在Hilbert空间中，对于凸优化，我们分析了三阶时间演化系统生成的轨迹在t趋于无穷大时的快速收敛性.要最小化的函数f应该是凸的、连续可微的，并且具有非空的最小值集。它通过梯度进入动态。基于这个新的动力系统，我们改进了[Attouch，Chbani，Riahi：Fast凸优化via a third order in time发展方程，Optimization 2020]中的结果。作为主要结果，当阻尼参数$α$满足$α> 3$时，我们证明了当t趋于无穷大时，阻尼参数$α$的值在1/t3阶收敛，以及轨迹的收敛.我们补充这些结果引入到动态的海森驱动的阻尼项，减少了振荡。在强凸函数f的情况下，我们证明了一个三阶自治演化系统的指数收敛速率。所有这些结果都对具有扩张实值的凸下连续函数的情形有自然扩张。把f换成它的莫罗信封。
英文摘要：
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.
The global linear convergence rate of the proximal version of the generalized alternating direction method of multipliers for separable convex programming
On Some Versions of Subspace Optimization Methods with Inexact Gradient Information
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Research Center for Artificial Intelligence, Innopolis University（因诺波利斯大学人工智能研究中心）；National Research University Higher School of Economics（国立高等经济大学）；V. Vernadsky Crimean Federal University（V.维尔纳茨基克里米亚联邦大学）
Domain-Driven Solver (DDS) Version 2.0: a MATLAB-based Software Package for Convex Optimization Problems in Domain-Driven Form
Domain-Driven Solver（DDS）是一个基于MATLAB的软件包，用于以Domain-Driven形式解决凸优化问题[Karimi and Tunçel，arXiv：1804.06925]。DDS的当前版本接受以下函数/集合约束的所有组合：（1）对称圆锥（LP，SOCP和SDP）;（2）SOCP可表示的二次约束;（3）定义为单变量凸函数的epigraphs的任意二维凸集集合的直和（包括作为特例的几何规划和熵规划）;（4）广义幂锥;（5）矩阵范数上图（包括作为线性子空间上的核范数的特殊情况的最小化）;（6）向量相对熵;（7）量子熵和量子相对熵的epigraphs;以及（8）涉及双曲多项式的约束。DDS是[Karimi and Tunçel，arXiv：1804.06925]中设计和分析的不可行启动原始对偶法的实际实现。这份手稿包含了用户指南，以及算法实现所需的理论结果。为了帮助用户，我们提供了许多示例。我们还讨论了一些实现细节和技术，我们用来提高效率和进一步扩展的软件，以涵盖新兴类的凸优化问题。
