In [19], a general, inexact, efficient proximal quasi-Newton algorithm for
composite optimization problems has been proposed and a sublinear global
convergence rate has been established. In this paper, we analyze the
convergence properties of this method, both in the exact and inexact setting,
in the case when the objective function is strongly convex. We also investigate
a practical variant of this method by establishing a simple stopping criterion
for the subproblem optimization. Furthermore, we consider an accelerated
variant, based on FISTA [1], to the proximal quasi-Newton algorithm. A similar
accelerated method has been considered in [7], where the convergence rate
analysis relies on very strong impractical assumptions. We present a modified
analysis while relaxing these assumptions and perform a practical comparison of
the accelerated proximal quasi- Newton algorithm and the regular one. Our
analysis and computational results show that acceleration may not bring any
benefit in the quasi-Newton setting.

本文针对复合优化问题中一般且高效的不精确近端拟牛顿算法，在强凸目标函数下分析了其精确和不精确执行的收敛性质，并建立了一个简单的停止标准来改善其实用性。同时，对基于 FISTA 的近端拟牛顿算法进行加速，并与传统算法进行比较和分析，结果表明加速并没有带来任何好处。

带有线性和加速亚线性收敛速率的正则化凸优化的近端拟牛顿方法

Proximal Quasi-Newton Methods for Regularized Convex Optimization with  Linear and Accelerated Sublinear Convergence Rates

We consider a bilevel optimisation approach for parameter learning in
higher-order total variation image reconstruction models. Apart from the least
squares cost functional, naturally used in bilevel learning, we propose and
analyse an alternative cost, based on a Huber regularised TV-seminorm.
Differentiability properties of the solution operator are verified and a
first-order optimality system is derived. Based on the adjoint information, a
quasi-Newton algorithm is proposed for the numerical solution of the bilevel
problems. Numerical experiments are carried out to show the suitability of our
approach and the improved performance of the new cost functional. Thanks to the
bilevel optimisation framework, also a detailed comparison between TGV$^2$ and
ICTV is carried out, showing the advantages and shortcomings of both
regularisers, depending on the structure of the processed images and their
noise level.

研究使用双层优化方法在高阶总变分图像重建模型中进行参数学习，提出并分析一种基于 Huber 正则化 TV 半范数的替代代价，验证解算符的可微性，并推导出一种一阶最优系统。基于伴随信息，提出了一个拟牛顿算法来数值解决双层问题。通过数值实验展示了我们的方法的适用性和新代价函数的改进性能。同时，利用双层优化框架，对比了 TGV^2 和 ICTV，在处理图像结构和噪声水平方面的优缺点。