论文标题
增强基于拉格朗日的一阶方法，用于具有弱符号目标的凸面限制程序
Augmented Lagrangian based first-order methods for convex-constrained programs with weakly-convex objective
论文作者
论文摘要
一阶方法（FOM）已被广泛用于解决大规模问题。大多数现有作品都集中在没有约束或简单约束的问题上。最近的一些作品研究了FOM的功能限制的问题。在本文中，我们设计了一种新型的基于Lagrangian（AL）的FOM，以解决具有非凸观目标和凸约束函数的问题。新方法遵循近端（PP）方法的框架。在大约求解PP子问题时，它将不精确方法（IALM）的用法和二次惩罚方法混合在一起，而后者始终由IALM用估计的乘数喂食。我们显示了$ o（\ varepsilon^{ - \ frac {5} {2}}} | \ log \ varepsilon |）$的复杂性结果。这是最著名的结果。从理论上讲，混合方法的迭代复杂性要求低于仅使用IALM来求解PP子问题的对应物，并且从数值上讲，它的性能比基于纯的基于纯的方法要好得多。数值实验是在线性约束二次程序和非convex QCQP上进行的。数值结果证明了所提出的方法比现有方法的效率。
First-order methods (FOMs) have been widely used for solving large-scale problems. A majority of existing works focus on problems without constraint or with simple constraints. Several recent works have studied FOMs for problems with complicated functional constraints. In this paper, we design a novel augmented Lagrangian (AL) based FOM for solving problems with non-convex objective and convex constraint functions. The new method follows the framework of the proximal point (PP) method. On approximately solving PP subproblems, it mixes the usage of the inexact AL method (iALM) and the quadratic penalty method, while the latter is always fed with estimated multipliers by the iALM. We show a complexity result of $O(\varepsilon^{-\frac{5}{2}}|\log\varepsilon|)$ for the proposed method to achieve an $\varepsilon$-KKT point. This is the best known result. Theoretically, the hybrid method has lower iteration-complexity requirement than its counterpart that only uses iALM to solve PP subproblems, and numerically, it can perform significantly better than a pure-penalty-based method. Numerical experiments are conducted on nonconvex linearly constrained quadratic programs and nonconvex QCQP. The numerical results demonstrate the efficiency of the proposed methods over existing ones.
