Mathematical Programming

, Volume 168, Issue 1–2, pp 123–175 | Cite as

Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity

  • Hedy Attouch
  • Zaki Chbani
  • Juan PeypouquetEmail author
  • Patrick Redont
Full Length Paper Series B


In a Hilbert space setting \({{\mathcal {H}}}\), we study the fast convergence properties as \(t \rightarrow + \infty \) of the trajectories of the second-order differential equation
$$\begin{aligned} \ddot{x}(t) + \frac{\alpha }{t} \dot{x}(t) + \nabla \Phi (x(t)) = g(t), \end{aligned}$$
where \(\nabla \Phi \) is the gradient of a convex continuously differentiable function \(\Phi : {{\mathcal {H}}} \rightarrow {{\mathbb {R}}}, \alpha \) is a positive parameter, and \(g: [t_0, + \infty [ \rightarrow {{\mathcal {H}}}\) is a small perturbation term. In this inertial system, the viscous damping coefficient \(\frac{\alpha }{t}\) vanishes asymptotically, but not too rapidly. For \(\alpha \ge 3\), and \(\int _{t_0}^{+\infty } t \Vert g(t)\Vert dt < + \infty \), just assuming that \({{\mathrm{argmin\,}}}\Phi \ne \emptyset \), we show that any trajectory of the above system satisfies the fast convergence property
$$\begin{aligned} \Phi (x(t))- \min _{{{\mathcal {H}}}}\Phi \le \frac{C}{t^2}. \end{aligned}$$
Moreover, for \(\alpha > 3\), any trajectory converges weakly to a minimizer of \(\Phi \). The strong convergence is established in various practical situations. These results complement the \({{\mathcal {O}}}(t^{-2})\) rate of convergence for the values obtained by Su, Boyd and Candès in the unperturbed case \(g=0\). Time discretization of this system, and some of its variants, provides new fast converging algorithms, expanding the field of rapid methods for structured convex minimization introduced by Nesterov, and further developed by Beck and Teboulle with FISTA. This study also complements recent advances due to Chambolle and Dossal.


Convex optimization Fast convergent methods Dynamical systems Gradient flows Inertial dynamics Vanishing viscosity Nesterov method 

Mathematics Subject Classification

34D05 49M25 65K05 65K10 90C25 90C30 


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  • Hedy Attouch
    • 1
  • Zaki Chbani
    • 2
  • Juan Peypouquet
    • 3
    Email author
  • Patrick Redont
    • 1
  1. 1.Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRSUniversité Montpellier 2Montpellier Cedex 5France
  2. 2.Laboratoire IBN AL-BANNA de Mathématiques et Applications (LIBMA), Faculty of Sciences Semlalia, MathematicsCadi Ayyad UniversityMarrakechMorocco
  3. 3.Universidad Técnica Federico Santa MaríaValparaisoChile

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