标题： 一种用于等式和不等式约束非凸优化的邻近增广拉格朗日方法 显示英文标题

标题： A proximal augmented Lagrangian method for nonconvex optimization with equality and inequality constraints

摘要： 我们提出了一种不精确的近似增广拉格朗日方法（P-ALM）用于非凸结构优化问题。 该方法的特点在于，不仅对于更新惩罚参数，而且对于自适应调整近似项都具有易于实现的规则。 它允许惩罚参数在早期阶段快速增长以加快进展，同时改善了后期迭代中病态条件的问题，这是传统线性增加惩罚参数方法的一个众所周知的缺点。 我们分析中的关键要素在于观察到，只要有一个初始可行点，就可以在迭代过程中有效地控制增广拉格朗日函数。 我们的分析虽然简单，但为P-ALM提供了一个新的理论视角，并且作为副产品，得出了其非近似变体——经典的增广拉格朗日方法（ALM）类似的收敛性质。 数值实验包括凸和非凸问题实例，证明了我们方法的有效性。 显示英文摘要

摘要： We propose an inexact proximal augmented Lagrangian method (P-ALM) for nonconvex structured optimization problems. The proposed method features an easily implementable rule not only for updating the penalty parameters, but also for adaptively tuning the proximal term. It allows the penalty parameter to grow rapidly in the early stages to speed up progress, while ameliorating the issue of ill-conditioning in later iterations, a well-known drawback of the traditional approach of linearly increasing the penalty parameters. A key element in our analysis lies in the observation that the augmented Lagrangian can be controlled effectively along the iterates, provided an initial feasible point is available. Our analysis, while simple, provides a new theoretical perspective about P-ALM and, as a by-product, results in similar convergence properties for its non-proximal variant, the classical augmented Lagrangian method (ALM). Numerical experiments, including convex and nonconvex problem instances, demonstrate the effectiveness of our approach.