This paper is concerned with a class of DC composite optimization problems which, as an extension of convex composite optimization problems and DC programs with nonsmooth components, often arises in robust factorization models of low-rank matrix recovery. For this class of nonconvex and nonsmooth problems, we propose an inexact linearized proximal algorithm (iLPA) by computing in each step an inexact minimizer of a strongly convex majorization constructed with a partial linearization of their objective functions, and establish the global convergence of the generated iterate sequence under the Kurdyka-\L\"ojasiewicz (KL) property of a potential function. In particular, by leveraging the composite structure, we provide a verifiable condition for the potential function to have the KL property of exponent $1/2$ at the limit point, so for the iterate sequence to have a local R-linear convergence rate, and clarify its relationship with the regularity used in the convergence analysis of algorithms for convex composite optimization. Finally, our iLPA is applied to a robust factorization model for matrix completions with outliers, and numerical comparison with the Polyak subgradient method confirms its superiority in computing time and quality of solutions.

翻译：本文关注一类DC复合优化问题，该问题作为凸复合优化问题和含非光滑项的DC规划问题的推广，常出现在低秩矩阵恢复的鲁棒分解模型中。针对这类非凸非光滑问题，我们提出一种不精确线性化近端算法（iLPA），通过每一步计算由目标函数部分线性化构造的强凸预函数的局部极小值，并基于势函数的Kurdyka-Łojasiewicz (KL)性质建立生成迭代序列的全局收敛性。特别地，利用复合结构，我们给出了势函数在极限点具有指数$1/2$的KL性质的可验证条件，从而确保迭代序列具有局部R-线性收敛速度，并阐明该条件与凸复合优化算法收敛分析中使用的正则性之间的关系。最后，将所提iLPA应用于含异常值矩阵补全的鲁棒分解模型，与Polyak次梯度方法的数值比较验证了其在计算时间和解质量方面的优越性。