Welcome to Manopt.jl

For a function $f:\mathcal M → ℝ$ defined on a Riemannian manifold $\mathcal M$ algorithms in this package aim to solve

\[\operatorname*{argmin}_{p ∈ \mathcal M} f(p),\]

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

Manopt.jl provides a framework for optimization on manifolds as well as a Library of optimization algorithms in Julia. It belongs to the “Manopt family”, which includes Manopt (Matlab) and pymanopt.org (Python).

If you want to delve right into Manopt.jl read 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 debug output throughout iterations.

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

    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 cite for example

    Author = {Ronny Bergmann},
    Copyright = {MIT License},
    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

Optimization algorithms (solvers)

For every optimization algorithm, a solver is implemented based on a AbstractManoptProblem that describes the problem to solve and its AbstractManoptSolverState that set up the solver, and stores values that are required between or for the next iteration. Together they form a plan.


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.

Functions on manifolds

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


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, for example to record a specified tuple of values per iteration, most prominently RecordCost and RecordIterate. Take a look at the 🏔️ Get started: optimize. tutorial on how to easily activate this.


If you want to get started with manifolds, one book is [Car92], and if you want do directly dive into optimization on manifolds, good references are [AMS08] and [Bou23], which are both available online for free

P.-A. Absil, R. Mahony and R. Sepulchre. Optimization Algorithms on Matrix Manifolds (Princeton University Press, 2008), open access.
N. Boumal. An Introduction to Optimization on Smooth Manifolds. First Edition (Cambridge University Press, 2023). Homepage to the book: nicolasboumal.net/book/index.html.
M. P. do Carmo. Riemannian Geometry. Mathematics: Theory & Applications (Birkhäuser Boston, Inc., Boston, MA, 1992); p. xiv+300.