CVXPYlayers

CVXPYlayers is a Python library for constructing differentiable convex optimization layers in PyTorch, JAX, and MLX using CVXPY. A convex optimization layer solves a parametrized convex optimization problem in the forward pass to produce a solution. It computes the derivative of the solution with respect to the parameters in the backward pass.

CVXPYlayers 1.0 supports GPU acceleration with the Moreau and CuClarabel backends.

This library accompanies our NeurIPS 2019 paper on differentiable convex optimization layers. For an informal introduction to convex optimization layers, see our blog post.

Our package uses CVXPY for specifying parametrized convex optimization problems.

• Installation
• Usage
• Examples
• Contributing
• Projects using CVXPYlayers
• License
• Citing

Installation

Use the package manager pip to install cvxpylayers.

pip install cvxpylayers

Our package includes convex optimization layers for PyTorch, JAX, and MLX; the layers are functionally equivalent. You will need to install PyTorch, JAX, or MLX separately, which can be done by following the instructions on their websites.

CVXPYlayers has the following dependencies:

• Python >= 3.11
• NumPy >= 1.22.4
• CVXPY >= 1.9.0
• diffcp >= 1.1.0

Additionally, install one of the following frameworks:

• PyTorch >= 2.0
• JAX >= 0.4.0
• MLX

GPU-accelerated pathway

For the best performance on CPU and GPU, install Moreau via the moreau extra. Before installing, review the Moreau installation guide for license terms and access requirements.

pip install cvxpylayers[moreau]

For GPU support, install the matching Moreau CUDA wheel (see the Moreau installation guide for details):

pip install "moreau[cuda12]"   # or moreau[cuda13]

As an open-source alternative, you can use CuClarabel for GPU acceleration. This requires installing Julia and several additional packages:

• Julia
• CuClarabel
• juliacall
• cupy
• diffqcp
• lineax from main (e.g., uv add "lineax @ git+https://github.com/patrick-kidger/lineax.git")

Usage

Below are usage examples of our PyTorch and JAX layers. Note that the parametrized convex optimization problems must be constructed in CVXPY, using DPP.

PyTorch

import cvxpy as cp
import torch

from cvxpylayers.torch import CvxpyLayer

n, m = 2, 3
x = cp.Variable(n)
A = cp.Parameter((m, n))
b = cp.Parameter(m)
constraints = [x >= 0]
objective = cp.Minimize(0.5 * cp.pnorm(A @ x - b, p=1))
problem = cp.Problem(objective, constraints)
assert problem.is_dpp()

layer = CvxpyLayer(problem, parameters=[A, b], variables=[x])
A_tch = torch.randn(m, n, requires_grad=True)
b_tch = torch.randn(m, requires_grad=True)

# solve the problem
(solution,) = layer(A_tch, b_tch)

# compute the gradient of the sum of the solution with respect to A, b
solution.sum().backward()

PyTorch on GPU with CuClarabel

import cvxpy as cp
import torch

from cvxpylayers.torch import CvxpyLayer

n, m = 2, 3
x = cp.Variable(n)
A = cp.Parameter((m, n))
b = cp.Parameter(m)
constraints = [x >= 0]
objective = cp.Minimize(0.5 * cp.pnorm(A @ x - b, p=1))
problem = cp.Problem(objective, constraints)
assert problem.is_dpp()

device = torch.device("cuda")
layer = CvxpyLayer(problem, parameters=[A, b], variables=[x], solver=cp.CUCLARABEL).to(device)
A_tch = torch.randn(m, n, requires_grad=True, device=device)
b_tch = torch.randn(m, requires_grad=True, device=device)

# solve the problem
(solution,) = layer(A_tch, b_tch)

# compute the gradient of the sum of the solution with respect to A, b
solution.sum().backward()

JAX

import cvxpy as cp
import jax

from cvxpylayers.jax import CvxpyLayer

n, m = 2, 3
x = cp.Variable(n)
A = cp.Parameter((m, n))
b = cp.Parameter(m)
constraints = [x >= 0]
objective = cp.Minimize(0.5 * cp.pnorm(A @ x - b, p=1))
problem = cp.Problem(objective, constraints)
assert problem.is_dpp()

layer = CvxpyLayer(problem, parameters=[A, b], variables=[x])
key = jax.random.PRNGKey(0)
key, k1, k2 = jax.random.split(key, 3)
A_jax = jax.random.normal(k1, shape=(m, n))
b_jax = jax.random.normal(k2, shape=(m,))

