Code for Nonlinear conjugate gradient methods: worst-case convergence rates via computer-assisted analyses

Usage • Language and solver installation • Citing • Reporting issues • Contact

The code in this repository can be used to reproduce and verify the results from the work:

Shuvomoy Das Gupta, Robert M. Freund, Xu Andy Sun, Adrien B. Taylor, "Nonlinear conjugate gradient methods: worst-case convergence rates via computer-assisted analyses", Mathematical Programming, 2024.

A preprint of the work is available on arXiv here.

Usage

The repository contains three folders: Code_for_Pseudogradient_Gradient_Ratio, Code_for_NCG_PEP, and Symbolic_Verifications.

Quick summary of the folders. If you are interested in verifying or reproducing the numerical results in the paper, then only 2. Using_the_saved_datasets_Julia.ipynb and 3. PEPIt_verification_Python.ipynb in the folders Code_for_Pseudogradient_Gradient_Ratio and Code_for_NCG_PEP will suffice. For verifying the symbolic calculations, the files Verify_PRP.ipynb and Verify_FR.ipynb in the folder Symbolic_Verifications will suffice. Please see the Language and solver installation section below for how to install the necessary solvers and languages.

The detailed usage of the files in the aforementioned folders is as follows.

Code_for_Pseudogradient_Gradient_Ratio

This folder has the following main contents.

• Ratio_of_pseudo_gradient_and_gradient.jl This file contains the Julia code to solve the performance estimation problem to compute the worst-case search directions (problem in Section 2.1) for Fletcher-Reeves (FR) and Polak-Ribière-Polyak (PRP), which are two famous nonlinear conjugate gradient methods.
• 1. Example_Julia.ipynb This Jupyter notebook is written in Julia programming language and it shows how to use Ratio_of_pseudo_gradient_and_gradient.jl to solve a specific instance of the problem in Section 2.1.
• 2. Using_the_saved_datasets_Julia.ipynb Using_the_saved_datasets_Julia.ipynb) This Jupyter notebook is written in Julia and it shows how to use the datasets saved in the subfolder Saved_Output_Files to reproduce Figure 5 and Figure 6 in Appendix A. It also shows how to extract relevant data to verify the results using the open-source Python package PEPit.
• 3. PEPIt_verification_Python.ipynb This Jupyter notebook is written in Python and it shows how to verify the lower-bounds associated with the datasets saved in the folder Saved_Output_Files.

Code_for_NCG_PEP

This folder has the following main contents.

• N_point_NCG_PEP.jl This file contains the Julia code to solve the performance estimation problems to compute the worst-case bounds (as defined by the problems and in Section 3.1) for Fletcher-Reeves (FR) and Polak-Ribière-Polyak (PRP).
• 1. Example_Julia.ipynb This Jupyter notebook is written in Julia and it shows how to use N_point_NCG_PEP.jl to solve a specific instances of and and in Section 3.1.
• 2. Using_the_saved_datasets_Julia.ipynb This Jupyter notebook is written in Julia and it shows how to use the datasets saved in the subfolder Saved_Output_Files to reproduce Figures 2, 3, and 4 in Sections 3.2 and 3.3. It also shows how to extract relevant data to verify the results using the open-source Python package PEPit.
• 3. PEPIt_verification_Python.ipynb This Jupyter notebook is written in Python and it shows how to verify the lower-bounds associated with the datasets saved in the folder Saved_Output_Files.

Symbolic_Verifications

This folder contains the following contents.

The following files written in Python can be used to verify the symbolic computations.

• Verify_PRP.ipynb This Jupyter notebook is written in Python using the Sympy package and is used to verify the algebra for establishing the worst-case bound for PRP (Section 2.2.1 of the paper).
• Verify_FR.ipynb This Jupyter notebook is written in Python using the Sympy package and is used to verify the algebra for establishing the worst-case bound for FR (Section 2.2.2 of the paper).

Alternatively, the following files written in the Wolfram language can be used to verify the symbolic computations as well.

• Verify_PRP_Wolfram_Language.ipynb This Jupyter notebook is written in the Wolfram language and is used to verify the algebra for establishing the worst-case bound for PRP (Section 2.2.1 of the paper).

• Verify_FR_Wolfram_Language.ipynb This Jupyter notebook is written in the Wolfram language and is used to verify the algebra for establishing the worst-case bound for FR (Section 2.2.2 of the paper).

Language and solver installation

Installation instructions for different languages and solvers can be found in the links below.

For numerical verifications

If you are interested in verifying or reproducing the results, then only 2. Using_the_saved_datasets_Julia.ipynb and 3. PEPIt_verification_Python.ipynb in the folders Code_for_Pseudogradient_Gradient_Ratio and Code_for_NCG_PEP will suffice. To run those files on your computer please install the following.

• Install Julia

• Install Jupyter to open the Jupyter notebooks ( .ipynb files)

  • Install the kernels for the Jupyter notebooks (required to run the .ipynb files)
  • Install IJulia (required to run Julia in the .ipynb files)
  • Install PEPit (required to verify the results)

• Install the optimization solver

  • Install Mosek

Optional

• To run the .jl files and the files named 1. Example_Julia.ipynb in Code_for_Pseudogradient_Gradient_Ratio and Code_for_NCG_PEP , please additionally install the following solvers:

  • Install Gurobi
  • Install KNITRO

• Please install following Julia packages by pressing ] and then running the following command in Julia REPL:

  add SCS, JuMP, MosekTools, Mosek, LinearAlgebra,  OffsetArrays,  Gurobi, Ipopt, JLD2, Distributions, OrderedCollections, BenchmarkTools, DiffOpt, SparseArrays, KNITRO

For platform specific instructions for these solvers, please see https://github.com/jump-dev/Gurobi.jl, https://github.com/jump-dev/KNITRO.jl, https://github.com/jump-dev/MosekTools.jl, and https://github.com/MOSEK/Mosek.jl.

For symbolic verifications

• Install Sympy (required to verify the symbolic calculations in the files Verify_PRP.ipynb and Verify_FR.ipynb )
• Install Wolfram language (required to verify the symbolic calculations in the files Verify_PRP_Wolfram_Language.ipynb and Verify_FR_Wolfram_Language.ipynb )

Citing

If you find the paper useful, we request you to cite the following paper.

@article{dasgupta2024NCGPEP,
author={Das Gupta, Shuvomoy and Freund, Robert M. and Sun, Xu Andy and Taylor, Adrien},
title={Nonlinear conjugate gradient methods: worst-case convergence rates via computer-assisted analyses},
journal={Mathematical Programming},
year={2024},
month={Aug},
day={22},
issn={1436-4646},
doi={10.1007/s10107-024-02127-7},
url={https://doi.org/10.1007/s10107-024-02127-7}
}

Reporting issues

Please report any issues via the Github issue tracker. All types of issues are welcome including bug reports, feature requests, implementation for a specific research problem and so on. Also, please feel free to send an email 📧 to [email protected] or [email protected] if you want to say hi 🚀!