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COIN-OR Interior Point Optimizer IPOPT

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Modeling language for Mathematical Optimization (linear, mixed-integer, conic, semidefinite, nonlinear)

Quick Overview

Ipopt (Interior Point OPTimizer) is an open-source software package for large-scale nonlinear optimization. It is designed to find (local) solutions of mathematical optimization problems and is particularly well-suited for large-scale nonlinear programming problems.

Pros

  • Highly efficient for large-scale nonlinear optimization problems
  • Robust and well-maintained, with regular updates and improvements
  • Supports various programming interfaces (C++, C, Fortran, AMPL, and more)
  • Extensive documentation and active user community

Cons

  • Steep learning curve for beginners in optimization
  • Can be computationally intensive for very large problems
  • Installation process can be complex, especially on Windows systems
  • Limited to finding local optima, not guaranteed to find global optima

Code Examples

  1. Basic usage in C++:
#include "IpIpoptApplication.hpp"
#include "IpTNLP.hpp"

using namespace Ipopt;

// Define your own class MyNLP : public TNLP { ... }

int main(int argc, char** argv) {
    SmartPtr<IpoptApplication> app = IpoptApplicationFactory();
    app->Options()->SetNumericValue("tol", 1e-7);
    app->Options()->SetStringValue("mu_strategy", "adaptive");
    
    SmartPtr<TNLP> mynlp = new MyNLP();
    ApplicationReturnStatus status = app->Initialize();
    status = app->OptimizeTNLP(mynlp);

    return (int) status;
}
  1. Setting options in Python (using cyipopt):
import cyipopt

nlp = cyipopt.Problem(...)
nlp.add_option('max_iter', 1000)
nlp.add_option('tol', 1e-8)
nlp.add_option('print_level', 5)

x, info = nlp.solve()
  1. Defining a simple problem in AMPL:
var x;
var y;

minimize obj: (x - 2)^2 + (y - 1)^2;

subject to con: x^2 + y^2 <= 1;

option solver ipopt;
solve;
display x, y;

Getting Started

To get started with Ipopt:

  1. Download and install Ipopt from the COIN-OR website or use a package manager.
  2. Include the necessary headers in your C++ project: 
    #include "IpIpoptApplication.hpp"
#include "IpTNLP.hpp"
  3. Create an instance of IpoptApplication and set options: 
    SmartPtr<IpoptApplication> app = IpoptApplicationFactory();
app->Options()->SetNumericValue("tol", 1e-7);
  4. Define your optimization problem by creating a class that inherits from TNLP.
  5. Initialize the application and solve your problem: 
    SmartPtr<TNLP> mynlp = new MyNLP();
ApplicationReturnStatus status = app->Initialize();
status = app->OptimizeTNLP(mynlp);

For other languages, consult the Ipopt documentation for specific instructions.

Competitor Comparisons

casadi logo

casadi

1,670

CasADi is a symbolic framework for numeric optimization implementing automatic differentiation in forward and reverse modes on sparse matrix-valued computational graphs. It supports self-contained C-code generation and interfaces state-of-the-art codes such as SUNDIALS, IPOPT etc. It can be used from C++, Python or Matlab/Octave.

Pros of CasADi

  • More versatile, supporting symbolic computation and automatic differentiation
  • Provides a high-level, user-friendly interface for optimization problems
  • Integrates multiple solvers, including Ipopt, offering flexibility

Cons of CasADi

  • Higher overhead due to its more general-purpose nature
  • Steeper learning curve for users familiar with direct solver interfaces
  • May be slower for simple problems that don't require its advanced features

Code Comparison

CasADi:

import casadi as ca

x = ca.SX.sym('x')
y = ca.SX.sym('y')
nlp = {'x': ca.vertcat(x, y), 'f': (1-x)**2 + 100*(y-x**2)**2}
solver = ca.nlpsol('solver', 'ipopt', nlp)
result = solver(x0=[2.5, 3.0])

Ipopt:

#include "IpIpoptApplication.hpp"
#include "MyNLP.hpp"

using namespace Ipopt;
SmartPtr<IpoptApplication> app = IpoptApplicationFactory();
app->Options()->SetNumericValue("tol", 1e-9);
app->Initialize();
SmartPtr<TNLP> mynlp = new MyNLP();
app->OptimizeTNLP(mynlp);
cvxpy logo

cvxpy

5,370

A Python-embedded modeling language for convex optimization problems.

