SIAM Journal on Optimization, Volume 30, Issue 3, Page 2134-2162, January 2020.
In a Hilbert space ${\mathcal H}$, we introduce a new class of proximal-gradient algorithms with finite convergence properties. These algorithms naturally occur as discrete temporal versions of an inertial differential inclusion which is stabilized under the joint action of three dampings: dry friction, viscous friction, and a geometric damping driven by the Hessian. The function $f:{\mathcal H} \to {\mathbb R}$ to be minimized is supposed to be differentiable (not necessarily convex) and enters the algorithm via its gradient. The dry friction damping function $\phi:{\mathcal H} \to {\mathbb R}_+$ is convex with a sharp minimum at the origin (typically $\phi(x) = r \|x\|$ with $r>0$). It enters the algorithm via its proximal mapping, which acts as a soft threshold operator on the velocities. It is the source of stabilization in a finite number of steps. The geometric damping driven by the Hessian intervenes in the dynamics in the form $\nabla^2 f (x(t)) \dot{x} (t)$. By treating this term as the time derivative of $ \nabla f (x (t)) $, this gives, in discretized form, first-order algorithms. The Hessian-driven damping allows one to control and to attenuate the oscillations. The convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence to zero. Replacing the potential function $f$ by its Moreau envelope, we extend the results to the case of a nonsmooth convex function $f$. In this case, the algorithm involves the proximal operators of $f$ and $\phi$ separately. Several variants of this algorithm are considered, including the case of the Nesterov accelerated gradient method. We then consider the extension to the case of additive composite optimization, thus leading to splitting methods. Numerical experiments are given for lasso-type problems. The performance profiles, as a comparison tool, highlight the effectiveness of a variant of the Nesterov accelerated method with dry friction and Hessian-driven damping.

中文翻译：

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干摩擦与Hessian驱动阻尼相结合的近梯度惯性算法的有限收敛性

SIAM优化杂志，第30卷，第3期，第2134-2162页，2020年1月。
在希尔伯特空间$ {\ mathcal H} $中，我们引入了具有有限收敛性的新型近端梯度算法。这些算法自然地作为惯性微分包含的离散时间形式出现，该惯性微分包含在三种阻尼的联合作用下保持稳定：干摩擦，粘滞摩擦和由Hessian驱动的几何阻尼。要最小化的函数$ f：{\ mathcal H} \ to {\ mathbb R} $是可微的（不一定是凸的），并通过其梯度进入算法。干摩擦阻尼函数$ \ phi：{\ mathcal H} \ to {\ mathbb R} _ + $是凸形的，在原点处具有极小的最小值（通常$ \ phi（x）= r \ | x \ | $ $ r> 0 $）。它通过其近端映射进入算法，该映射充当速度的软阈值运算符。它是有限步骤中稳定的来源。由Hessian驱动的几何阻尼以$ \ nabla ^ 2 f（x（t））\ dot {x}（t）$的形式干预动力学。通过将此项视为$ \ nabla f（x（t））$的时间导数，可以离散化形式给出一阶算法。Hessian驱动的阻尼使人们可以控制并减弱振荡。在其渐近收敛为零的唯一假设下，收敛结果容忍错误的存在。用其Moreau包络代替潜在函数$ f $，我们将结果扩展到非光滑凸函数$ f $的情况。在这种情况下，该算法分别涉及$ f $和$ \ phi $的近端运算符。考虑了该算法的几种变体，包括Nesterov加速梯度方法的情况。然后，我们考虑扩展到加法复合优化的情况，从而导致拆分方法。给出了套索类型问题的数值实验。性能配置文件作为比较工具，突出了具有干摩擦和Hessian驱动阻尼的Nesterov加速方法的变体的有效性。