Fast convex optimization via inertial dynamics with Hessian driven damping

We first study the fast minimization properties of the trajectories of the second-order evolution equation $\ddot{x}(t) + \fracα{t} \dot{x}(t) + β\nabla^2 Φ(x(t))\dot{x} (t) + \nabla Φ(x(t)) = 0,$ where $Φ:\mathcal H\to\mathbb R$ is a smooth convex function acting on a real Hilbert space $\mathcal H$, and $α$, $β$ are positive parameters. This inertial system combines an isotropic viscous damping which vanishes asymptotically, and a geometrical Hessian driven damping, which makes it naturally related to Newton's and Levenberg-Marquardt methods. For $α\geq 3$, $β>0$, along any trajectory, fast convergence of the values $Φ(x(t))- \min_{\mathcal H}Φ=\mathcal O\left(t^{-2}\right)$ is obtained, together with rapid convergence of the gradients $\nablaΦ(x(t))$ to zero. For $α>3$, just assuming that $Φ$ has minimizers, we show that any trajectory converges weakly to a minimizer of $Φ$, and $Φ(x(t))-\min_{\mathcal H}Φ= o(t^{-2})$. Strong convergence is established in various practical situations. For the strongly convex case, convergence can be arbitrarily fast depending on the choice of $α$. More precisely, we have $Φ(x(t))- \min_{\mathcal H}Φ= \mathcal O(t^{-\frac{2}{3}α})$. We extend the results to the case of a general proper lower-semicontinuous convex function $Φ: \mathcal H \rightarrow \mathbb R \cup \{+\infty \}$. This is based on the fact that the inertial dynamic with Hessian driven damping can be written as a first-order system in time and space. By explicit-implicit time discretization, this opens a gate to new $-$ possibly more rapid $-$ inertial algorithms, expanding the field of FISTA methods for convex structured optimization problems.

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