2
$\begingroup$

I'm attempting to use some quadratic programming techniques to solve a particular optimization problem and my chosen Objective Function is indefinite.

I've found some texts online which regard splitting the objective function into two components of which one is positive semi-definite and one is negative semi-definite.

All these concerns aside, my question becomes this:

If an nxn matrix Q is indefinite, is there an algorithm which produces matrices A, and B, both nxn such that A and B are both positive semidefinite and Q = A - B?

Thanks,

James

$\endgroup$
3
$\begingroup$

You can just set $A = Q + kI$ where $k$ is larger than all the negative eigenvalues of $Q$, and put $B = kI$.

$\endgroup$
1
$\begingroup$

I suppose that your matrix is symmetric. Note that a symmetric matrix can always be diagonalized.

Say, $PQP^{-1}=D$ where $D$ is diagonal.

You can take $PAP^{-1}$ to be the diagonal matrix consisting of all positive entries in $D$ and $0$ in rest of the places, and $PBP^{-1}=D-PAP^{-1}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.