IPOpt

Ipopt, short for Interior Point Optimizer, is an open-source software library for large-scale nonlinear optimization. It is designed to solve problems of the form minimize f(x) subject to nonlinear inequality and equality constraints, as well as bound constraints on the decision variables. Ipopt is part of the COIN-OR project and is widely used in both academic research and industry for solving complex optimization problems.

The algorithmic core of Ipopt is a primal-dual interior-point method for nonlinear programming. It constructs and solves a sequence of sparse Newton systems to approach optimality, applying a line search or a filter strategy to ensure global convergence. Ipopt can utilize exact second-order information via the Hessian of the Lagrangian supplied by the user, or it can work with a quasi-Newton approximation such as BFGS. It also supports the use of first-order derivatives when second-order information is unavailable. The solver is designed to handle large, sparse problems efficiently by exploiting structure in the Jacobian and Hessian.

Ipopt provides interfaces to multiple modeling environments and programming languages. It accepts models described in the NL format or via modeling tools that translate to NL, such as AMPL, Pyomo, or JuMP. The core is implemented in C++, with additional bindings and interfaces that connect to various optimization workflows and solvers. It supports integration with a range of sparse linear solvers to handle the Newton systems.

As an open-source project, Ipopt has seen broad community involvement and is used across disciplines that require robust nonlinear optimization capabilities, from engineering and economics to machine learning and operations research.