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 the Polyak subgradient method confirms its superiority in terms of computing time and quality of solutions.

翻译：本文研究一类DC复合优化问题，作为凸复合优化问题与非光滑DC规划问题的扩展，该问题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，我们提出了一种非精确线性化近端算法（iLPA）：每步迭代通过在当前点对目标函数进行部分线性化构造强凸预优函数，并计算其非精确极小点。在势函数满足Kurdyka-Łojasiewicz（KL）性质的条件下，建立了迭代序列的收敛性。特别地，利用复合结构，我们给出了势函数在极限点处具有指数1/2的KL性质的可验证条件，从而保证迭代序列具有局部R-线性收敛速度。最后，将所提出的iLPA应用于含异常值和非均匀采样的鲁棒矩阵补全分解模型，并与Polyak次梯度方法进行数值比较，验证了该方法在计算时间和解质量方面的优越性。