Error: The solution of this Sylvester equation does not exist or is not unique.

[Ziyuan Lin]

I've got these messages when optimizing a problem:

Error using lyap (line 69)

The solution of this Sylvester equation does not exist or is not unique.

Error in symfixedrankYYfactory/projection (line 44)

        Omega = lyap(SS, -AS);

Error in symfixedrankYYfactory>@(Y1,Y2,d)projection(Y2,d) (line 96)

M.transp = @(Y1, Y2, d) projection(Y2, d);

Error in productmanifold/transp (line 217)

            v.(elems{i}) = elements.(elems{i}).transp(x1.(elems{i}), ...

Error in conjugategradient (line 274)

        oldgrad = problem.M.transp(x, newx, grad);

My manifold is

elements = struct();

elements.p = symfixedrankYYfactory(5, 2);

elements.alpha = sympositivedefinitefactory(1);

manifold = productmanifold(elements);

I have several runs, each of which is initialized with a different random seed. 

Other runs are OK other than this particular seed. So I think my algorithm is fine.

Any ideas about what was going on? Thank you.

[BM]

Hello Ziyuan,

Strange results, indeed. Could you check whether ptp is becoming rank deficient or not in your case? Also when does the error show up in the iterations, in the beginning or in the middle?

Regards,

Bamdev

[Ziyuan Lin]

Hi BM,

For the second question, it happened in the middle, after 377 iterations, where the cost val=+5.8572e-01 and grad. norm=8.8901e-05.

Best,

Ziyuan

[BM]

Hmm... what about the eigenvalues of ptp? 

[BM]

The reason why I am asking this is that the error comes from Matlab's Lyap which solves the equation SS \Omega + \Omega SS = AS, where SS and AS are given, and we solve for \Omega. Also SS = ptp for our case. 

Lyap throws the error when some eigenvalues of SS or ptp are zeros or close to zeros, http://www.mathworks.nl/help/control/ref/lyap.html

It means that p is becoming rank deficient.

[Nicolas Boumal]

I agree with Bamdev, this error seems to indicate an ill conditioned matrix p.

Did you get a better look into when this happens? Could it be that it makes sense for your problem that the solution would be rank deficient?

The underlying issue is that the Riemannian geometry breaks when we reach rank deficient matrices: there are no tangent spaces anymore, only tangent cones.