Analysis of continuous dynamical system models with Hessians derived from optimization methods (Q6601496)

The authors consider continuous-time dynamical systems involving Hessians, derived from unconstrained continuous optimization methods,\N\(x_{\ast}\in\arg\min_{x\in \mathbb{R}^{n}}f(x)\), of the form: \N\[\Na_{1}\ddot{x}+a_{2}\dot{x}+a_{3}\nabla^{2}f\dot{x}+a_{4}\nabla f=0 \tag{1}\N\]\Nwhere \(a_{i}:\mathbb{R}^{+}\rightarrow \mathbb{R}\) \((i=1,2,3,4)\) are some functions of \(t\), and \(a_{3}\) and \(a_{4}\) are not zeros. They classify \((1)\) into three cases, and for each case, they exhibit a Lyapunov function and obtain the best rate of convergence when using this Lyapunov function. However, in some cases, this rate is the same as either the simplest gradient flow, or the continuous dynamical system corresponding to Nesterov's accelerated gradient method.\NTherefore in these cases, the incorporation of the Hessian does not bring any improvement. \N\NTheir results extend previous results in [\textit{H. Attouch} et al., J. Differ. Equations 261, No. 10, 5734--5783 (2016; Zbl 1375.49028); \textit{B. Shi} et al., Math. Program. 195, No. 1--2 (A), 79--148 (2022; Zbl 1500.65026); \textit{J. J. Suh} et al., ``Continuous-time analysis of accelerated gradient methods via conservation laws in dilated coordinate systems'', Preprint, \url{arXiv:2202.05501}].