Strickly positive and orthogonal constraints on variables

[grandowife]

Hi there,

Recently I come across a problem that constrains its variable, e.g., matrix U,  non-negative and orthogonality.

I have look for the tutorial of manopt，and found that the Grassmannfactory/stiefelfactory which can  meet the orthogonality

and the Matrices with strictly positive entries which is able to hold non-negative.

However, I am wondering how to use them both, may I have an example of this?

Best

qiuying Shi

[Nicolas Boumal]

Hello,

In general, combining constraints (that is, taking the intersection of two manifolds) does not yield a manifold, which is why there is no general tool to do this in manopt.

You could try two things:

1) Try MADMM-style algorithms, where you optimize over both X in Stiefel and Y positive, and impose the linear equality constraints X = Y.

2) Try an augmented Lagrangian approach as in this paper [https://arxiv.org/abs/1901.10000] : it reduces optimization over Stiefel with non-negativity constraints to a sequence of unconstrained optimization problems on Stiefel.

Best,

Nicolas