This paper concerns 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 at the current iterate, and establish the 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. Finally, we apply the proposed iLPA to a robust factorization model for matrix completions with outliers and non-uniform sampling, and numerical comparison with a proximal alternating minimization (PAM) method confirms iLPA yields the comparable relative errors or NMAEs within less running time, especially for large-scale real data.

翻译：本文研究一类DC复合优化问题，该类问题作为凸复合优化问题和非光滑DC规划问题的推广，常见于低秩矩阵恢复的鲁棒分解模型中。针对此类非凸非光滑问题，我们提出一种不精确线性化近端算法（iLPA），该算法在每一步计算由当前迭代点处目标函数的部分线性化构建的强凸优函数的不精确极小值点，并基于势函数的Kurdyka-Łojasiewicz（KL）性质建立了生成迭代序列的收敛性。特别地，通过利用复合结构，我们给出了势函数在极限点处具有指数为$1/2$的KL性质的可验证条件，从而使迭代序列具有局部R线性收敛速度。最后，我们将所提出的iLPA应用于含离群值和非均匀采样的矩阵补全鲁棒分解模型，与近端交替最小化（PAM）方法的数值对比表明，iLPA能够在更短运行时间内获得可比的相对误差或NMAE值，尤其适用于大规模真实数据。