Riemannian and Euclidian optimization

[Petrichor]

Hello,

So here is my optimisation problem:

The first variable belongs to a manifold, the second one is a vector.

Can I still do Riemannian optimisation?

Thank you!

[Nicolas Boumal]

Yes: linear spaces are manifolds, and products of manifolds are manifolds.

In Manopt, you can use the tool productmanifold as documented here:

https://www.manopt.org/manifolds.html#product-manifolds

[Petrichor]

Thank you, is it advisable to consider that they live in a new manifold (complexe cirlce x the space of my vector), or will the problem be harder? (knowing that my objective function is non convexe)

[Nicolas Boumal]

I did not understand the question, could you reformulate?

The variables do live on a manifold; it just happens to be a product manifold. The fact that the cost function is non-convex also does not affect the situation much, since circles are already non-convex, so regardless of the cost function the optimization problem as a whole is non-convex.

[Petrichor]

I wanted to do a block coordiante, and alternate between an optimization on a Riemannian space (the complexe circle) and an optimization on a Euclidian space (vector space). (So that I can avoid having a product of manifold, but I think that I do not have a good reasoning)