Continuous Newton-like Methods Featuring Inertia and Variable Mass
提出一种结合惯性与牛顿法的二阶动力系统，通过可变质量参数控制牛顿与惯性行为，在强凸优化中实现非渐近控制与加速收敛，为设计融合一阶和二阶方法优势的算法提供基础。
We introduce a new dynamical system, at the interface between second-order dynamics with inertia and Newton's method. This system extends the class of inertial Newton-like dynamics by featuring a time-dependent parameter in front of the acceleration, called variable mass. For strongly convex optimization, we provide guarantees on how the Newtonian and inertial behaviors of the system can be non-asymptotically controlled by means of this variable mass. A connection with the Levenberg--Marquardt (or regularized Newton's) method is also made. We then show the effect of the variable mass on the asymptotic rate of convergence of the dynamics, and in particular, how it can turn the latter into an accelerated Newton method. We provide numerical experiments supporting our findings. This work represents a significant step towards designing new algorithms that benefit from the best of both first- and second-order optimization methods.