SIAM Journal on Optimization, Volume 30, Issue 4, Page 3069-3097, January 2020.
This paper deals with a general framework for inexact forward--backward algorithms aimed at minimizing the sum of an analytic function and a lower semicontinuous, subanalytic, convex term. Such a framework relies on an implementable inexactness condition for the computation of the proximal operator and on a linesearch procedure, which is possibly performed whenever a variable metric is allowed into the forward--backward step. The main focus of this work is the convergence of the considered scheme without additional convexity assumptions on the objective function. Toward this aim, we employ the recent concept of forward--backward envelope to define a continuously differentiable surrogate function, which coincides with the objective at its stationary points and satisfies the so-called Kurdyka--Łojasiewicz (KL) property on its domain. We adapt the abstract convergence scheme usually exploited in the KL framework to our inexact forward--backward scheme, prove the convergence of the iterates to a stationary point of the problem, and prove the convergence rates for the function values. Finally, we show the effectiveness and the flexibility of the proposed framework on a large-scale image restoration test problem.

中文翻译：

---

使用前向-后向包络的不精确前向-后向算法的收敛

SIAM优化杂志，第30卷，第4期，第3069-3097页，2020年1月。
本文讨论了不精确的正向-向后算法的通用框架，该框架旨在最小化解析函数和下半连续，亚解析凸项的总和。这样的框架依赖于可实现的不精确性条件来计算近端算子，并且依赖于线搜索程序，只要向前或向后步骤中允许使用可变度量，就可以执行该程序。这项工作的主要重点是考虑的方案的收敛，而对目标函数没有附加的凸性假设。为了实现这一目标，我们采用了最新的前向后向包络概念来定义一个连续可区分的替代函数，该函数与目标在固定点相吻合，并满足其域上所谓的Kurdyka-Łojasiewicz（KL）性质。我们将通常在KL框架中使用的抽象收敛方案调整为我们不精确的前向后方案，证明迭代的收敛到问题的平稳点，并证明函数值的收敛速度。最后，我们展示了所提出框架在大规模图像恢复测试问题上的有效性和灵活性。