论文标题
基于ADMM的增强线性和圆锥优化的内点方法
An Enhanced ADMM-based Interior Point Method for Linear and Conic Optimization
论文作者
论文摘要
基于ADMM的内部点(Abip,Lin等,2021)方法是一种混合算法,可有效结合内部点方法(IPM)和一阶方法,以实现大规模线性优化的性能提升。与依赖于计算密集型牛顿步骤的传统IPM不同,ABIP方法应用了乘数的交替方向方法(ADMM)来大致解决障碍惩罚问题。但是,类似于其他一阶方法,此技术仍然对条件数和逆精度敏感。在本文中,我们提供了一种增强的ABIP方法,具有多种改进。首先,我们开发了一种ABIP方法来解决一般线性圆锥优化并建立相关的迭代复杂性。其次,受到一些现有方法的启发,我们为ABIP方法开发了不同的实现策略,从而在线性优化中显着提高了其性能。最后,我们在合成数据集和现实数据集中进行了广泛的数值实验,以证明我们的发展的经验优势。特别是,增强的ABIP方法可在Netlib $ 105 $选定的LP实例上的运行时间的几何平均值减少5.8倍,并且在某些结构化问题(例如SVM和Pagerank)中具有优势。但是,增强的ABIP方法仍然落后于许多基准,尤其是在需要高精度的情况下。我们认为,它可以作为互补工具与建立良好的求解器一起使用。
The ADMM-based interior point (ABIP, Lin et al. 2021) method is a hybrid algorithm that effectively combines interior point method (IPM) and first-order methods to achieve a performance boost in large-scale linear optimization. Different from traditional IPM that relies on computationally intensive Newton steps, the ABIP method applies the alternating direction method of multipliers (ADMM) to approximately solve the barrier penalized problem. However, similar to other first-order methods, this technique remains sensitive to condition number and inverse precision. In this paper, we provide an enhanced ABIP method with multiple improvements. Firstly, we develop an ABIP method to solve the general linear conic optimization and establish the associated iteration complexity. Secondly, inspired by some existing methods, we develop different implementation strategies for ABIP method, which substantially improve its performance in linear optimization. Finally, we conduct extensive numerical experiments in both synthetic and real-world datasets to demonstrate the empirical advantage of our developments. In particular, the enhanced ABIP method achieves a 5.8x reduction in the geometric mean of run time on $105$ selected LP instances from Netlib, and it exhibits advantages in certain structured problems such as SVM and PageRank. However, the enhanced ABIP method still falls behind commercial solvers in many benchmarks, especially when high accuracy is desired. We posit that it can serve as a complementary tool alongside well-established solvers.