SIAM Journal on Optimization, Volume 31, Issue 2, Page 1215-1241, January 2021.
We present a forward-backward-based algorithm to minimize a sum of a differentiable function and a nonsmooth function, both being possibly nonconvex. The main contribution of this work is to consider the challenging case where the nonsmooth function corresponds to a sum of nonconvex functions, resulting from composition between a strictly increasing, concave, differentiable function and a convex nonsmooth function. The proposed variable metric composite function forward-backward (C2FB) algorithm circumvents the explicit, and often challenging, computation of the proximity operator of the composite functions through a majorize-minimize approach. Precisely, each composite function is majorized using a linear approximation of the differentiable function, which allows one to apply the proximity step only to the sum of the nonsmooth functions. We prove the convergence of the algorithm iterates to a critical point of the objective function leveraging the Kurdyka--Ł ojasiewicz inequality. The convergence is guaranteed even if the proximity operators are computed inexactly, considering relative errors. We show that the proposed approach is a generalization of reweighting methods, with convergence guarantees. In particular, applied to the log-sum function, our algorithm reduces to a generalized version of the celebrated reweighted $\ell_1$ method. Finally, we show through simulations on an image processing problem that the proposed C2FB algorithm necessitates fewer iterations to converge and leads to better critical points compared with traditional reweighting methods and classic forward-backward algorithms.

中文翻译：

---

变度量逆向复合最小化算法

SIAM优化杂志，第31卷，第2期，第1215-1241页，2021年1月。
我们提出了一种基于前向后向的算法，以最小化可微函数和非平滑函数之和，二者均可能是非凸的。这项工作的主要贡献是考虑具有挑战性的情况，其中非光滑函数对应于非凸函数的总和，这是由严格增加的，凹的，可微函数和凸的非光滑函数之间的组合导致的。所提出的可变度量复合函数前后（C2FB）算法通过一种主观最小化方法来绕开对复合函数的接近算子进行显式且通常具有挑战性的计算。精确地，使用可微函数的线性逼近来对每个复合函数进行主化，这使得一个复合函数只能将非近似函数应用于非平滑函数的总和。我们证明了算法的收敛性利用Kurdyka-Łojasiewicz不等式迭代到目标函数的临界点。即使考虑了相对误差，即使对接近算子进行了不精确的计算，也可以保证收敛。我们表明，所提出的方法是加权加权的通用化，具有收敛性保证。特别地，应用于对数和函数后，我们的算法简化为著名的重新加权$ \ ell_1 $方法的广义版本。最后，我们通过对图像处理问题的仿真表明，与传统的加权方法和经典的前向后退算法相比，所提出的C2FB算法收敛所需的迭代次数更少，并且临界点更好。