Continuous Newton-like Inertial Dynamics for Monotone Inclusions

Abbas, B., Attouch, H., Svaiter, B.F.: Newton-like dynamics and forward-backward methods for structured monotone inclusions in Hilbert spaces. J. Optim. Theory Appl. 161(2), 331–360 (2014)

Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Analysis 9(1-2), 3–11 (2001)

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. 81(8), 747–779 (2002)

Apidopoulos, V., Aujol, J.-F., Dossal, C h: Convergence rate of inertial Forward-Backward algorithm beyond Nesterov’s rule. Math. Program. 180, 137–156 (2020)

Attouch, H., Cabot, A.: Convergence of a relaxed inertial proximal algorithm for maximally monotone operators. Math Program. https://doi.org/10.1007/s10107-019-01412-0 (2019)

Attouch, H., Cabot, A.: Convergence rates of inertial forward-backward algorithms. SIAM J. Optim. 28(1), 849–874 (2018)

Attouch, H., Chbani, Z., Fadili, J., Riahi, H.: First-order algorithms via inertial systems with Hessian driven damping. HAL-02193846 (2019)

Attouch, H., Chbani, Z., Peypouquet, J., Redont, P.: Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity. Math. Program. B 168, 123–175 (2018)

Attouch, H., Chbani, Z., Riahi, H.: Fast proximal methods via time scaling of damped inertial dynamics. SIAM J. Optim. 29(3), 2227–2256 (2019)

Attouch, H., Chbani, Z., Riahi, H.: Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3. ESAIM-COCV 25, Article number 2 (2019)

Attouch, H., Goudou, X., Redont, P.: The heavy ball with friction method. The continuous dynamical system, global exploration of the local minima of a real-valued function by asymptotical analysis of a dissipative dynamical system. Commun. Contemp. Math. 2(1), 1–34 (2000)

Attouch, H., László, S.C.: Newton-like inertial dynamics and proximal algorithms governed by maximally monotone operators. HAL-02549730 (2020)

Attouch, H., Maingé, P.E.: Asymptotic behavior of second order dissipative evolution equations combining potential with non-potential effects. ESAIM Control Optim. Calc. Var. 17(3), 836–857 (2011)

Attouch, H., Maingé, P.E., Redont, P.: A second-order differential system with Hessian-driven damping; Application to non-elastic shock laws. Differ. Equ. Appl. 4(1), 27–65 (2012)

Attouch, H., Marques Alves, M., Svaiter, B.F.: A dynamic approach to a proximal-Newton method for monotone inclusions in Hilbert Spaces, with complexity $\mathcal {O}(1/n^{2})$. J. Convex Anal. 23(1), 139–180 (2016)

Attouch, H., Peypouquet, J.: Convergence of inertial dynamics and proximal algorithms governed by maximal monotone operators. Math. Program. 174(1-2), 391–432 (2019)

Attouch, H., Peypouquet, J.: The rate of convergence of Nesterov’s accelerated forward-backward method is actually faster than 1/k2. SIAM J. Optim. 26(3), 1824–1834 (2016)

Attouch, H., Peypouquet, J., Redont, P.: Fast convex minimization via inertial dynamics with Hessian driven damping. J. Differ. Equ. 261(10), 5734–5783 (2016)

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(3), 624–650 (2013)

Attouch, H., Svaiter, B.F.: A continuous dynamical Newton-Like approach to solving monotone inclusions. SIAM J. Control Optim. 49(2), 574–598 (2011)

Bauschke, H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert spaces. CMS Books in Mathematics. Springer (2011)

Brézis, H.: Opérateurs maximaux monotones dans les espaces de Hilbert et équations d’évolution. Lecture Notes 5 North Holland (1972)

Boţ, R.I., Csetnek, E.R.: Second order forward-backward dynamical systems for monotone inclusion problems. SIAM J. Control Optim. 54, 1423–1443 (2016)

Boţ, R. I., Csetnek, E.R., László, S.C.: A second-order dynamical approach with variable damping to nonconvex smooth minimization. Appl. Anal. 99(3), 361–378 (2020)

Boţ, R.I., Csetnek, E.R., László, S.C.: Tikhonov regularization of a second order dynamical system with Hessian damping. Mathematical Programming. https://doi.org/10.1007/s10107-020-01528-8 (2020)

Castera, C., Bolte, J., Févotte, C., Pauwels, E.: An Inertial Newton Algorithm for Deep Learning. HAL-02140748 (2019)

Chambolle, A., Dossal, C h: On the convergence of the iterates of the Fast Iterative Shrinkage Thresholding Algorithm. J. Optim. Theory Appl. 166, 968–982 (2015)

Haraux, A.: Systèmes dynamiques dissipatifs et applications. RMA 17 Masson (1991)

Kim, D.: Accelerated Proximal Point Method for Maximally Monotone Operators. arXiv:1905.05149v3 (2020)

László, S.C.: Convergence rates for an inertial algorithm of gradient type associated to a smooth nonconvex minimization. Mathematical Programming. https://doi.org/10.1007/s10107-020-01534-w (2020)

Lin, T., Jordan, M.I.: A Control-Theoretic Perspective on Optimal High-Order Optimization. arXiv:1912.07168v1 (2019)

May, R.: Asymptotic for a second-order evolution equation with convex potential and vanishing damping term. Turkish J. Math. 41(3), 681–685 (2017)

Nesterov, Y.: A method for solving the convex programming problem with convergence rate o(1/k2). (Russian) Dokl. Akad. Nauk SSSR 269(3), 543–547 (1983)

Polyak, B.T.: Some methods of speeding up the convergence of iterative methods. Z. Vylist. Math. Fiz. 4, 1–17 (1964)

Shi, B., Du, S.S., Jordan, M.I., Su, W.J.: Understanding the acceleration phenomenon via high-resolution differential equations. arXiv:submit/2440124[cs.LG] (2018)

Su, W.J., Boyd, S., Candès, E.J.: A differential equation for modeling Nesterov’s accelerated gradient method: theory and insights. Neural Information Processing Systems 27, 2510–2518 (2014)