We propose a new majorization-minimization (MM) method for non-smooth and
non-convex programs, which is general enough to include the existing MM
methods. Besides the local majorization condition, we only require that the
difference between the directional derivatives of the objective function and
its surrogate function vanishes when the number of iterations approaches
infinity, which is a very weak condition. So our method can use a surrogate
function that directly approximates the non-smooth objective function. In
comparison, all the existing MM methods construct the surrogate function by
approximating the smooth component of the objective function. We apply our
relaxed MM methods to the robust matrix factorization (RMF) problem with
different regularizations, where our locally majorant algorithm shows
advantages over the state-of-the-art approaches for RMF. This is the first
algorithm for RMF ensuring, without extra assumptions, that any limit point of
the iterates is a stationary point.

本文提出了一种新的主化 - 最小化（MM）方法，适用于非光滑和非凸规划，其广泛性足以涵盖现有的 MM 方法。与现有的 MM 方法相比，该方法构建替代函数的方式更优越，相应的算法在抗干扰矩阵分解问题（RMF）中得到应用，并非仅仅基于平滑部分逼近的代理函数，而是直接逼近非平滑目标函数。