arXiv 1602.019732016-02-05math.OC

结合快速惯性动力学求解凸优化和Tikhonov正则化

Combining fast inertial dynamics for convex optimization with Tikhonov regularization

Hedy Attouch, Zaki Chbani

摘要：

在Hilbert空间中，研究了二阶微分方程\begin{equation*} \mbox{(AVD)}_{α, ε} \quad \quad \ddot{x}(t) + \fracα{t} \dot{x}(t) + \nabla Φ(x(t)) + ε(t) x(t) =0, \end{equation*}的轨线作为$t \to + \infty$的收敛性质，其中$\nablaΦ$是凸连续可微函数$Φ: \mathcal H \to \mathbb R$的梯度，$α$是正参数，$ε(t) x(t)$是Tikhonov正则项，带$\lim_{t \to \infty}ε(t) =0$ .在该阻尼惯性系统中，阻尼系数$\fracα{t}$渐近地消失，但不会太快，这是获得值的快速收敛的关键特性。在$ε(\cdot) \equiv 0$的例子中，苏、博伊德和坎迪斯最近强调了这种动态，认为它是涅斯捷罗夫加速方法的一个连续版本。根据$ε(t)$收敛到零的速度，我们分析了$\mbox{(AVD)}_{α, ε}$轨迹的收敛性质。我们得到的结果范围从$Φ(x(t))$的快速收敛$\min Φ$时$ε(t)$迅速减少到零，强遍历收敛的轨迹的元素的最小范数的最小的$Φ$的集合，当$ε(t)$慢慢趋于零。

英文摘要：

In a Hilbert space setting $\mathcal H$, we study the convergence properties as $t \to + \infty$ of the trajectories of the second-order differential equation \begin{equation*} \mbox{(AVD)}_{α, ε} \quad \quad \ddot{x}(t) + \fracα{t} \dot{x}(t) + \nabla Φ(x(t)) + ε(t) x(t) =0, \end{equation*} where $\nablaΦ$ is the gradient of a convex continuously differentiable function $Φ: \mathcal H \to \mathbb R$, $α$ is a positive parameter, and $ε(t) x(t)$ is a Tikhonov regularization term, with $\lim_{t \to \infty}ε(t) =0$. In this damped inertial system, the damping coefficient $\fracα{t}$ vanishes asymptotically, but not too quickly, a key property to obtain rapid convergence of the values. In the case $ε(\cdot) \equiv 0$, this dynamic has been highlighted recently by Su, Boyd, and Candès as a continuous version of the Nesterov accelerated method. Depending on the speed of convergence of $ε(t)$ to zero, we analyze the convergence properties of the trajectories of $\mbox{(AVD)}_{α, ε}$. We obtain results ranging from the rapid convergence of $Φ(x(t))$ to $\min Φ$ when $ε(t)$ decreases rapidly to zero, up to the strong ergodic convergence of the trajectories to the element of minimal norm of the set of minimizers of $Φ$, when $ε(t)$ tends slowly to zero.

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摘要：

英文摘要：

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