, pp. 667-681 | Zbl

[24] Ming Yan A new primal-dual algorithm for minimizing the sum of three functions with a linear operator, J. Sci. Comput., Volume 76 (2018) no. 3, pp. 1698-1717 | Zbl

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