Line search procedures are often employed in primal-dual methods for bilinear saddle point problems, especially when the norm of the linear operator is large or difficult to compute. In this paper, we demonstrate that line search is unnecessary by introducing a novel primal-dual method, the auto-conditioned primal-dual hybrid gradient (AC-PDHG) method, which achieves optimal complexity for solving bilinear saddle point problems. AC-PDHG is fully adaptive to the linear operator, using only past iterates to estimate its norm. We further tailor AC-PDHG to solve linearly constrained problems, providing convergence guarantees for both the optimality gap and constraint violation. Moreover, we explore an important class of linearly constrained problems where both the objective and constraints decompose into two parts. By incorporating the design principles of AC-PDHG into the preconditioned alternating direction method of multipliers (ADMM), we propose the auto-conditioned alternating direction method of multipliers (AC-ADMM), which guarantees convergence based solely on one part of the constraint matrix and fully adapts to it, eliminating the need for line search. Finally, we extend both AC-PDHG and AC-ADMM to solve bilinear problems with an additional smooth term. By integrating these methods with a novel acceleration scheme, we attain optimal iteration complexities under the single-oracle setting.

翻译：在求解双线性鞍点问题的原始-对偶方法中，线搜索过程常被采用，特别是当线性算子的范数较大或难以计算时。本文通过提出一种新型原始-对偶方法——自适应条件化原始-对偶混合梯度法（AC-PDHG），证明了线搜索的非必要性。该方法在求解双线性鞍点问题时能达到最优复杂度。AC-PDHG对线性算子具有完全自适应性，仅利用历史迭代点估计其范数。我们进一步将AC-PDHG定制用于求解线性约束问题，同时为最优性间隙和约束违反量提供收敛性保证。此外，我们研究了一类重要的线性约束问题，其目标函数与约束条件均可分解为两部分。通过将AC-PDHG的设计原理融入预条件化交替方向乘子法（ADMM），我们提出了自适应条件化交替方向乘子法（AC-ADMM）。该方法仅需依赖约束矩阵的一部分即可保证收敛，并对其实现完全自适应，从而消除了线搜索的需求。最后，我们将AC-PDHG与AC-ADMM扩展至求解带附加光滑项的双线性问题。通过将这些方法与新型加速方案相结合，我们在单预言机设置下获得了最优迭代复杂度。