SIAM Journal on Optimization, Volume 31, Issue 1, Page 653-685, January 2021.
We introduce Bella, a locally superlinearly convergent Bregman forward-backward splitting method for minimizing the sum of two nonconvex functions, one of which satisfies a relative smoothness condition and the other one is possibly nonsmooth. A key tool of our methodology is the Bregman forward-backward envelope (BFBE), an exact and continuous penalty function with favorable first- and second-order properties, which enjoys a nonlinear error bound when the objective function satisfies a Łojasiewicz-type property. The proposed algorithm is of linesearch type over the BFBE along user-defined update directions and converges subsequentially to stationary points and globally under the Kurdyka--Łojasiewicz condition. Moreover, when the update directions are superlinear in the sense of Facchinei and Pang [Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I, Springer, New York, 2003], owing to the given nonlinear error bound unit stepsize is eventually always accepted and the algorithm attains superlinear convergence rates even when the limit point is a nonisolated minimum.

中文翻译：

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非凸复合优化的Bregman向前-向后线性搜索算法：非孤立局部极小值的超线性收敛

SIAM优化杂志，第31卷，第1期，第653-685页，2021年1月。
我们介绍了Bella，这是一种局部超线性收敛的Bregman前向后向拆分方法，用于最小化两个非凸函数之和，其中一个满足相对平滑条件，而另一个则可能不光滑。我们的方法学的关键工具是Bregman前向后包络（BFBE），它是精确且连续的惩罚函数，具有良好的一阶和二阶属性，当目标函数满足Łojasiewicz类型的属性时，它将具有非线性误差范围。所提出的算法是沿着BFBE沿着用户定义的更新方向进行线搜索的，并且随后在Kurdyka-Łojasiewicz条件下收敛到固定点并全局收敛。而且，