Abstract Forward–backward methods are valid tools to solve a variety of optimization problems where the objective function is the sum of a smooth, possibly nonconvex term plus a convex, possibly nonsmooth function. The corresponding iteration is built on two main ingredients: the computation of the gradient of the smooth part and the evaluation of the proximity (or resolvent) operator associated with the convex term. One of the main difficulties, from both implementation and theoretical point of view, arises when the proximity operator is computed in an inexact way. The aim of this paper is to provide new convergence results about forward–backward methods with inexact computation of the proximity operator, under the assumption that the objective function satisfies the Kurdyka–Łojasiewicz property. In particular, we adopt an inexactness criterion which can be implemented in practice, while preserving the main theoretical properties of the proximity operator. The main result is the proof of the convergence of the iterates generated by the forward–backward algorithm in Bonettini et al. (2017) to a stationary point. Convergence rate estimates are also provided. At the best of our knowledge, there exists no other inexact forward–backward algorithm with proved convergence in the nonconvex case and equipped with an explicit procedure to inexactly compute the proximity operator.

中文翻译：

---

不精确变量度量前向后向方法的新收敛结果

摘要 前向-后向方法是解决各种优化问题的有效工具，其中目标函数是平滑的、可能非凸的项加上凸的、可能不平滑的函数的总和。相应的迭代建立在两个主要成分上：平滑部分的梯度计算和与凸项相关的邻近（或解析）算子的评估。从实现和理论的角度来看，主要困难之一出现在以不精确的方式计算邻近算符时。本文的目的是在目标函数满足 Kurdyka-Łojasiewicz 性质的假设下，提供关于具有不精确计算接近算子的前向 - 后向方法的新收敛结果。特别是，我们采用了一种可以在实践中实现的不精确性标准，同时保留了邻近算子的主要理论特性。主要结果是 Bonettini 等人的前向-后向算法生成的迭代收敛的证明。(2017) 到一个静止点。还提供了收敛率估计。据我们所知，不存在其他不精确的前向后向算法在非凸情况下证明收敛并配备显式程序来不精确地计算邻近算子。还提供了收敛率估计。据我们所知，不存在其他不精确的前向后向算法在非凸情况下证明收敛并配备显式程序来不精确地计算邻近算子。还提供了收敛率估计。据我们所知，不存在其他不精确的前向后向算法在非凸情况下证明收敛并配备显式程序来不精确地计算邻近算子。