Unfortunately, nothing comes to mind at the moment. I certainly believe this is a good topic for research!

Best,

Nicolas

Problems of Information Transmission, Vol. 50, No. 1, pp. 27––56 (2014) have now retracted their claim. Please see their erratum at DOI: 10.1134/S0032946016020083

I apologize if I created confusion! ]]>

That is a good question, and one that comes up quite a bit in applications actually. At the moment, I do not know about any paper that treats the general problem of constrained optimization over manifolds, though such work would certainly interest a lot of people.

A natural framework to try and generalize to manifolds might be the Augmented Lagrangian Method, which is a type of quadratic penalty method (in terms of the constraint violation), but with an extra linear term whose main positive effect (from a numerical standpoint) is that it allows to get good constraint satisfaction without increasing the penalty parameter all the way to infinity.

I had a few home trials with ALM on a sphere the other day (for equality constraints on top of the sphere constraints, where the latter was enforced via Riemannian optimization and the former were (approximately) enforced by ALM), and this worked excellently. I conducted these tests using Manopt, it took just an hour to set it up (http://www.manopt.org).

So in short: I don’t know of any existing paper that does constrained Riemannian optimization (though it is possibly an oversight on my part), but I do think many things should be doable.

]]>I don’t have an example problem to offer at this time, but perhaps I will think of one. What can I say, I like incorporating constraints, of all kinds, into optimization problems.

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