# Augmented Lagrangian
[](https://pypi.org/project/augmented-lagrangian/)
[](https://pypi.org/project/augmented-lagrangian/)
[](LICENSE)
A Python implementation of the Augmented Lagrangian method for constrained optimization problems.
## Overview
The Augmented Lagrangian method is a powerful technique for solving constrained optimization problems of the form:
```
minimize f(x)
subject to c_i(x) = 0 for i = 1, ..., m
```
This package provides a flexible and easy-to-use implementation that can handle both single and multiple equality constraints.
## Features
- **Multiple backends**: Support for SciPy (BFGS) and PyTorch (SGD) optimization backends
- **Flexible constraint handling**: Support for single or multiple equality constraints
- **Customizable parameters**: Control penalty parameters, tolerances, and iteration limits
- **Convergence monitoring**: Track optimization progress with detailed history
- **Easy-to-use API**: Simple interface for defining objective and constraint functions
- **GPU acceleration**: PyTorch backend supports GPU acceleration when available
## Installation
```bash
# Basic installation (SciPy backend only)
pip install augmented-lagrangian
# With PyTorch backend support
pip install augmented-lagrangian[pytorch]
```
## Quick Start
Here's a simple example of using the Augmented Lagrangian solver:
```python
import numpy as np
from aug_lag import AugmentedLagrangian
# Define objective function: minimize (x1 - 1)^2 + (x2 - 2)^2
def objective(x):
return (x[0] - 1)**2 + (x[1] - 2)**2
# Define constraint: x1 + x2 - 3 = 0
def constraint(x):
return x[0] + x[1] - 3
# Create solver instance
solver = AugmentedLagrangian(
objective_func=objective,
constraint_funcs=constraint,
tolerance=1e-6,
verbose=True
)
# Solve the problem
x0 = np.array([0.0, 0.0]) # Initial guess
result = solver.solve(x0)
print(f"Solution: x = {result['x']}")
print(f"Objective value: {result['fun']}")
print(f"Constraint violation: {result['constraint_violation']}")
```
## API Reference
### AugmentedLagrangian Class
#### Constructor Parameters
- `objective_func`: Function to minimize f(x)
- `constraint_funcs`: Single constraint function or list of constraint functions
- `backend`: Optimization backend - "scipy" (default) or "pytorch"
- `mu_0`: Initial penalty parameter (default: 1.0)
- `tolerance`: Convergence tolerance (default: 1e-6)
- `rho`: Factor to increase penalty parameter (default: 1.5)
- `max_mu`: Maximum penalty parameter value (default: 1000.0)
- `constraint_tolerance`: Tolerance for constraint satisfaction (default: 1e-4)
- `max_outer_iterations`: Maximum outer iterations (default: 20)
- `max_inner_iterations`: Maximum inner iterations per subproblem (default: 50)
- `verbose`: Whether to print optimization progress (default: True)
#### Methods
- `solve(x0, max_outer_iterations=100, tolerance=1e-6)`: Solve the optimization problem
- `set_functions(objective_func, constraint_funcs)`: Set objective and constraint functions
## PyTorch Backend Example
The PyTorch backend uses SGD optimization and supports GPU acceleration:
```python
import numpy as np
from aug_lag import AugmentedLagrangian
# Define objective and constraint functions (same as before)
def objective(x):
return (x[0] - 1)**2 + (x[1] - 2)**2
def constraint(x):
return x[0] + x[1] - 3
# Create solver with PyTorch backend
solver = AugmentedLagrangian(
objective_func=objective,
constraint_funcs=constraint,
backend="pytorch", # Use PyTorch backend
max_inner_iterations=200, # More epochs for SGD
tolerance=1e-6,
verbose=True
)
