Let [tex]P^ + ,P^ - ,I,Q \in R^{n\times n}, K\in R^{n\times 1}, M\in R^{1 \times n}[/tex], and assume that [tex]Q[/tex] is positive definite, [tex]P^ -[/tex] is positive semidefinite whence [tex](MP^ - M^T + Q)^{ - 1}[/tex] exists (where [tex]T[/tex] denotes transpose).(adsbygoogle = window.adsbygoogle || []).push({});

In what sense does [tex]K = P^ - M^T(MP^ - M^T + Q)^{ - 1}[/tex] minimize the quadratic expression [tex]P^ + : = (I - KM)P^ - (I - KM)^T + KQK^T[/tex], over [tex]K[/tex]?

Is this minimization of [tex]P^ +[/tex] over all vectors [tex]K[/tex] with respect to the usual ordering for positive semidefinite matrices [tex]A\le B[/tex] iff [tex]B - A [/tex]is positive semidefinite?

Next consider the extension [tex]P^ + : = (I - KAM)P^ - (I - KAM)^T + KAQA^TK^T[/tex], where [tex]A\in R^{n\times 1}, K\in R^{n\times n}[/tex] and [tex]K[/tex] diagonal, where all other matrices are as above.

What is the minimum over [tex]K[/tex] (with respect to the previous ordering or something )??

Any help will be deeply appreciated.

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# Simple quadratic optimization problem

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