AI 中文总结
针对线性约束凸复合问题，提出三种保留标准AL子问题结构的非精确无参数增广拉格朗日方法，达到凸设定下最优$\fancyscript{O}(ε^{-1})$非遍历原-对偶复杂度，强凸下接近最优，实验较PAL方法提速5至50倍。
AI 中文摘要
增广拉格朗日（AL）方法是求解约束优化的经典框架，但对于可直接验证的近似KKT点，标准非精确AL方法已知的一阶复杂度界并非最优，而已知最优的近端增广拉格朗日（PAL）界仍带有额外的对数因子。我们研究带有光滑凸项以及定义域紧的可能非光滑闭正常凸项的线性约束凸复合问题。我们提出了三种非精确AL方案，它们保留了标准AL子问题的结构，在凸设定下达到最优原-对偶复杂度$\boldsymbol{\fancyscript{O}}(\boldsymbol{\fancyscript{ε}}^{-1})$，改进了此前$\boldsymbol{\fancyscript{O}}(\boldsymbol{\fancyscript{ε}}^{-4/3})$、$\boldsymbol{\fancyscript{O}}(\boldsymbol{\fancyscript{ε}}^{-7/4})$和$\boldsymbol{\fancyscript{O}}(\boldsymbol{\fancyscript{ε}}^{-2})$的AL界，并消除了PAL保证中的对数因子。其中两种变体是无参数的，且三种方法均具备非遍历保证，其中一种变体还拥有更强的最后迭代保证。\n这些结果表明，近端正则化、遍历平均以及对问题相关常数的先验知识，并非在标准AL框架内达到最优可验证原-对偶复杂度的内在要求。关键技术是一种无参数加速方法，它能以最优复杂度为标准无正则化AL子问题计算可验证的平稳性证书。在强凸设定下，我们的方法达到接近最优的复杂度$\boldsymbol{\fancyscript{O}}(\boldsymbol{\fancyscript{ε}}^{-1/2}\boldsymbol{\fancyscript{log}}(\boldsymbol{\fancyscript{ε}}^{-1}))$，且包含两种无参数变体。在包括弹性网最小二乘回归、组稀疏Huber化支持向量机以及量子半定规划（SDP）在内的六类问题上的数值实验表明，该方法相比代表性PAL方法具有显著计算优势，加速比通常在5到50倍之间。
英文摘要
Augmented Lagrangian (AL) methods are a classical framework for constrained optimization, but for directly verifiable approximate KKT points, known first-order complexity bounds for standard inexact AL methods are suboptimal, while the best known proximal augmented Lagrangian (PAL) bounds retain an additional logarithmic factor. We consider linearly constrained convex composite problems with a smooth convex term and a possibly nonsmooth closed proper convex term with compact domain. We develop three inexact AL schemes that preserve the standard AL subproblem structure and attain the optimal primal-dual complexity $\mathcal O(ε^{-1})$ in the convex setting, improving prior AL bounds of $\mathcal O(ε^{-4/3})$, $\mathcal O(ε^{-7/4})$, and $\mathcal O(ε^{-2})$, and removing the logarithmic factor from PAL guarantees. Two variants are parameter-free, and all three admit nonergodic guarantees, including a stronger last-iterate guarantee for one variant. These results show that proximal regularization, ergodic averaging, and prior knowledge of problem-dependent constants are not intrinsic requirements for attaining optimal verifiable primal-dual complexity within the standard AL framework. A key ingredient is a parameter-free accelerated method that computes verifiable stationarity certificates for the standard, unregularized AL subproblems with optimal complexity. In the strongly convex setting, our methods attain near-optimal complexity $\mathcal O(ε^{-1/2}\log(ε^{-1}))$, with two parameter-free variants. Numerical experiments on six problem classes, including elastic-net least-squares regression, group-sparse Huberized support vector machines, and a quantum semidefinite program (SDP), demonstrate substantial computational advantages over a representative PAL method, with speedups frequently ranging from $5$ to $50$ times.
