On the geometry and refined rate of primal–dual hybrid gradient for linear programming
Name
10107_2024_Article_2109.pdf
Size
1.05 MB
Format
Adobe PDF
Checksum (MD5)
6e2ecb481f46f532a8bb4a0bcecbf1aa
Author(s)
Lu, Haihao
Yang, Jinwen
Date Issued
July 17, 2024
Journal
Mathematical Programming
Publisher
Springer Science and Business Media LLC
Citation
Lu, H., Yang, J. On the geometry and refined rate of primal–dual hybrid gradient for linear programming. Math. Program. (2024).
Version
Final published version
Abstract
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.
MIT Department
Sloan School of Management
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s10107-024-02109-9
