Article
| Title: | On the structural monotone inclusion of continuous second-order dynamics combining dry friction and Hessian-driven damping (English) |
| Author: | Tang, Yan |
| Author: | Huang, Yanqin |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 71 |
| Issue: | 3 |
| Year: | 2026 |
| Pages: | 299-326 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | To obtain the null point of the sum for a maximally monotone operator $A$ and a cocoercive operator $B$, this article is concerned with the asymptotic properties of the trajectories to a second-order continuous damped autonomous dynamical systems with dry friction, which plays a crucial role for achieving stability within a finite time. Additionally, Hessian-driven damping is incorporated, which helps mitigating oscillations for inertial effects. The combination of dry friction and Hessian-driven damping in the context of split monotone inclusions represents a novel contribution, as it simultaneously ensures finite-time convergence and suppresses oscillatory behavior. The existence and uniqueness of global solutions, as well as the convergence of system trajectories are analyzed, the performance of the iterative algorithm generated by the second-order dynamic system discretely are also verified through several numerical experiments. (English) |
| Keyword: | monotone inclusion problem |
| Keyword: | dynamical system |
| Keyword: | dry friction |
| Keyword: | Lyapunov analysis |
| MSC: | 37N40 |
| MSC: | 65K05 |
| MSC: | 90C25 |
| DOI: | 10.21136/AM.2026.0178-25 |
| . | |
| Date available: | 2026-07-02T06:50:36Z |
| Last updated: | 2026-08-19 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153669 |
| . | |
| Reference: | [1] Adly, S., Attouch, H.: Finite-time stabilization of continuous inertial dynamics combining dry friction with Hessian-driven damping.J. Convex Anal. 28 (2021), 281-310. Zbl 1470.37118, MR 4245869 |
| Reference: | [2] Adly, S., Attouch, H.: First-order inertial algorithms involving dry friction damping.Math. Program. 193 (2022), 405-445. Zbl 1497.37120, MR 4412398, 10.1007/s10107-020-01613-y |
| Reference: | [3] Adly, S., Attouch, H., Le, M. H.: A doubly nonlinear evolution system with threshold effects associated with dry friction.J. Optim. Theory Appl. 203 (2024), 1188-1218. Zbl 1558.37091, MR 4822736, 10.1007/s10957-024-02417-2 |
| Reference: | [4] Alvarez, F., Attouch, H., Bolte, J., Redont, P.: A second-order gradient-like dissipative dynamical system with Hessian-driven damping: Application to optimization and mechanics.J. Math. Pures Appl. (9) 81 (2002), 747-779. Zbl 1036.34072, MR 1930878, 10.1016/S0021-7824(01)01253-3 |
| Reference: | [5] Attouch, H., Balhag, A., Chbani, Z., Riahi, H.: Damped inertial dynamics with vanishing Tikhonov regularization: Strong asymptotic convergence towards the minimum norm solution.J. Differ. Equations 311 (2022), 29-58. Zbl 1489.37113, MR 4354851, 10.1016/j.jde.2021.12.005 |
| Reference: | [6] Attouch, H., Boţ, R. I., Csetnek, E. R.: Fast optimization via inertial dynamics with closed-loop damping.J. Eur. Math. Soc. 25 (2023), 1985-2056. Zbl 1523.37103, MR 4592864, 10.4171/JEMS/1231 |
| Reference: | [7] Attouch, H., Cabot, A.: Convergence of damped inertial dynamics governed by regularized maximally monotone operators.J. Differ. Equations 264 (2018), 7138-7182. Zbl 1406.37068, MR 3779633, 10.1016/j.jde.2018.02.017 |
| Reference: | [8] Attouch, H., Fadili, J.: From the ravine method to the Nesterov method and vice versa: A dynamical system perspective.SIAM J. Optim. 32 (2022), 2074-2101. Zbl 1503.37098, MR 4468624, 10.1137/22M1474357 |
| Reference: | [9] Attouch, H., László, S. C.: Continuous Newton-like inertial dynamics for monotone inclusions.Set-Valued Var. Anal. 29 (2021), 555-581. Zbl 1477.37101, MR 4295322, 10.1007/s11228-020-00564-y |
| Reference: | [10] Attouch, H., Maingé, P.-É.: Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects.ESAIM, Control Optim. Calc. Var. 17 (2011), 836-857. Zbl 1230.34051, MR 2826982, 10.1051/cocv/2010024 |
| Reference: | [11] Attouch, H., Peypouquet, J.: The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than $1/k^2$.SIAM J. Optim. 26 (2016), 1824-1834. Zbl 1346.49048, MR 3543173, 10.1137/15M1046095 |
