近点法的检验和非线性预处理
Testing and non-linear preconditioning of the proximal point method
摘要：
利用非线性预处理的思想和经典邻近点法的检验方法，将最优化方法的收敛速度和收敛性证明中的常见问题形式化为一个简单的迭代不等式的验证问题.当应用于不动点算子时，后者可以看作是公司非扩张性或$α$平均属性的推广。这项工作的主要目的是提供抽象的背景理论，我们的同伴论文“块邻近方法与空间适应加速”。在本帐户中，我们证明了几个经典算法的一般方法的有效性，以及它们的随机变量。当然，除了邻近点法之外，这些方法包括梯度下降法、向前-向后分裂法、Douglas-Rachford分裂法、牛顿法，以及几种求解鞍点问题的方法，如乘子交替方向法和Chambolle-Pock方法。
英文摘要：
Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple iteration-wise inequality. When applied to fixed point operators, the latter can be seen as a generalisation of firm non-expansivity or the $α$-averaged property. The main purpose of this work is to provide the abstract background theory for our companion paper "Block-proximal methods with spatially adapted acceleration". In the present account we demonstrate the effectiveness of the general approach on several classical algorithms, as well as their stochastic variants. Besides, of course, the proximal point method, these method include the gradient descent, forward--backward splitting, Douglas--Rachford splitting, Newton's method, as well as several methods for saddle-point problems, such as the Alternating Directions Method of Multipliers, and the Chambolle--Pock method.
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