Global or local Minimum?

[Mohamed Elsayed]

Dear Nicolas, 

I have a question regarding the optimality of the solution reached via Riemannian Manifold Optimization.
How can I ensure that the objective F(x), where x lies on a certain manifold, is conversed to a global minimum?

[Nicolas Boumal]

In general, this is very difficult to do.

[Mohamed Elsayed]

Then, the obtained is local minimum? If so, I found the authors in the paper below said that the optimization algorithm using manifold achieves a "near-global local minima" without any proof. Is this claim true? or is it not a general rule?

"F. Liu, C. Masouros, A. Li, H. Sun and L. Hanzo, "MU-MIMO Communications With MIMO Radar: From Co-Existence to Joint Transmission," in IEEE Transactions on Wireless Communications, vol. 17, no. 4, pp. 2755-2770, April 2018, doi: 10.1109/TWC.2018.2803045." 

[Nicolas Boumal]

I do not know this paper. Two quick points:

 (a) in general, there is no guarantee that optimization algorithms such as the one mentioned there converge to local minimizers (although we certainly do expect them to do so in most cases), so the claim should be accompanied by a mathematical argument (and there does not seem to be one).

 (b) the term "near-global" is not defined, so I suppose that part of the claim is neither incorrect nor informative.

-- To re-iterate, I am not familiar with that paper, I only looked at the claim you highlighted. The above is only the prior I would have after reading this bit and before taking a closer look.

[Mohamed Elsayed]

Thank you very much.
Best regards.

--
http://www.manopt.org
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