# Enlsip.jl documentation

## Introduction

Package `Enlsip.jl`

is the Julia version of an eponymous Fortran77 library (ENLSIP standing for Easy Nonlinear Least Squares Inequalities Program) designed to solve nonlinear least-squares problems under nonlinear constraints.

The optimization method implemented in `Enlsip.jl`

was conceived in the early 1980s by two Swedish authors named Per Lindström and Per Åke Wedin ^{[LW88]}.

It is designed for solve nonlinear least-squares problems subject to (s.t.) nonlinear constraints, which can be modeled as the following optimization problem:

\[\begin{aligned} \min_{x \in \mathbb{R}^n} \quad & \dfrac{1}{2} \|r(x)\|^2 \\ \text{s.t.} \quad & c_i(x) = 0, \quad i \in \mathcal{E} \\ & c_i(x) \geq 0, \quad i \in \mathcal{I}, \\ \end{aligned}\]

where:

- the residuals $r_i:\mathbb{R}^n\rightarrow\mathbb{R}$ and the constraints $c_i:\mathbb{R}^n\rightarrow\mathbb{R}$ are assumed to be $\mathcal{C}^2$ functions;
- norm $\|\cdot\|$ denotes the Euclidean norm.

Note that box constraints are modeled as general inequality constraints.

## How to install

To add Enlsip, use Julia's package manager by typing the following command inside the REPL:

```
using Pkg
Pkg.add("Enlsip")
```

## How to use

Using `Enlsip.jl`

to solve optimization problems consists in, first, instantiating a model and then call the solver on it.

Details and examples with problems from the literature can be found in the Usage page.

## Description of the algorithm

`Enlsip.jl`

incorporates an iterative method computing a first order critical point of the problem. A brief description of the method and the stopping criteria is given in Method.

## Bug reports and contributions

As this package is a conversion from Fortran77 to Julia, there might be some bugs that we did not encountered yet, so if you think you found one, you can open an issue to report it.

Issues can also be opened to discuss about eventual suggestions of improvement.

- LW88P. Lindström and P.Å. Wedin,
*Gauss-Newton based algorithms for constrained nonlinear least squares problems*, Institute of Information processing, University of Umeå Sweden, 1988.