# Welcome to Manopt.jl

For a function $f:\mathcal M → ℝ$ defined on a Riemannian manifold $\mathcal M$ we aim to solve

$$$\operatorname*{argmin}_{x ∈ \mathcal M} f(x),$$$

or in other words: find the point $x$ on the manifold, where $f$ reaches its minimal function value.

Manopt.jl provides a framework for optimization on manifolds. Based on Manopt and MVIRT, both implemented in Matlab, this toolbox provide an easy access to optimization methods on manifolds for Julia, including example data and visualization methods.

If you want to delve right into Manopt.jl check out the Get started: Optimize! tutorial.

Manopt.jl makes it easy to use an algorithm for your favourite manifold as well as a manifold for your favourite algorithm. It already provides many manifolds and algorithms, which can easily be enhanced, for example to record certain data or debout outpuy throughout iterations.

If you use Manopt.jlin your work, please cite the following

@article{Bergmann2022,
Author    = {Ronny Bergmann},
Doi       = {10.21105/joss.03866},
Journal   = {Journal of Open Source Software},
Number    = {70},
Pages     = {3866},
Publisher = {The Open Journal},
Title     = {Manopt.jl: Optimization on Manifolds in {J}ulia},
Volume    = {7},
Year      = {2022},
}

To refer to a certain version or the source code in general we recommend to cite for example

@software{manoptjl-zenodo-mostrecent,
Author = {Ronny Bergmann},
Doi = {10.5281/zenodo.4290905},
Publisher = {Zenodo},
Title = {Manopt.jl},
Year = {2022},
}

for the most recent version or a corresponding version specific DOI, see the list of all versions. Note that both citations are in BibLaTeX format.

## Main Features

### Functions on Manifolds

Several functions are available, implemented on an arbitrary manifold, cost functions, differentials, and gradients as well as proximal maps, but also several jacobi Fields and their adjoints.

### Optimization Algorithms (Solvers)

For every optimization algorithm, a solver is implemented based on a Problem that describes the problem to solve and its Options that set up the solver, store interims values. Together they form a plan.

### Visualization

To visualize and interpret results, Manopt.jl aims to provide both easy plot functions as well as exports. Furthermore a system to get debug during the iterations of an algorithms as well as record capabilities, i.e. to record a specified tuple of values per iteration, most prominently RecordCost and RecordIterate. Take a look at the Get started: Optimize! tutorial how to easily activate this.

## Manifolds

This project is build upon ManifoldsBase.jl, a generic interface to implement manifolds. Certain functions are extended for specific manifolds from Manifolds.jl, but all other manifolds from that package can be used here, too.

The notation in the documentation aims to follow the same notation from these packages.

## Literature

If you want to get started with manifolds, one book is [do Carmo, 1992], and if you want do directly dive into optimization on manifolds, my favourite reference is [Absil, Mahony, Sepulchre, 2008], which is also available online for free.