A positive semidefinite programming problem

2012-01-06T05:49:52Z

Dear all,

I've got a SDP problem as follows:

$\min_{{\bf H}\succeq0}\quad trace({\bf H}) - {\bf a}^{\top}{\bf H}{\bf b}$,

where ${\bf a}$ and ${\bf b}$ are two constant vectors. May somebody tell me how to solve this SDP problem? Thank you very much in advance.

[Added] Thanks for Suvrit to point out some issues. I add one more parameter $\lambda$ (to be pre-defined) and assume ${\bf b}={\bf a}$ for the second term in the above problem as:

$\min_{{\bf H}\succeq0}\quad trace({\bf H}) - \lambda{\bf a}^{\top}{\bf H}{\bf a}$.

Answer by S. Sra for A positive semidefinite programming problem

2012-01-06T11:06:57Z

Your problem has no solution. Here is why.

Let $H$ be $2 \times 2$. Let $a=(2, 0)$ and $b=(1, 0)$. Then, since $a^THb=\mbox{tr}(Hab^T)$, the objective function of your problem can be rewritten as $\mbox{tr}(H-Hab^T) = \mbox{tr}(HC)$, where $$C = I-ab^T = \begin{bmatrix} -1 & 0\\ 0 & 1\end{bmatrix}.$$ Now you can see that if we set \begin{equation*} H=\begin{bmatrix} \alpha & 0\\ 0 & 0 \end{bmatrix}, \end{equation*} then as $\alpha\to\infty$, your objective function goes to $-\infty$. Thus, in general, there is no solution.

Even if you let $a=b$, the same example above shows that there is no solution. You need to restrict $H$ to lie in a compact set.

A positive semidefinite programming problem - MathOverflow

A positive semidefinite programming problem

Answer by S. Sra for A positive semidefinite programming problem