manifolds subject to equality constraints

[Matthias F]

Dear ManoptTeam,

I optimize a function using manopt (very successfully - thanks for this great lib!) .

The function is defined on a product manifold that is constructed from the Stiefel manifold (orthogonal matrices more precisely) among others.  Now I would like to impose a zero constraint on  elements of the orthogonal matrix. 

I already came across 

Optimization-on-manifolds-with-extra-constraints: The code for the paper <Simple algorithms for optimization on Riemannian manifolds with extra constraints

Would this be the way to go? Or there other direct way of achieving this? My problem is not really big (3x3) and it is a single zero constraint. 

Any hints computional ones  or as regards the literature would be much appreciated ! 

Thanks indeed!

Matthias 

[Ronny Bergmann]

Yes, Changshous repository should work fine, I think, if you load both Manopt and use his code, that is any of the solvers in

https://github.com/losangle/Optimization-on-manifolds-with-extra-constraints/tree/master/solvers

Those are even the codes corresponding to the paper introducing both algorithms

I did not try those exact codes myself, but recreated his results in Manopt.jl when reimplementing the solvers there as well.

Best

Ronny

[Nicolas Boumal]

This constraint might actually define a manifold (to be checked), in which case it could also be implemented with its own retraction etc. as a factory.

From what I understand, you are considering 3x3 orthogonal matrices (let's say with determinant +1), with a single entry forced to zero (without loss of generality, let's say it is the bottom-left entry of the matrix that is zero).

Then:

* the first column is really just a unit-norm vector in the XY plane (a point on the unit circle in that plane).

* the second column is a unit norm vector orthogonal to the first column.

* the third column is entirely determined by the first two, simply as their cross product.

Thus, your set is in direct correspondence with { (u, v) : |u|^2 = 1, |v|^2 = 1, <u, v> = 0 } where u is in the XY plane of R^3 and v is in R^3.

The defining equation is:

0 = h(u, v) = (|u|^2 - 1, |v|^2 - 1, <u, v>)

The differential is:

Dh(u, v)[a, b] = (2<u, a>, 2<v, b>, <a, v> + <u, b>)

To show that the set under consideration is a smooth manifold, it is sufficient to show that for all (u, v) in the set the map Dh(u, v) is surjective, that is: for all z in R³, is it possible to choose a and b (where a in the in XY plane and b is free) in order to have Dh(u, v)[a, b] = z ?

If so, then it should be possible to write a factory file to optimize over that set directly, without constraints (other than being on the smooth manifold). This won't be "flexible" though: if you later need to be able to have more than one entry equal to zero, on larger orthogonal groups, etc., you mind find it easier to just proceed with a constrained-optimization approach.

It could be that there is some trouble specifically when v = (0, 0, 1) or v = (0, 0, -1), but it would require a check.

Best,

Nicolas

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