Refrence

[Hamide Zebardast]

Hi,

Hope everything goes well,

I am using "trustregions", "steepestdescent", and "rlbfgs" solvers for my problem on "complexcirclefactory" manifold. My optimization problem is based on multiple vectors, so I am using '"productmanifoldK" to construct manifold. I will be so grateful if you introduce    references for each solver in this case. Actually, I want to derive the equations of each algorithm in this case.

Best Regards,

Hamideh

[Ronny Bergmann]

Hi,

for the Riemannian BFGS see https://github.com/NicolasBoumal/manopt/blob/d88fc5365b592df9fab2c02658d6e82e69197736/manopt/solvers/bfgs/rlbfgs.m#L119-L128

for trust regions the documentation page lists 3 references https://www.manopt.org/solver_documentation_trustregions.html

for gradient descent it might also be enough to check chapter 4 in Absil, Mahony, Sepulchre: Optimisation on Matrix Manifolds (Princeton Press, 2008) might be enough?

All of them are for general manifolds, so this includes product manifolds for sure :)

Best regards

Ronny

[Hamide Zebardast]

Thanks a lot for your suggestions. Actually, my question is about the Wolfe conditions in the attached file. I will be so grateful if you answer my question. 

Best Regard

Hamideh

[Ronny Bergmann]

Hm, now we are from references to a specific other question, but ok.

I do not understand all of your notation, I am sorry; also this seems to be a problem (as stated) not on manifolds but constrained on complex matrices? Then any classical textbook (Nocedal & Wright for example) should be fine. In your notation, the X has columns x_1,...,x_M, but they do not depend on i so what is x_m(i) then? The with entry of x_m? but then your matrix would just contain values +1 and -1? That would be tough to optimise since it would be a discrete domain. For the <A,B> I think you mean <a_i, b_i> in the definition of the inner product? 

Concerning Wolfe – you switched the sign in the second, but that should be fine, the rest looks like https://en.wikipedia.org/wiki/Wolfe_conditions, but we are then no linger in the constrained setting?

[Hamide Zebardast]

Thanks a lot.

Actually, x_m  with N elements is the m'th column of the matrix X. The cost function is written as f(x_1,x_2,...x_M), so I considered the constraint on the vectors x_m. The vector x_m has complex elements with unit amplitude (its phase is in (0,2pi)). I have considered the search space as the product of M manifolds, while x_m is in manifold m (each manifold m is an N-dimensional complex circle). 

<A,B>=\sum_{i=1}^M  (a_i,b_i)

[Ronny Bergmann]

Ah, sorry, I forgot the complex part there. The problem when switching from this constraint problem to the unconstraint case is that within the Wolfe conditions you can no longer to a + (since your space is no longer Euclidean, cf. for example https://www.math.fsu.edu/~whuang2/pdf/BFGSNonsmooth_techrep.pdf

[Hamideh Zebardast]

Ok, thanks a lot. I got it. Is it correct to consider the armijo condition to both RBFGS and Riemannian steepest descent? 

Best regards

Hamideh

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[Nicolas Boumal]

Dear Hamideh,

These are more general optimization questions that you can read about in standard textbooks. I would recommend Nocedal & Wright's book for example: they have detailed discussions of line-search algorithms. They work the same on manifolds.

(Also, this is getting far from the original topic of this post, which was about references.)

Best,

Nicolas

[Hamideh Zebardast]

Ok, thanks a lot.

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