# Simple quadratic optimization problem

1. Feb 27, 2009

### 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 $$B - A$$is 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.

2. Feb 28, 2009

### 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+.

3. Feb 28, 2009

### nfrer

That is correct $$Q > 0$$, a scalar.

Please help with this.

4. Feb 28, 2009

### maze

What are your thoughts on the problem so far?

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