Does the inclusion of linear constraints in the problem preserve the manifold nature of the feasible space?

[Tiebin Mi]

Hi, I've noticed that manopt is incredibly powerful. However, I'm curious: if additional constraints are introduced, such as linear constraints, does the feasible set is still a manifold structure? I think the answer is likely no. How can I expand this function? What are the primary challenges? Sorry for boring. 

[Nicolas Boumal]

Hello,

Indeed, in general, the intersection of two manifolds is not a manifold, even if one of them is linear.

Stated differently: even if your initial search space is a smooth manifold, adding constraints to your problem may break that property.

Manopt only handles smooth manifolds "as is", but you can always try to handle extra constraints in different ways.

See for example this paper (among many): https://arxiv.org/abs/1901.10000.

Best,

Nicolas