NLPModels.jl documentation

This package provides general guidelines to represent optimization problems in Julia and a standardized API to evaluate the functions and their derivatives. The main objective is to be able to rely on that API when designing optimization solvers in Julia.

Introduction

The general form of the optimization problem is

\[\begin{aligned} \min \quad & f(x) \\ & c_i(x) = 0, \quad i \in E, \\ & c_{L_i} \leq c_i(x) \leq c_{U_i}, \quad i \in I, \\ & \ell \leq x \leq u, \end{aligned}\]

where $f:\mathbb{R}^n\rightarrow\mathbb{R}$, $c:\mathbb{R}^n\rightarrow\mathbb{R}^m$, $E\cup I = \{1,2,\dots,m\}$, $E\cap I = \emptyset$, and $c_{L_i}, c_{U_i}, \ell_j, u_j \in \mathbb{R}\cup\{\pm\infty\}$ for $i = 1,\dots,m$ and $j = 1,\dots,n$.

For computational reasons, we write

\[\begin{aligned} \min \quad & f(x) \\ & c_L \leq c(x) \leq c_U \\ & \ell \leq x \leq u, \end{aligned}\]

defining $c_{L_i} = c_{U_i}$ for all $i \in E$. The Lagrangian of this problem is defined as

\[L(x,\lambda,z^L,z^U;\sigma) = \sigma f(x) + c(x)^T\lambda + \sum_{i=1}^n z_i^L(x_i-l_i) + \sum_{i=1}^nz_i^U(u_i-x_i),\]

where $\sigma$ is a scaling parameter included for computational reasons. Notice that, for the Hessian, the variables $z^L$ and $z^U$ are not used.

Optimization problems are represented by an instance/subtype of AbstractNLPModel. Such instances are composed of

• an instance of NLPModelMeta, which provides information about the problem, including the number of variables, constraints, bounds on the variables, etc.
• other data specific to the provenance of the problem.

Nonlinear Least Squares

A special type of NLPModels are the NLSModels, i.e., Nonlinear Least Squares models. In these problems, the function $f(x)$ is given by $\tfrac{1}{2}\Vert F(x)\Vert^2$, where $F$ is referred as the residual function. The individual value of $F$, as well as of its derivatives, is also available.

Tools

There are a few tools to use on NLPModels, for instance to query whether the problem is constrained or not, and to get the number of function evaluations. See Tools.

Install

Install NLPModels.jl with the following command.

pkg> add NLPModels

This will enable the use of the API and the tools described here, and it allows the creation of a manually written model. Look into Models for more information on that subject, and on a list of packages implementing ready-to-use models.

Usage

See the Models, the Tools, or the API.

Attributes

NLPModelMeta objects have the following attributes:

License

This content is released under the MPL2.0 License.

Contents

• API
• Reference guide
• API for NLSModels
• AbstractNLPModel functions
• AbstractNLSModel
• Internal
• Guidelines for creating models
• Bare minimum
• Expected behaviour
• Show
• Advanced counters
• Advanced tests
• NLPModels.jl documentation
• Introduction
• Nonlinear Least Squares
• Tools
• Install
• Usage
• Attributes
• License
• Contents
• Models
• Packages
• Reference
• Tools
• Functions evaluations
• Querying problem type
• Docs