Symmetric positive semidefinite, fixed-rank with unit trace manifold

[Mohamed Elsayed]

Dear all,

I have a problem with the " Symmetric positive semidefinite, fixed-rank with unit trace " manifold. The problem is mainly that I want to replace the unit trace of the matrix with a bounded trace value, for example, trace(X)<= 10. Can you help me with this issue?

Best regards.

[Nicolas Boumal]

Hello,

Unfortunately the inequality constraint does not yield a smooth manifold. Here is one way to think about it:  for a vector x in R^n, the constraints ||x|| = 1 defines a sphere, which is a smooth surface; but the constraint ||x|| <= 1 defines a ball: that has a boundary (the sphere), which makes it not smooth.  (Of course, it is still a "manifold with boundary", but Manopt does not have built-in tools for that.)

Best,

Nicolas

[Nicolas Boumal]

To optimize on a ball (rather than a sphere), wen ow have lifts:

https://www.manopt.org/lifts.html#balls-and-disks