论文标题
快速凸优化通过时间演化方程式：tog-v tog tog的改进版本
Fast convex optimization via a third-order in time evolution equation: TOGES-V an improved version of TOGES
论文作者
论文摘要
在希尔伯特空间设置h中，为了进行凸优化，我们分析了快速收敛性能，因为t倾向于无穷大，这是时间演化系统中三阶生成的轨迹。最小化的函数f被认为是凸的，不断差异，并具有一组非空的最小化器。它通过其梯度进入动态。基于这个新的动力学系统，我们改善了[Attouch，Chbani，Riahi：通过时间演化方程式的快速凸优化，优化2020]。作为主要结果，当阻尼参数$α$满足$α> 3 $时，我们表明，当t到t到Infinity时，值在1/T3上的收敛性以及轨迹的收敛性。我们通过引入动态的Hessian驱动阻尼项来补充这些结果，从而减少了振荡。在强凸功能f的情况下，我们显示了以指数收敛速率的第三顺序的自主演化系统。所有这些结果对凸较低的半连续函数的情况具有自然扩展，并具有扩展的实际值。只需用莫罗的信封代替f即可。
In a Hilbert space setting H, for convex optimization, we analyze the fast convergence properties as t tends to infinity of the trajectories generated by a third-order in time evolution system. The function f to minimize is supposed to be convex, continuously differentiable, with a nonempty set of minimizers. It enters into the dynamic through its gradient. Based on this new dynamical system, we improve the results obtained by [Attouch, Chbani, Riahi: Fast convex optimization via a third-order in time evolution equation, Optimization 2020]. As a main result, when the damping parameter $α$ satisfies $α> 3$, we show that the convergence of the values at the order 1/t3 as t goes to infinity, as well as the convergence of the trajectories. We complement these results by introducing into the dynamic an Hessian driven damping term, which reduces the oscillations. In the case of a strongly convex function f, we show an autonomous evolution system of the third order in time with an exponential rate of convergence. All these results have natural extensions to the case of a convex lower semicontinuous function with extended real values. Just replace f with its Moreau envelope.
