论线性规划的初等-双重混合梯度的几何形状和精炼率

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2024-07-17 DOI:10.1007/s10107-024-02109-9
Haihao Lu, Jinwen Yang
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引用次数: 0

摘要

我们研究了求解线性规划(LP)的原始双混合梯度(PDHG)的收敛行为。PDHG 是一种新型通用一阶法 LP 求解器 PDLP 的基础算法,其目的是利用现代计算架构的优势来扩展 LP。尽管在数值上取得了成功,但人们对 PDHG 用于 LP 的理论理解仍然非常有限;之前的复杂度结果依赖于 KKT 系统的全局霍夫曼常数,而众所周知,该常数非常松散,且信息量不大。在这项工作中,我们旨在从根本上理解 LP 的 PDHG 收敛行为,并开发出一种不依赖全局霍夫曼常数的精细复杂度率。我们证明了 LP 的 PDHG 有两个主要阶段:在第一阶段,PDHG 识别活动变量,第一阶段的长度受某个量的驱动,该量衡量 LP 实例的非退化部分与退化的接近程度;在第二阶段,PDHG 有效地求解了一个同质线性不等式系统,第二阶段的复杂度受该系统的一个良好的局部锐度常数的驱动。这一发现与非光滑优化中的局部光滑性概念密切相关,也是第一个没有非退化假设的有限时间辨识复杂性结果。我们的结果还有一个有趣的含义,即退化本身并不会减慢 LP 的 PDHG 收敛速度,但接近退化却会。
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On the geometry and refined rate of primal–dual hybrid gradient for linear programming

We study the convergence behaviors of primal–dual hybrid gradient (PDHG) for solving linear programming (LP). PDHG is the base algorithm of a new general-purpose first-order method LP solver, PDLP, which aims to scale up LP by taking advantage of modern computing architectures. Despite its numerical success, the theoretical understanding of PDHG for LP is still very limited; the previous complexity result relies on the global Hoffman constant of the KKT system, which is known to be very loose and uninformative. In this work, we aim to develop a fundamental understanding of the convergence behaviors of PDHG for LP and to develop a refined complexity rate that does not rely on the global Hoffman constant. We show that there are two major stages of PDHG for LP: in Stage I, PDHG identifies active variables and the length of the first stage is driven by a certain quantity which measures how close the non-degeneracy part of the LP instance is to degeneracy; in Stage II, PDHG effectively solves a homogeneous linear inequality system, and the complexity of the second stage is driven by a well-behaved local sharpness constant of the system. This finding is closely related to the concept of partial smoothness in non-smooth optimization, and it is the first complexity result of finite time identification without the non-degeneracy assumption. An interesting implication of our results is that degeneracy itself does not slow down the convergence of PDHG for LP, but near-degeneracy does.

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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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