Solving convex optimization problems via a second order dynamical system with implicit Hessian damping and Tikhonov regularization
研究了一个带有Tikhonov正则化的二阶动力系统，用于最小化凸函数，通过显式离散化得到惯性梯度算法，并分析了目标函数值的快速收敛性和轨迹的弱/强收敛性。
Abstract This paper deals with a second order dynamical system with a Tikhonov regularization term in connection to the minimization problem of a convex Fréchet differentiable function. The fact that beside the asymptotically vanishing damping we also consider an implicit Hessian driven damping in the dynamical system under study allows us, via straightforward explicit discretization, to obtain inertial algorithms of gradient type. We show that the value of the objective function in a generated trajectory converges rapidly to the global minimum of the objective function and depending the Tikhonov regularization parameter the generated trajectory converges weakly to a minimizer of the objective function or the generated trajectory converges strongly to the element of minimal norm from the $$\mathop {\text {argmin}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>argmin</mml:mtext> </mml:math> set of the objective function. We also obtain the fast convergence of the velocities towards zero and some integral estimates. Our analysis reveals that the Tikhonov regularization parameter and the damping parameters are strongly correlated, there is a setting of the parameters that separates the cases when weak convergence of the trajectories to a minimizer and strong convergence of the trajectories to the minimal norm minimizer can be obtained.