| Reference: | [12] Attouch, H., Peypouquet, J.: Convergence of inertial dynamics and proximal algorithms governed by maximally monotone operators.Math. Program. 174 (2019), 391-432. Zbl 1412.37083, MR 3935085, 10.1007/s10107-018-1252-x |
| Reference: | [13] Attouch, H., Peypouquet, J., Redont, P.: Fast convex optimization via inertial dynamics with Hessian driven damping.J. Differ. Equations 261 (2016), 5734-5783. Zbl 1375.49028, MR 3548269, 10.1016/j.jde.2016.08.020 |
| Reference: | [14] Attouch, H., Redont, P., Svaiter, B. F.: Global convergence of a closed-loop regularized Newton method for solving monotone inclusions in Hilbert spaces.J. Optim. Theory Appl. 157 (2013), 624-650. Zbl 1290.90061, MR 3047022, 10.1007/s10957-012-0222-3 |
| Reference: | [15] Bauschke, H. H., Combettes, P. L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces.CMS Books in Mathematics. Springer, Cham (2017). Zbl 1359.26003, MR 3616647, 10.1007/978-3-319-48311-5 |
| Reference: | [16] Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems.SIAM J. Imaging Sci. 2 (2009), 183-202. Zbl 1175.94009, MR 2486527, 10.1137/080716542 |
| Reference: | [17] Bellman, R.: The stability of solutions of linear differential equations.Duke Math. J. 10 (1943), 643-647. Zbl 0061.18502, MR 0009408, 10.1215/S0012-7094-43-01059-2 |
| Reference: | [18] Boţ, R. I., Csetnek, E. R.: Second order forward-backward dynamical systems for monotone inclusion problems.SIAM J. Control Optim. 54 (2016), 1423-1443. Zbl 1339.34070, MR 3507089, 10.1137/15M1012657 |
| Reference: | [19] Boţ, R. I., Hulett, D. A.: Second order splitting dynamics with vanishing damping for additively structured monotone inclusions.J. Dyn. Differ. Equations 36 (2024), 727-756. Zbl 1543.37088, MR 4710796, 10.1007/s10884-022-10160-3 |
| Reference: | [20] Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert.North-Holland Mathematics Studies 5. Elsevier, Amsterdam (1973), French. Zbl 0252.47055, MR 0348562 |
| Reference: | [21] Byrd, R. H., Lu, P., Nocedal, J., Zhu, C.: A limited memory algorithm for bound constrained optimization.SIAM J. Sci. Comput. 16 (1995), 1190-1208. Zbl 0836.65080, MR 1346301, 10.1137/0916069 |
| Reference: | [22] Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging.J. Math. Imaging Vis. 40 (2011), 120-145. Zbl 1255.68217, MR 2782122, 10.1007/s10851-010-0251-1 |
| Reference: | [23] Combettes, P. L., Wajs, V. R.: Signal recovery by proximal forward-backward splitting.Multiscale Model. Simul. 4 (2005), 1168-1200. Zbl 1179.94031, MR 2203849, 10.1137/050626090 |
| Reference: | [24] Daubechies, I., Defrise, M., Mol, C. De: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint.Commun. Pure Appl. Math. 57 (2004), 1413-1457. Zbl 1077.65055, MR 2077704, 10.1002/cpa.20042 |
| Reference: | [25] Eckstein, J., Bertsekas, D. P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators.Math. Program., Ser. A 55 (1992), 293-318. Zbl 0765.90073, MR 1168183, 10.1007/BF01581204 |
| Reference: | [26] Nesterov, Y. E.: A method for solving the convex programming problem with convergence rate $O(1/k^2)$.Sov. Math., Dokl. 27 (1983), 372-376. Zbl 0535.90071, MR 0701288 |
| Reference: | [27] O'Donoghue, B., Candès, E.: Adaptive restart for accelerated gradient schemes.Found. Comput. Math. 15 (2015), 715-732. Zbl 1320.90061, MR 3348171, 10.1007/s10208-013-9150-3 |
| Reference: | [28] Polyak, B. T.: Some methods of speeding up the convergence of iteration methods.U.S.S.R. Comput. Math. Math. Phys. 4 (1964), 1-17. Zbl 0147.35301, MR 0169403, 10.1016/0041-5553(64)90137-5 |
| Reference: | [29] Rockafellar, R. T.: Monotone operators and the proximal point algorithm.SIAM J. Control Optim. 14 (1976), 877-898. Zbl 0358.90053, MR 0410483, 10.1137/0314056 |
| Reference: | [30] Shehu, Y., Dong, Q.-L., Liu, L.-L., Yao, J.-C.: New strong convergence method for the sum of two maximal monotone operators.Optim. Eng. 22 (2021), 2627-2653. Zbl 07460650, MR 4333482, 10.1007/s11081-020-09544-5 |
| Reference: | [31] Su, W., Boyd, S., Candès, E. J.: A differential equation for modeling Nesterov's accelerated gradient method: Theory and insights.J. Mach. Learn. Res. 17 (2016), Article ID 153, 43 pages. Zbl 1391.90667, MR 3555044 |
| Reference: | [32] Wright, S. J.: Coordinate descent algorithms.Math. Program. 151 (2015), 3-34. Zbl 1317.49038, MR 3347548, 10.1007/s10107-015-0892-3 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
