how to deal with this problem using manopt?

[jifa zhang]

I want to deal with this problem using manopt. However, the feasible region is not a manifold owing to (4). So how can i transform this problem? I try to move (4) into the objective function  as a penalty term, but the obtained solution may not satisfy (4). Can you help me? thank you.

[BM]

Hello Jifa,

You could try the approaches suggested in [1901.10000] Simple algorithms for optimization on Riemannian manifolds with constraints (arxiv.org) to deal with the inequality constraint (4).

Regards,

Bamdev

[jifa zhang]

thank you very much, I will read this paper carefully. 

[Ronny Bergmann]

As an add-on to man opt you can find the algorithms at https://github.com/losangle/Optimization-on-manifolds-with-extra-constraints

or if you feel adventurous, they are both available in Julia (https://manoptjl.org/stable/solvers/exact_penalty_method/, https://manoptjl.org/stable/solvers/exact_penalty_method/, disclaimer – I helped implementing these, so I might be biased)

Best

Ronny