SDPRLayer

Introduction

The SDPRLayer is an optimization layer for semidefinite program relaxations (SDPR) of non-convex polynomial problems designed for certifiable optimization. It is effectively a wrapper for CVXPYLayers that is built specifically for semidefinite programs (SDPs).

When the SDP relaxation is tight to the original problem, the output of the layer and the gradients produced during backpropagation are guaranteed to be correct.

Please see our paper for more details.

Usage

The user is responsible for converting a given polynomial problem to a Quadratically Constrained Quadratic Problem (QCQP). The SDPRLayer can accept the QCQP in homogenized or non-homogenized form. If the problem will be provided in non-homogenized form, the user must set the "homogenize" flag to true during initialization of the layer.

The input vectors/matrices that characterize the cost and constraints are provided to the SDPRLayer on initialization of the layer. The cost must be provided either as a 3-tuple (non-homogenized) or a matrix (homogenized). The constraints are provided as a list of either 3-tuples (non-homogenized) or matrices (homogenized).

Importantly, setting the cost input or members of the constraint list to None flags them as inputs that will require gradients and will change throughout tuning/training. These flagged inputs are then provided to the forward function of the SDPRLayer as a list of PyTorch tensors in the same order that they appeared in the initialization (cost first, then constraints in list order).

The SDPRLayer returns both the vector solution to the original problem and the matrix solution to the SDP relaxation. Both outputs are PyTorch tensors.

Please see _scripts directory for example usages. An example of how to use the tightening tools is given here.

To see an example usage of the SDPRLayer in a full robotics pipeline, see our robot localization example (note that this is not on the main branch).

Installation

The SDPRLayer relies on our PolyMatrix package as well as few modifications to the CVXPYLayers and diffcp libraries. Prior to setting up the Conda environment, clone the following repositories to the parent folder of the SDPRLayer repo:

git clone git@github.com:utiasASRL/poly_matrix.git
git clone git@github.com:holmesco/diffcp.git
git clone git@github.com:holmesco/cvxpylayers.git

Once these repositories have been downloaded, the environment can be created.

conda env create -f environment.yml

The following test functions can be used to verify the installation:

conda activate sdprlayer
pytest _test/test_sdprlayer.py
pytest _test/test_sdpr_poly6.py
pytest _test/test_stereo_tune.py

Multithread Library Issue

We have noticed that there are sometimes issues with the Conda environment due to collisions between the Intel OpenMP ('libiomp') and LLVM OpenMP ('libomp'). This can cause crashing or deadlock. If observed, defining the following environment variable can fix the issue:

export MKL_THREADING_LAYER=GNU

The issue is well known and other workarounds are given here.

Citation

If you use any of the code or ideas from this repo, please cite our work:

@article{holmesSDPRLayers2024,
  title = {{SDPRLayers}: {Certifiable Backpropagation Through Polynomial Optimization Problems} in {Robotics}},
  author = {Holmes, Connor and D{\"u}mbgen, Frederike and Barfoot, Timothy D.},
  year = {2024},
  eprint = {2405.19309},
  publisher = {arXiv},
  journal = {arXiv:2405.19309}
  url={https://arxiv.org/abs/2405.19309}
}

utiasASRL / sdprlayer