Introduction

IPOPT (Interior Point Optimizer, pronounced ``I-P-Opt'') is an open source software package for large-scale nonlinear optimization. It can be used to solve general nonlinear programming problems of the form

$$\min_{x\in{\mathbb{R}}^n} f(x)$$ (1)

$$\; g^L \leq g(x) \leq g^U$$ (2)

$$x^L \leq x \leq x^U,$$ (3)

where $x \in {\mathbb{R}}^n$ are the optimization variables (possibly with lower and upper bounds, $x^L\in({\mathbb{R}}\cup\{-\infty\})^n$ and $x^U\in({\mathbb{R}}\cup\{+\infty\})^n$ ), $f:{\mathbb{R}}^n\longrightarrow{\mathbb{R}}$ is the objective function, and $g:{\mathbb{R}}^n\longrightarrow {\mathbb{R}}^m$ are the general nonlinear constraints. The functions $f(x)$ and $g(x)$ can be linear or nonlinear and convex or non-convex (but should be twice continuously differentiable). The constraints, $g(x)$ , have lower and upper bounds, $g^L\in({\mathbb{R}}\cup\{-\infty\})^m$ and $g^U\in({\mathbb{R}}\cup\{+\infty\})^m$ . Note that equality constraints of the form $g_i(x)=\bar g_i$ can be specified by setting $g^L_{i}=g^U_{i}=\bar g_i$ .