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https://github.com/thchang/projection-failure

Demo of failure cases for various QP codes when solving the projection problem from ACM TOMS 1012
https://github.com/thchang/projection-failure

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

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# 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

   ```