Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

First-Order Methods for Linear Programming

Haihao Lu

TL;DR

This article provides an overview of the recent development of first-order methods for solving large-scale linear programming.

Abstract

Linear programming is the seminal optimization problem that has spawned and grown into today's rich and diverse optimization modeling and algorithmic landscape. This article provides an overview of the recent development of first-order methods for solving large-scale linear programming.

First-Order Methods for Linear Programming

TL;DR

This article provides an overview of the recent development of first-order methods for solving large-scale linear programming.

Abstract

Linear programming is the seminal optimization problem that has spawned and grown into today's rich and diverse optimization modeling and algorithmic landscape. This article provides an overview of the recent development of first-order methods for solving large-scale linear programming.
Paper Structure (14 sections, 8 theorems, 28 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 14 sections, 8 theorems, 28 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. PDHG for LP
  3. Initial attempts
  4. Theory of PDHG on LP
  5. Optimal FOM for LP
  6. Infeasibility detection
  7. PDLP
  8. Algorithmic enhancements in PDLP
  9. Numerical performance of PDLP
  10. Case studies of large LP
  11. Personalized marketing
  12. PageRank
  13. Robust production inventory problem
  14. Open questions

Key Result

Theorem 1

Consider the iterates $\{z^k\}_{k=0}^{\infty}$ of PDHG alg:pdhg for solving eq:minmax. Denote $\bar{z}^k=(\bar{x}^k,\bar{y}^k)=\frac{1}{k}\sum_{i=1}^k z^i$ as the average iterates. Then it holds for any $0<s\le \frac{1}{\|A\|_2}$, $k\ge 1$, and $z=(x,y)\in \mathcal{Z}$ that

Figures (3)

  • Figure 1: Trajectories of GDA, PPM and PDHG to solve a simple bilinear problem \ref{['eq:bilinear']} with initial solution $(2, 2)$ and step-size $\eta=\sigma=0.2$.
  • Figure 2: Summary of relative impact of PDLP's improvements
  • Figure 3: Number of problems solved for MIP Relaxations (left), LP benchmark (middle), and Netlib (right) datasets within one hour time limit.

Theorems & Definitions (11)

  • Theorem 1: Average iterate convergence of PDHG chambolle2011firstchambolle2016ergodiclu2023unified
  • Theorem 2: Last iterate convergence of PDHG lu2022infimal
  • Definition 1: Normalized duality gap applegate2022faster
  • Proposition 1: applegate2022faster
  • Theorem 3: lu2022infimal
  • Theorem 4: applegate2022faster
  • Definition 2
  • Theorem 5: Lower complexity bound applegate2022faster
  • Definition 3
  • Theorem 6: Behaviors of PDHG for infeasible LP applegate2021infeasibility
  • ...and 1 more