交替方向法作为求解矩阵补全问题的经典方法之一, 具有能够将一个极小化问题分解成多个规模更小、更容易求解的子问题的优势, 近年来在图像处理和数据分析等领域备受青睐。本文采用交替方向法的框架, 结合惯性策略, 提出了一种求解矩阵补全问题的惯性加速交替方向法。新算法在每步迭代中, 对于其中一部分变量利用交替方向法的前两次迭代点外推得到新一步的迭代点, 从而提高计算效率。本文在合理的假设条件下, 证明了新算法的收敛性。最后, 通过随机矩阵补全的数值实验及图像修复的实例验证了新算法的有效性和可行性。

The alternating direction method of multiplier is attractive for solving matrix completion problems, which has the advantage of being able to decompose a minimization problem into many smaller and easier subproblems. Therefore, it is popular in the field of image processing and data analysis in recent years. Based on the framework of the alternating direction method of multiplier, we combine the inertial strategy and propose an inertial accelerated alternating direction method of multiplier to solve the matrix completion problems in this paper. In each iteration of the new method, for some of variables, the iterative points of the previous two iterations of the alternating direction method of multiplier are extrapolated to obtain the new iteration points, which improve the computational efficiency of the new method. Under the reasonable assumptions, we prove convergence of the new method. Finally, we also verify the effectiveness and feasibility of the new method by numerical experiments of random matrix completion and an example of image restoration.