(solution,) = layer(A_jax, b_jax)

# compute the gradient of the summed solution with respect to A, b
dlayer = jax.grad(lambda A, b: sum(layer(A, b)[0]), argnums=[0, 1])
gradA, gradb = dlayer(A_jax, b_jax)

Dual variables

CVXPYlayers can return constraint dual variables (Lagrange multipliers) alongside the primal solution:

import cvxpy as cp
import torch
from cvxpylayers.torch import CvxpyLayer

x = cp.Variable(2)
c = cp.Parameter(2)
b = cp.Parameter()

eq_con = cp.sum(x) == b
prob = cp.Problem(cp.Minimize(c @ x), [eq_con, x >= 0])

# Request both primal and dual variables
layer = CvxpyLayer(prob, parameters=[c, b], variables=[x, eq_con.dual_variables[0]])

c_tch = torch.tensor([1.0, 2.0], requires_grad=True)
b_tch = torch.tensor(1.0, requires_grad=True)

x_star, eq_dual = layer(c_tch, b_tch)

Log-log convex programs

CVXPYlayers can also differentiate through log-log convex programs (LLCPs), which generalize geometric programs. Use the keyword argument gp=True when constructing a CvxpyLayer for an LLCP. Below is a simple usage example

import cvxpy as cp
import torch
from cvxpylayers.torch import CvxpyLayer

x = cp.Variable(pos=True)
y = cp.Variable(pos=True)
z = cp.Variable(pos=True)

a = cp.Parameter(pos=True, value=2.)
b = cp.Parameter(pos=True, value=1.)
c = cp.Parameter(value=0.5)

objective_fn = 1/(x*y*z)
objective = cp.Minimize(objective_fn)
constraints = [a*(x*y + x*z + y*z) <= b, x >= y**c]
problem = cp.Problem(objective, constraints)
assert problem.is_dgp(dpp=True)

layer = CvxpyLayer(problem, parameters=[a, b, c],
                   variables=[x, y, z], gp=True)
a_tch = torch.tensor(a.value, requires_grad=True)
b_tch = torch.tensor(b.value, requires_grad=True)
c_tch = torch.tensor(c.value, requires_grad=True)

x_star, y_star, z_star = layer(a_tch, b_tch, c_tch)
sum_of_solution = x_star + y_star + z_star
sum_of_solution.backward()

Solvers

CVXPYlayers supports multiple solvers including Moreau (recommended), Clarabel, SCS, and CuClarabel.

Passing arguments to the solvers

One can pass arguments to solvers by adding the argument as a key-value pair in the solver_args argument. For example, to increase the tolerance of SCS to 1e-8 one would write:

layer(*parameters, solver_args={"eps": 1e-8})

If SCS is not converging, we highly recommend using the following arguments to SCS:

solver_args={"eps": 1e-8, "max_iters": 10000, "acceleration_lookback": 0}

Examples

Our examples subdirectory contains simple applications of convex optimization layers.

Contributing

Pull requests are welcome. For major changes, please open an issue first to discuss what you would like to change.

Please make sure to update tests as appropriate.

Running tests

CVXPYlayers uses the pytest framework for running tests. To install pytest, run:

pip install pytest

Execute the tests from the main directory of this repository with:

pytest tests/

Projects using CVXPYlayers

Below is a list of projects using CVXPYlayers. If you have used CVXPYlayers in a project, you're welcome to make a PR to add it to this list.

• Learning Convex Optimization Control Policies
• Learning Convex Optimization Models
• DeepDow - Portfolio optimization with deep learning
• NeuroMANCER - PNNL's PyTorch library for constrained optimization, physics-informed system identification, and model predictive control

License

CVXPYlayers carries an Apache 2.0 license.

Citing

If you use CVXPYlayers for research, please cite our accompanying NeurIPS paper:

@inproceedings{cvxpylayers2019,
  author={Agrawal, A. and Amos, B. and Barratt, S. and Boyd, S. and Diamond, S. and Kolter, Z.},
  title={Differentiable Convex Optimization Layers},
  booktitle={Advances in Neural Information Processing Systems},
  year={2019},
}

If you use CVXPYlayers to differentiate through a log-log convex program, please cite the accompanying paper:

@article{agrawal2020differentiating,
  title={Differentiating through log-log convex programs},
  author={Agrawal, Akshay and Boyd, Stephen},
  journal={arXiv},
  archivePrefix={arXiv},
  eprint={2004.12553},
  primaryClass={math.OC},
  year={2020},
}