Pros of CVXPY

  • Higher-level abstraction, making it easier to formulate optimization problems
  • Supports a wide range of problem types, including convex and non-convex optimization
  • Automatic problem transformation and solver selection

Cons of CVXPY

  • May be slower for large-scale problems compared to Ipopt
  • Less control over the underlying optimization algorithm
  • Limited to Python, while Ipopt can be used with multiple programming languages

Code Comparison

CVXPY example:

import cvxpy as cp

x = cp.Variable()
objective = cp.Minimize((x - 2)**2)
constraints = [x >= 0]
prob = cp.Problem(objective, constraints)
prob.solve()

Ipopt example (using Python interface):

from ipopt import problem

def objective(x):
    return (x[0] - 2)**2

def gradient(x):
    return [2 * (x[0] - 2)]

nlp = problem(n=1, m=0, problem_obj=MyProblem(), lb=[0], ub=[None])
x, info = nlp.solve([1.0])
scipy logo

scipy

12,986

SciPy library main repository

Pros of SciPy

  • Broader scope: SciPy is a comprehensive scientific computing library, offering a wide range of tools beyond optimization
  • Extensive documentation and community support
  • Seamless integration with other Python scientific libraries like NumPy and Matplotlib

Cons of SciPy

  • Less specialized for large-scale nonlinear optimization problems
  • May not perform as well as Ipopt for certain complex optimization tasks
  • Fewer options for fine-tuning optimization algorithms

Code Comparison

SciPy optimization example:

from scipy.optimize import minimize

def objective(x):
    return x[0]**2 + x[1]**2

result = minimize(objective, [1, 1], method='BFGS')

Ipopt optimization example:

import pyipopt

nlp = pyipopt.create(n, x_L, x_U, m, g_L, g_U, nnzj, nnzh, eval_f, eval_grad_f, eval_g, eval_jac_g)
x, obj = nlp.solve([1.0, 1.0])

Note: The Ipopt example requires more setup and problem definition, which is not shown here for brevity.

jump-dev logo

JuMP.jl

2,230

Modeling language for Mathematical Optimization (linear, mixed-integer, conic, semidefinite, nonlinear)

Pros of JuMP.jl

  • Written in Julia, offering better performance and integration with the Julia ecosystem
  • Provides a high-level modeling language for mathematical optimization problems
  • Supports a wide range of solvers and problem types

Cons of JuMP.jl

  • Requires knowledge of Julia programming language
  • May have a steeper learning curve for users familiar with traditional optimization software

Code Comparison

Ipopt (C++):

#include "IpIpoptApplication.hpp"
#include "IpTNLP.hpp"

// Problem formulation and solver setup

JuMP.jl (Julia):

using JuMP
using Ipopt

model = Model(Ipopt.Optimizer)
@variable(model, x >= 0)
@objective(model, Min, x^2)
optimize!(model)

JuMP.jl provides a more concise and intuitive syntax for defining optimization problems, while Ipopt requires more low-level implementation details. JuMP.jl's integration with various solvers, including Ipopt, allows for easier experimentation and solver switching. However, Ipopt's C++ implementation may offer better performance for specific use cases and provides more direct control over the optimization process.

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README

Ipopt

Introduction

Ipopt (Interior Point OPTimizer, pronounced eye-pea-Opt) is a software package for large-scale nonlinear optimization. It is designed to find (local) solutions of mathematical optimization problems of the form

$$\begin{align} \min_{x \in R^n}\ & f(x), \ \text{s.t.}\ & g_L \le g(x) \le g_U, \ & x_L \le x \le x_U, \end{align}$$

where $f: R^n \rightarrow R$ is the objective function, and $g: R^n \rightarrow R^m$ are the constraint functions. The vectors $g_L$ and $g_U$ denote the lower and upper bounds on the constraints, and the vectors $x_L$ and $x_U$ are the bounds on the variables $x$. The functions $f(x)$ and $g(x)$ can be nonlinear and nonconvex, but should be twice continuously differentiable. Note that equality constraints can be formulated in the above formulation by setting the corresponding components of $g_L$ and $g_U$ to the same value.

Ipopt is part of the COIN-OR Initiative. The Ipopt project webpage is https://github.com/coin-or/Ipopt.

Background

Ipopt is written in C++ and is released as open source code under the Eclipse Public License (EPL). The code has been written by Andreas Wächter and Carl Laird. The COIN-OR project managers for Ipopt are Andreas Wächter und Stefan Vigerske. For a list of all contributors, see the AUTHORS file.

The C++ version has first been released on Aug 26, 2005 as version 3.0.0. The previously released pre-3.0 Fortran version is no longer maintained.

