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Topic: Problem in matrix theory
Replies: 2   Last Post: Dec 13, 1996 11:47 AM

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 Tobias Ryden Posts: 5 Registered: 12/7/04
Problem in matrix theory
Posted: Dec 11, 1996 10:34 AM

Hello,

Does anybody has a solution or reference to the following problem?

Let G be a symmetric and positive definite matrix, block partitioned
as

[G_11 G_12 G_13 ... G_1n]
[G_21 G_22 .... G_2n]
[... ]
[... ]
[G_n1 G_n2 .... G_nn]

where all blocks are square and equally sized, and G_ii is positive
semi-definite for each i (which might be obvious?)

The problem is to minimize

Q(a) = (\sum_{i=1}^n a_i G_ii)^{-1}

* (\sum_{i,j} a_i a_j G_ij)

* (\sum_{i=1}^n a_i G_ii)^{-1}

over all vectors a={a_i}, i=1,...,n, satisfying a_1+...+a_n=1 and a_i>=0
for each i. The minimization should be done in the sense of "definiteness",
i.e. if a* is the optimal vector and a is any other vector, then Q(a)-Q(a*)
is positive semi-definite.

The middle part of the expression can be viewed as a quadratic form in
the matrix blocks, while the outer parts, that are inverted, are linear
combinations of the diagonal blocks.

The questions are if a vector a* that is optimal in the sense above exists,
and, if so, if there is an algorithm to compute it?

Best wishes,

Tobias Ryden
--
-- Tobias RydÃ©n E-mail: tobias@maths.lth.se
Dept. of Mathematical Statistics Tel: int+46-46 222 4778
Lund University Fax: int+46-46 222 4623
-- Box 118, S-221 00 Lund, Sweden WWW: www.maths.lth.se/matstat

Date Subject Author
12/11/96 Tobias Ryden
12/12/96 Peter Spellucci
12/13/96 Henry Wolkowicz