thchang / projection-failure

Demo of failure cases for various QP codes when solving the projection problem from ACM TOMS 1012

Geometric Projection Failures for Remark on ACM TOMS Algorithm 1012

This repository contains demonstrations of failure cases for various quadratic program (QP) solver codes on the projection problem from ACM TOMS 1012.

Problem Details

At certain points in ACM TOMS Algorithm 1012, one needs to compute the projection of an extrapolation point $z$ onto the convex hull of the training points $P$, denoted $CH(P)$.

This is done by solving

$\min_{x\in R^n} || Wx - z ||^2$

such that

$\sum x = 1; x \geq 0$

where $P$ consists of $n$ points in $R^d$, and $W = [p_1 | p_2 | ... | p_n]$ for $p_i \in P$.

Note that this problem can be posed as an equality-constrained nonnegative least-squares problem, which is a special case of a convex QP.

Due to the unusual shape of the matrix $W$, this is a nonstandard use-case for many QP solvers and an extremely difficult problem when the number of training points ($n$) is large. This results in most open-source solvers failing to achieve the necessary accuracy.

We have proposed the usage of the BQPD solver of R. Fletcher to solve this problem, using a dot-product kernel that exploits the structure of the problem. The purpose of this repository is to demonstrate this solution compared to other open-source QP solvers and the SLATEC DWNNLS solver, which was used in the original DelaunaySparse code.

Setup

Before attempting to reproduce our results, take the following steps to install and build dependencies:

• Make sure that you have a copy of gfortran installed on your machine (using the shell command gfortran to compile);
• Fetch the old and new versions of DelaunaySparse into the experiments subdirectory using the commands

  git submodule init
  git submodule update --recursive --remote
  

• Make sure that you have python 3.8 or newer;
• We recommend that you set up a local python virtual environment for reproducibility

  python3 -m venv env
  source env/bin/activate
  

• Install requirements with python -m pip install -r REQUIREMENTS.txt in the base directory;
• Build and test DelaunaySparse's shared object libraries, use the command

  pushd experiments/ds_v1/DelaunaySparse/python && python example.py && popd
  pushd experiments/ds_v2/DelaunaySparse/python && python example.py && popd
  

Reproducing Results

After following the above steps:

• The experiments subdirectory contains the script and dependencies that reproduce our results. To run it: cd experiments && python projection.py
• the data subdirectory contains a list of csv data files containing the test problems used. The first line is the name of the problem, the second line contains the integer values d,n,m in that order. The next n lines contain the data points (defining the convex hull) as comma-separated row-vectors. The last m lines contain the points to project onto the convex hull as comma-separated row-vectors.
• The generative experiments take approximately 50 hours to run on a 10 core 2021 Macbook Pro M1 laptop and can be executed and analyzed with the commands

  pushd experiments
  python run_generative_tests.py
  python process_generative_test_results.py
  popd
  

• Additionally some simple visualizations are provided that help provide intuition into how projection onto the Delaunay mesh works. First the following commands are needed to fix one of the required libraries:

  pushd env/lib/python3.11/site-packages/plotly/grid_objs
  cp grid_objs.py old.grid_objs.py
  python -c "import fileinput, re; [print(re.sub('from collections import MutableSequence', 'from collections.abc import MutableSequence', line), end='') for line in fileinput.input()]" < old.grid_objs.py > grid_objs.py
  popd
  

  Then different explorations can be done with the following:

  python explore_sample_projection.py
  python explore_sample_lattice_extrapolation.py
  python explore_sample_delaunay.py