Simple quadratic optimization problem

[nfreris2]

Let P^ + ,P^ - ,I,Q \in R^{n\times n}, K\in R^{n\times 1}, M\in R^{1 \times n}, and assume that Q is positive definite, P^ - is positive semidefinite whence (MP^ - M^T + Q)^{ - 1} exists (where T denotes transpose).

In what sense does K = P^ - M^T(MP^ - M^T + Q)^{ - 1} minimize the quadratic expression P^ + : = (I - KM)P^ - (I - KM)^T + KQK^T, over K?
Is this minimization of P^ + over all vectors K with respect to the usual ordering for positive semidefinite matrices A\le B iff B - Ais positive semidefinite?

Next consider the extension P^ + : = (I - KAM)P^ - (I - KAM)^T + KAQA^TK^T, where A\in R^{n\times 1}, K\in R^{n\times n} and K diagonal, where all other matrices are as above.
What is the minimum over K (with respect to the previous ordering or something )??

Any help will be deeply appreciated.

 

[maze]

I think maybe there is a typo, because if K is a column vector (in Rnx1) and Q a square matrix then KQK* is meaningless, you can't do that matrix multiplication. However, if you meant that K is a row vector, then that implies KQK* is a scalar, so it cannot possibly add to anything to equal the nxn matrix P+.

 

[nfrer]

That is correct Q > 0, a scalar.

Please help with this.

 

[maze]

What are your thoughts on the problem so far?