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.

Revised:

Accepted:

Published online:

^{1}; Philippe Mahey

^{2}; Eladio Ocaña

^{1}

@article{OJMO_2023__4__A3_0, author = {Ernesto Or\'e and Philippe Mahey and Eladio Oca\~na}, title = {Inertial-relaxed splitting for composite monotone inclusions}, journal = {Open Journal of Mathematical Optimization}, eid = {3}, pages = {1--20}, publisher = {Universit\'e de Montpellier}, volume = {4}, year = {2023}, doi = {10.5802/ojmo.22}, language = {en}, url = {https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.22/} }

TY - JOUR AU - Ernesto Oré AU - Philippe Mahey AU - Eladio Ocaña TI - Inertial-relaxed splitting for composite monotone inclusions JO - Open Journal of Mathematical Optimization PY - 2023 SP - 1 EP - 20 VL - 4 PB - Université de Montpellier UR - https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.22/ DO - 10.5802/ojmo.22 LA - en ID - OJMO_2023__4__A3_0 ER -

%0 Journal Article %A Ernesto Oré %A Philippe Mahey %A Eladio Ocaña %T Inertial-relaxed splitting for composite monotone inclusions %J Open Journal of Mathematical Optimization %D 2023 %P 1-20 %V 4 %I Université de Montpellier %U https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.22/ %R 10.5802/ojmo.22 %G en %F OJMO_2023__4__A3_0

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. https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.22/

[1] 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] 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] 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] Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, 2011 | DOI

[5] 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] Primal-Dual Partial Inverse Splitting for Constrained Monotone Inclusions (2020) (https://arxiv.org/abs/2007.01983v1)

[7] Primal-dual splittings as fixed-point iterations in the range of linear operators (2019) (https://arxiv.org/abs/1910.02329v1)

[8] 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] 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] Proximal splitting algorithms : a tour of recent advances with new twists (2020) (https://arxiv.org/abs/1912.00137)

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

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

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

[17] 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] 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] A unified splitting algorithm for composite monotone inclusions, J. Convex Anal., Volume 27 (2020) no. 3, pp. 893-922 | Zbl

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

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

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

[24] 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

*Cited by Sources: *