ABSTRACT

In this paper, we propose a new framework to implement interior point method (IPM) in order to solve some very large-scale linear programs (LPs). Traditional IPMs typically use Newton's method to approximately solve a subproblem that aims to minimize a log-barrier penalty function at each iteration. Due its connection to Newton's method, IPM is often classified as second-order method – a genre that is attached with stability and accuracy at the expense of scalability. Indeed, computing a Newton step amounts to solving a large system of linear equations, which can be efficiently implemented if the input data are reasonably sized and/or sparse and/or well-structured. However, in case the above premises fail, then the challenge still stands on the way for a traditional IPM. To deal with this challenge, one approach is to apply the iterative procedure, such as preconditioned conjugate gradient method, to solve the system of linear equations. Since the linear system is different in each iteration, it is difficult to find good pre-conditioner to achieve the overall solution efficiency. In this paper, an alternative approach is proposed. Instead of applying Newton's method, we resort to the alternating direction method of multipliers (ADMM) to approximately minimize the log-barrier penalty function at each iteration, under the framework of primal–dual path-following for a homogeneous self-dual embedded LP model. The resulting algorithm is an ADMM-Based Interior Point Method, abbreviated as ABIP in this paper. The new method inherits stability from IPM and scalability from ADMM. Because of its self-dual embedding structure, ABIP is set to solve any LP without requiring prior knowledge about its feasibility. We conduct extensive numerical experiments testing ABIP with large-scale LPs from NETLIB and machine learning applications. The results demonstrate that ABIP compares favourably with other LP solvers including SDPT3, MOSEK, DSDP-CG and SCS.

摘要

在本文中，我们提出了一个新的框架来实现内部点方法（IPM），以解决一些非常大规模的线性程序（LP）。传统的IPM通常使用牛顿方法来近似解决一个子问题，该子问题旨在使每次迭代的对数障碍惩罚函数最小化。由于其与牛顿法的联系，IPM通常被归类为二阶方法–以稳定性和准确性为代价的流派，但以可伸缩性为代价。实际上，计算牛顿步就等于求解大型线性方程组，如果输入数据的大小和/或稀疏和/或结构合理，则可以有效地实现该方程。但是，如果上述前提失败，那么传统IPM仍将面临挑战。为了应对这一挑战，一种方法是应用迭代过程（例如预处理的共轭梯度法）来求解线性方程组。由于线性系统在每次迭代中都不同，因此很难找到好的预调节器来实现整体求解效率。本文提出了一种替代方法。与其应用牛顿方法，我们采用交替方向乘数法（ADMM）在均质自对偶嵌入式LP模型的原始-对偶路径遵循的框架下，在每次迭代中将对数壁垒罚函数近似最小化。生成的算法是基于ADMM的内部点方法，缩写为ABIP在本文中。新方法继承了IPM的稳定性和ADMM的可伸缩性。由于其具有自我对偶的嵌入结构，ABIP可以解决任何LP，而无需事先了解其可行性。我们进行广泛的数值实验测试ABIPNETLIB和机器学习应用程序的大型LP。结果表明ABIP 与其他LP解算器相比，包括 SDPT3， MOSEK， DSDP-CG 和 SCS。