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Cooper is a toolkit for Lagrangian-based constrained optimization in Pytorch. This library aims to encourage and facilitate the study of constrained optimization problems in machine learning.

Cooper is (almost!) seamlessly integrated with Pytorch and preserves the usual loss -> backward -> step workflow. If you are already familiar with Pytorch, using Cooper will be a breeze! 🙂

Cooper was born out of the need to handle constrained optimization problems for which the loss or constraints are not necessarily “nicely behaved” or “theoretically tractable”, e.g. when no (efficient) projection or proximal are available. Although assumptions of this kind have enabled the development of great Pytorch-based libraries such as CHOP and GeoTorch, they are seldom satisfied in the context of many modern machine learning problems.

Many of the structural design ideas behind Cooper are heavily inspired by the TensorFlow Constrained Optimization (TFCO) library. We highly recommend TFCO for TensorFlow-based projects and will continue to integrate more of TFCO’s features in future releases.

⚠️ This library is under active development. Future API changes might break backward compatibility. ⚠️

Getting Started

Here we consider a simple convex optimization problem to illustrate how to use Cooper. This example is inspired by this StackExchange question:

I am trying to solve the following problem using Pytorch: given a 6-sided die whose average roll is known to be 4.5, what is the maximum entropy distribution for the faces?

import torch
import cooper

class MaximumEntropy(cooper.ConstrainedMinimizationProblem):
    def __init__(self, mean_constraint):
        self.mean_constraint = mean_constraint

    def closure(self, probs):
        # Verify domain of definition of the functions
        assert torch.all(probs >= 0)

        # Negative signed removed since we want to *maximize* the entropy
        entropy = torch.sum(probs * torch.log(probs))

        # Entries of p >= 0 (equiv. -p <= 0)
        ineq_defect = -probs

        # Equality constraints for proper normalization and mean constraint
        mean = torch.sum(torch.tensor(range(1, len(probs) + 1)) * probs)
        eq_defect = torch.stack([torch.sum(probs) - 1, mean - self.mean_constraint])

        return cooper.CMPState(loss=entropy, eq_defect=eq_defect, ineq_defect=ineq_defect)

# Define the problem and formulation
cmp = MaximumEntropy(mean_constraint=4.5)
formulation = cooper.LagrangianFormulation(cmp)

# Define the primal parameters and optimizer
probs = torch.nn.Parameter(torch.rand(6)) # Use a 6-sided die
primal_optimizer = cooper.optim.ExtraSGD([probs], lr=3e-2, momentum=0.7)

# Define the dual optimizer. Note that this optimizer has NOT been fully instantiated
# yet. Cooper takes care of this, once it has initialized the formulation state.
dual_optimizer = cooper.optim.partial_optimizer(cooper.optim.ExtraSGD, lr=9e-3, momentum=0.7)

# Wrap the formulation and both optimizers inside a ConstrainedOptimizer
coop = cooper.ConstrainedOptimizer(formulation, primal_optimizer, dual_optimizer)

# Here is the actual training loop.
# The steps follow closely the `loss -> backward -> step` Pytorch workflow.
for iter_num in range(5000):
    lagrangian = formulation.composite_objective(cmp.closure, probs)
    coop.step(cmp.closure, probs)


Basic Installation

pip install git+

Development Installation

First, clone the repository, navigate to the Cooper root directory and install the package in development mode by running:





pip install --editable ".[dev, tests]"

Editable mode. Matches test environment.


pip install --editable ".[docs]"

Used to re-generate the documentation.


pip install --editable ".[examples]"

Install dependencies for running examples

No Tests

pip install --editable .

Editable mode, without tests.

Package structure

  • cooper - base package

    • problem - abstract class for representing ConstrainedMinimizationProblems (CMPs)

    • constrained_optimizer - torch.optim.Optimizer-like class for handling CMPs

    • lagrangian_formulation - Lagrangian formulation of a CMP

    • multipliers - utility class for Lagrange multipliers

    • optim - aliases for Pytorch optimizers and extra-gradient versions of SGD and Adam

  • tests - unit tests for cooper components

  • tutorials - source code for examples contained in the tutorial gallery


Please read our CONTRIBUTING guide prior to submitting a pull request. We use black for formatting, isort for import sorting, flake8 for linting, and mypy for type checking.

We test all pull requests. We rely on this for reviews, so please make sure any new code is tested. Tests for cooper go in the tests folder in the root of the repository.


Cooper is distributed under an MIT license, as found in the LICENSE file.


Cooper supports the use of extra-gradient style optimizers for solving the min-max Lagrangian problem. We include the implementations of the extra-gradient version of SGD and Adam by Hugo Berard.

We thank Manuel del Verme for insightful discussions during the early stages of this library.

This README follows closely the style of the NeuralCompression repository.

How to cite this work?

If you find Cooper useful in your research, please consider citing it using the snippet below:

    author={Gallego-Posada, Jose and Ramirez, Juan},
    title={{Cooper: a toolkit for Lagrangian-based constrained optimization}},