The Ipopt distribution can be used to generate a library that can be linked to one's own C++, C, Fortran, or Java code, as well as a solver executable for the AMPL modeling environment. The package includes an interface to the R programming environment. IPOPT can be used on Linux/UNIX, macOS, and Windows platforms.

As open source software, the source code for Ipopt is provided without charge. You are free to use it, also for commercial purposes. You are also free to modify the source code (with the restriction that you need to make your changes public if you decide to distribute your version in any way, e.g. as an executable); for details see the EPL license. And we are certainly very keen on feedback from users, including contributions!

In order to compile Ipopt, certain third party code is required (such as some linear algebra routines). Those are available under different conditions/licenses.

If you want to learn more about Ipopt, you can find references in the bibliography of the documentation.

For information on projects or papers that use Ipopt, refer to the Ipopt usage stories and papers discussion.

Getting Started

Please consult the detailed installation instructions in the Ipopt documentation. In the following, we only summarize some main points.

Dependencies

Ipopt requires at least one of the following solvers for systems of linear equations:

A fast implementation of BLAS and LAPACK is required by Ipopt.

To build the AMPL interface of Ipopt, the AMPL Solver Library (ASL) is required. It is recommended to use project ThirdParty-ASL to build a ASL library for use by Ipopt.

Build

After installation of dependencies, an Ipopt build and installation follows these 4 steps:

  1. Run ./configure. Use ./configure --help to see available options.

  2. Run make to build the Ipopt libraries. If ASL was made available, also Ipopt executables will be build.

  3. Run make test to test the Ipopt build.

  4. Run make install to install Ipopt (libraries, executables, and header files).

It is suggested to use the same installation prefix (--prefix option of configure) when configuring the build of ThirdParty-ASL, ThirdParty-HSL, ThirdParty-MUMPS, and Ipopt.

Using coinbrew

An alternative to the above steps is to use the coinbrew script from https://coin-or.github.io/coinbrew/. coinbrew automates the download of the source code for ASL, MUMPS, and Ipopt and the sequential build and installation of these three packages.

After obtaining the coinbrew script, run

/path/to/coinbrew fetch Ipopt --no-prompt
/path/to/coinbrew build Ipopt --prefix=/dir/to/install --test --no-prompt --verbosity=3
/path/to/coinbrew install Ipopt --no-prompt

More details on using coinbrew can be found at the instructions on Getting Started with the COIN-OR Optimization Suite.

Precompiled binaries

Some precompiled binaries of Ipopt are also available:

Getting Help

  • Ipopt Documentation with installation instructions, options reference, and more
  • Issue tracking system: If you believe you found a bug in the code, please use the issue tracking system. Please include as much information as possible, and if possible some (ideally simple) example code so that we can reproduce the error.
  • Discussions: ask questions, share ideas, engage with the Ipopt community
  • Mailing list archive (2002-2020): predecessor of Discussions
  • External resources:

Please Cite Us

We provide this program in the hope that it may be useful to others, and we would very much like to hear about your experience with it. If you found it helpful and are using it within our software, we encourage you to add your feedback to the Ipopt usage stories and papers discussion.

Since a lot of time and effort has gone into Ipopt's development, please cite the following publication if you are using Ipopt for your own research:

Versioning

Ipopts version numbers have the form x.y.z. x.y specifies the major and minor version number of Ipopt. An increase in x or y can mean the addition or removal of features, backward-incompatible API changes, etc. Increases in y indicate less severe changes than increases in x. For example, the change from Ipopt 2 to Ipopt 3 came due to a complete rewrite of Ipopt in a different programming language. z specifies the release number of Ipopt. An increase in z usually means bugfixes or additions of small feature. Changes to the API, if any, are done in a backward-compatible way. However, the ABI may changed in a backward-incompatible way.

Source code is organized in branches named stable/x.y. Development towards a next x.y.z release is happening on the stable/x.y branch. The code on branch stable/x.y already caries a x.y.z version number, which can correspond to the next x.y.z release that will be made from this branch. The default branch of the repository is the latest stable/x.y branch, even if x.y is still in beta testing.

An Ipopt x.y.z release is associated with a tag releases/x.y.z on branch stable/x.y. Releases are fixed and don't change.

Additional branches may exist where development of bugfixes or features is taking place. A branch devel may collect development for the next Ipopt x.y version. It will be renamed to stable/x.y when it is considered stable enough for beta testing.

If you want to contribute a bugfix or small feature, please create a pull-request to the latest stable/x.y branch. If you want to contribute a larger feature or something else that changes the API, please create a pull-request to branch devel, if existing, and latest stable/x.y otherwise.