Primal–dual hybrid gradient (PDHG) and alternating direction method of multipliers (ADMM) are popular first-order optimization methods. They are easy to implement and have diverse applications. As first-order methods, however, they are sensitive to problem conditions and can struggle to reach the desired accuracy. To improve their performance, researchers have proposed techniques such as diagonal preconditioning and inexact subproblems. This paper realizes additional speedup about one order of magnitude. Specifically, we choose general (non-diagonal) preconditioners that are much more effective at reducing the total numbers of PDHG/ADMM iterations than diagonal ones. Although the subproblems may lose their closed-form solutions, we show that it suffices to solve each subproblem approximately with a few proximal-gradient iterations or a few epochs of proximal block-coordinate descent, which are simple and have closed-form steps. Global convergence of this approach is proved when the inner iterations are fixed. Our method opens the choices of preconditioners and maintains both low per-iteration cost and global convergence. Consequently, on several typical applications of primal–dual first-order methods, we obtain 4–95\(\times \) speedup over the existing state-of-the-art.

原始-双重混合梯度（PDHG）和乘数交替方向方法（ADMM）是流行的一阶优化方法。它们易于实现并且具有多种应用。但是，作为一阶方法，它们对问题情况很敏感，并且可能难以达到所需的精度。为了提高其性能，研究人员提出了诸如对角线预处理和不精确的子问题之类的技术。本文实现了大约一个数量级的额外加速。具体来说，我们选择通用的（非对角线）预处理器，其在减少PDHG / ADMM迭代总数上比对角线预处理器更有效。尽管子问题可能会丢失其封闭式解决方案，我们表明，用一些近端梯度迭代或近端块坐标下降的几个时期来解决每个子问题就足够了，这很简单并且具有封闭形式的步骤。当内部迭代固定时，证明了该方法的全局收敛性。我们的方法打开了预处理器的选择，并保持了较低的每次迭代成本和全球收敛性。因此，在原始-对偶一阶方法的几种典型应用中，我们得到4-95\（\ times \）加速了现有的最新技术。