A new result in convex analysis on the calculation of proximity operators in
certain scaled norms is derived. We describe efficient implementations of the
proximity calculation for a useful class of functions; the implementations
exploit the piece-wise linear nature of the dual problem. The second part of
the paper applies the previous result to acceleration of convex minimization
problems, and leads to an elegant quasi-Newton method. The optimization method
compares favorably against state-of-the-art alternatives. The algorithm has
extensive applications including signal processing, sparse recovery and machine
learning and classification.

本文提出了一种计算特定尺度规范下近似算子的凸分析方法，并利用对偶问题的分段线性特性描述了一类函数的高效实现，之后将该方法应用于凸最小化问题加速中并得到了优雅的拟牛顿方法，该算法在信号处理、稀疏恢复和机器学习分类等领域中具有广泛应用，并与现有方案相比具有较高的性能。

一种拟牛顿近端分裂方法

A quasi-Newton proximal splitting method

We propose a primal-dual splitting algorithm for solving monotone inclusions
involving a mixture of sums, linear compositions, and parallel sums of
set-valued and Lipschitzian operators. An important feature of the algorithm is
that the Lipschitzian operators present in the formulation can be processed
individually via explicit steps, while the set-valued operators are processed
individually via their resolvents. In addition, the algorithm is highly
parallel in that most of its steps can be executed simultaneously. This work
brings together and notably extends various types of structured monotone
inclusion problems and their solution methods. The application to convex
minimization problems is given special attention.

提出一种原始 - 对偶分裂算法用于解决包含和的混合、线性组合和并行和的集合值和 Lipschitzian 算子的单调包含问题，其重要特征在于该算法可以通过显式步骤分别处理配方中的 Lipschitzian 算子和集合值算子的解算器，且大多数步骤可以同时执行。该算法结合了各种类型的结构化单调包含问题及其解法，特别关注于凸最小化问题的应用。