This paper is concerned with a class of DC composite optimization problems which, as an extension of the convex composite optimization problem and the DC program with nonsmooth components, often arises from robust factorization models of low-rank matrix recovery. For this class of nonconvex and nonsmooth problems, we propose an inexact linearized proximal algorithm (iLPA) which in each step computes an inexact minimizer of a strongly convex majorization constructed by the partial linearization of their objective functions. The generated iterate sequence is shown to be convergent under the Kurdyka-{\L}ojasiewicz (KL) property of a potential function, and the convergence admits a local R-linear rate if the potential function has the KL property of exponent $1/2$ at the limit point. For the latter assumption, we provide a verifiable condition by leveraging the composite structure, and clarify its relation with the regularity used for the convex composite optimization. Finally, the proposed iLPA is applied to a robust factorization model for matrix completions with outliers, DC programs with nonsmooth components, and $\ell_1$-norm exact penalty of DC constrained programs, and numerical comparison with the existing algorithms confirms the superiority of our iLPA in computing time and quality of solutions.

翻译：本文研究一类DC复合优化问题，该问题作为凸复合优化问题与含非光滑分量的DC规划问题的推广，常见于低秩矩阵恢复的鲁棒因子分解模型中。针对此类非凸非光滑问题，我们提出一种非精确线性化邻近算法(iLPA)，该算法每一步通过目标函数的部分线性化构造强凸上界函数，并计算其非精确极小点。在势函数的Kurdyka-Łojasiewicz(KL)性质假设下，证明了生成的迭代序列收敛，且若势函数在极限点处具有指数1/2的KL性质，则收敛速率达到局部R-线性收敛。针对后一假设，我们借助复合结构给出了可验证条件，并阐明了其与凸复合优化中正则性的关联。最后，将所提出的iLPA应用于含离群点的矩阵填充鲁棒因子分解模型、含非光滑分量的DC规划问题以及DC约束问题的ℓ₁范数精确罚函数模型，数值比较结果表明iLPA在计算时间和解的质量上均优于现有算法。