The first-order primal-dual algorithms have received much considerable attention in the literature due to their quite promising performance in solving large-scale image processing models. In this paper, we consider a general saddle point problem and propose a double extrapolation primal-dual algorithm, which employs the efficient extrapolation strategy for both primal and dual variables. It is remarkable that the proposed algorithm enjoys a unified framework including several existing efficient solvers as special cases. Another exciting property is that, under quite flexible requirements on the involved extrapolation parameters, our algorithm is globally convergent to a saddle point of the problem under consideration. Moreover, the worst case \({{\mathcal {O}}}(1/t)\) convergence rate in both ergodic and nonergodic senses, and the linear convergence rate can be established for more general cases, where t counts the iteration. Some computational results on solving image deblurring, image inpainting and the nearest correlation matrix problems further show that the proposed algorithm is efficient, and performs better than some existing first-order solvers in terms of taking less iterations and computing time in some cases.

一阶本原对偶算法由于在解决大规模图像处理模型方面的出色表现而备受关注。在本文中，我们考虑了一般的鞍点问题，并提出了一种双重外推的原始对偶算法，该算法对原始变量和对偶变量均采用了有效的外推策略。值得注意的是，所提出的算法具有统一的框架，其中包括作为特例的几种现有的高效求解器。另一个令人兴奋的特性是，在对涉及的外推参数非常灵活的要求下，我们的算法可以全局收敛到所考虑问题的鞍点。而且，最坏的情况\（{{\ mathcal {O}}}（1 / t）\）遍历和非遍历意义上的收敛速度，可以为更一般的情况建立线性收敛速度，其中t计算迭代次数。在解决图像去模糊，图像修复和最近相关矩阵问题方面的一些计算结果进一步表明，该算法是有效的，并且在某些情况下，在减少迭代次数和计算时间方面比某些现有的一阶求解器更好。