# Solve the problem
x0 = np.array([0.0, 0.0])
result = solver.solve(x0)
print(f"Solution: x = {result['x']}")
print(f"Backend used: {solver.backend}")
```
**Backend Comparison:**
- **SciPy backend**: Uses BFGS algorithm, typically faster convergence, CPU-only
- **PyTorch backend**: Uses SGD algorithm, supports GPU acceleration, good for large-scale problems
## Multiple Constraints Example
```python
import numpy as np
from augmented_lagrangian import AugmentedLagrangian
# Objective function
def objective(x):
return x[0]**2 + x[1]**2
# Multiple constraints
def constraint1(x):
return x[0] + x[1] - 1
def constraint2(x):
return x[0] - x[1]
# Create solver with multiple constraints
solver = AugmentedLagrangian(
objective_func=objective,
constraint_funcs=[constraint1, constraint2]
)
# Solve
x0 = np.array([0.0, 0.0])
result = solver.solve(x0)
```
## Algorithm Details
The Augmented Lagrangian method combines the objective function with penalty terms for constraint violations:
```
L_A(x, λ, μ) = f(x) - Σ λ_i * c_i(x) + (μ/2) * Σ c_i(x)²
```
Where:
- `f(x)` is the objective function
- `c_i(x)` are the constraint functions
- `λ_i` are the Lagrange multipliers
- `μ` is the penalty parameter
The algorithm iteratively:
1. Minimizes the augmented Lagrangian with respect to x
2. Updates the Lagrange multipliers: λ := λ - μ * c(x)
3. Increases the penalty parameter if needed
4. Repeats until convergence
## Requirements
- Python >= 3.8
- NumPy >= 1.20.0
- SciPy >= 1.7.0
## License
This project is licensed under the MIT License - see the [LICENSE](LICENSE) file for details.
## Contributing
Contributions are welcome! Please feel free to submit a Pull Request.
## Citation
If you use this package in your research, please consider citing:
```bibtex
@software{augmented_lagrangian,
title={Augmented Lagrangian: A Python Implementation},
author={Hongwei Jin},
year={2025},
url={https://github.com/cshjin/augmented-lagrangian}
}
```
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"description": "# Augmented Lagrangian\n\n[](https://pypi.org/project/augmented-lagrangian/)\n[](https://pypi.org/project/augmented-lagrangian/)\n[](LICENSE)\n\n\nA Python implementation of the Augmented Lagrangian method for constrained optimization problems.\n\n## Overview\n\nThe Augmented Lagrangian method is a powerful technique for solving constrained optimization problems of the form:\n\n```\nminimize f(x)\nsubject to c_i(x) = 0 for i = 1, ..., m\n```\n\nThis package provides a flexible and easy-to-use implementation that can handle both single and multiple equality constraints.\n\n## Features\n\n- **Multiple backends**: Support for SciPy (BFGS) and PyTorch (SGD) optimization backends\n- **Flexible constraint handling**: Support for single or multiple equality constraints\n- **Customizable parameters**: Control penalty parameters, tolerances, and iteration limits\n- **Convergence monitoring**: Track optimization progress with detailed history\n- **Easy-to-use API**: Simple interface for defining objective and constraint functions\n- **GPU acceleration**: PyTorch backend supports GPU acceleration when available\n\n## Installation\n\n```bash\n# Basic installation (SciPy backend only)\npip install augmented-lagrangian\n\n# With PyTorch backend support\npip install augmented-lagrangian[pytorch]\n```\n\n## Quick Start\n\nHere's a simple example of using the Augmented Lagrangian solver:\n\n```python\nimport numpy as np\nfrom aug_lag import AugmentedLagrangian\n\n# Define objective function: minimize (x1 - 1)^2 + (x2 - 2)^2\ndef objective(x):\n return (x[0] - 1)**2 + (x[1] - 2)**2\n\n# Define constraint: x1 + x2 - 3 = 0\ndef constraint(x):\n return x[0] + x[1] - 3\n\n# Create solver instance\nsolver = AugmentedLagrangian(\n objective_func=objective,\n constraint_funcs=constraint,\n tolerance=1e-6,\n verbose=True\n)\n\n# Solve the problem\nx0 = np.array([0.0, 0.0]) # Initial guess\nresult = solver.solve(x0)\n\nprint(f\"Solution: x = {result['x']}\")\nprint(f\"Objective value: {result['fun']}\")\nprint(f\"Constraint violation: {result['constraint_violation']}\")\n```\n\n## API Reference\n\n### AugmentedLagrangian Class\n\n#### Constructor Parameters\n\n- `objective_func`: Function to minimize f(x)\n- `constraint_funcs`: Single constraint function or list of constraint functions\n- `backend`: Optimization backend - \"scipy\" (default) or \"pytorch\"\n- `mu_0`: Initial penalty parameter (default: 1.0)\n- `tolerance`: Convergence tolerance (default: 1e-6)\n- `rho`: Factor to increase penalty parameter (default: 1.5)\n- `max_mu`: Maximum penalty parameter value (default: 1000.0)\n- `constraint_tolerance`: Tolerance for constraint satisfaction (default: 1e-4)\n- `max_outer_iterations`: Maximum outer iterations (default: 20)\n- `max_inner_iterations`: Maximum inner iterations per subproblem (default: 50)\n- `verbose`: Whether to print optimization progress (default: True)\n\n#### Methods\n\n- `solve(x0, max_outer_iterations=100, tolerance=1e-6)`: Solve the optimization problem\n- `set_functions(objective_func, constraint_funcs)`: Set objective and constraint functions\n\n## PyTorch Backend Example\n\nThe PyTorch backend uses SGD optimization and supports GPU acceleration:\n\n```python\nimport numpy as np\nfrom aug_lag import AugmentedLagrangian\n\n# Define objective and constraint functions (same as before)\ndef objective(x):\n return (x[0] - 1)**2 + (x[1] - 2)**2\n\ndef constraint(x):\n return x[0] + x[1] - 3\n\n# Create solver with PyTorch backend\nsolver = AugmentedLagrangian(\n objective_func=objective,\n constraint_funcs=constraint,\n backend=\"pytorch\", # Use PyTorch backend\n max_inner_iterations=200, # More epochs for SGD\n tolerance=1e-6,\n verbose=True\n)\n\n# Solve the problem\nx0 = np.array([0.0, 0.0])\nresult = solver.solve(x0)\n\nprint(f\"Solution: x = {result['x']}\")\nprint(f\"Backend used: {solver.backend}\")\n```\n\n**Backend Comparison:**\n- **SciPy backend**: Uses BFGS algorithm, typically faster convergence, CPU-only\n- **PyTorch backend**: Uses SGD algorithm, supports GPU acceleration, good for large-scale problems\n\n## Multiple Constraints Example\n\n```python\nimport numpy as np\nfrom augmented_lagrangian import AugmentedLagrangian\n\n# Objective function\ndef objective(x):\n return x[0]**2 + x[1]**2\n\n# Multiple constraints\ndef constraint1(x):\n return x[0] + x[1] - 1\n\ndef constraint2(x):\n return x[0] - x[1]\n\n# Create solver with multiple constraints\nsolver = AugmentedLagrangian(\n objective_func=objective,\n constraint_funcs=[constraint1, constraint2]\n)\n\n# Solve\nx0 = np.array([0.0, 0.0])\nresult = solver.solve(x0)\n```\n\n## Algorithm Details\n\nThe Augmented Lagrangian method combines the objective function with penalty terms for constraint violations:\n\n```\nL_A(x, \u03bb, \u03bc) = f(x) - \u03a3 \u03bb_i * c_i(x) + (\u03bc/2) * \u03a3 c_i(x)\u00b2\n```\n\nWhere:\n- `f(x)` is the objective function\n- `c_i(x)` are the constraint functions\n- `\u03bb_i` are the Lagrange multipliers\n- `\u03bc` is the penalty parameter\n\nThe algorithm iteratively:\n1. 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