This problem is resubmitted a second time:
<http://answers.google.com/answers/threadview?id=565636>

I have the folowing problem.
Given hermitian matrix positive semi-definite G (eigenvalues >=0), the
question is to find the hermitian positive semi-definite "Block
Diagonal Matrix" M, which maximize:

        det(I+M*G) with tr(M)<=1.

with I identity matrix.
we can write M as:

    |M1  0 0 0 0 0|
    |0  M2 0 0 0 0|
M = |0  0  M3  0 .|
    |0  0  0 ... .|
    |0  0  ...  Mn|

with Mi hermitian positive semi-definite matrix.
I need a formulation of Mi in function of G or in function of the
eigenvectors and eigenvalues of G (EVD). I have already an iterative
solution but i
need a direct formulation of each Mi.

Thank you.

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Clarification of Question by torrent-ga on 26 Oct 2005 17:22 PDT

Hi mathtalk,

i read your comment about the M diagonal case.
Indeed, it is a good start, evenif your example is simple.
I wonder if the number of m_i is large, if the polynomial solution is
easy to find as you said.
I need a solution as simple as possible. For example m_i is the
solution of a polynom order 40 is not really easy, i think.
You can consider a "direct formulation" a solution like water-filling,
which means that it is not find analytically but which is simple to
find numerically.
Water-filling is less complicated than finding zeros of polynom of high order.
Please continu your idea, and i hope you can precise to me how you can
use the diagonal case to solve the block diagonal case.

Thanks.

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Clarification of Question by torrent-ga on 26 Oct 2005 17:24 PDT

Hi fanfantm-ga,

Thank you for your comment.
I have already a Matlab source of it.
What i need is an analytic solution using as simple as possible an
"iterative algorithm", but not the hole solution. For example
water-filling has a part of the solution an iterative method but not
the hole solution.

Thank you.

Hi, torrent-ga:

I would appreciate any remarks you may have on my solution to the case
where M is required specifically to be diagonal (rather than block
diagonal), posted as a Comment to the earlier Question.  In particular
I'm wondering how close such a solution comes to your criteria for
"direct formulation".

As you may note from my earlier Comments on that thread, the diagonal
M case may be an effective building block for the general block
diagonal M cases.  If this is of interest, I will continue to outline
my thoughts in that vein.

regards, mathtalk-ga

Hi torrent-ga,

I read your comment about already having an "iterative" solution, but
would you be interested in a MATLAB implementation of a routine that
numerically solves your problem based on the use of (convex)
semidefinite optimization techniques ?

For example, the optimal solution for matrix G=[3 2 3;2 6 4;3 6 5]
with a (1,2) block-diagonal M is found to be equal to [9/40 0 0;0 9/20
-3/20;0 -3/20 13/40].

Hi, fanfantm-ga:

Thanks for posting the computational example.

As stated G is not quite symmetric, and if the last two entries of G's
second row were swapped to make it symmetric, then it would not be
positive definite:

  | 4  6 |
  | 6  5 |

is 20 - 36 < 0.  I'm guessing you meant the 6 in the third row to be 4.


regards, mathtalk-ga