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:


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



A positive semidefinite programming problem
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.


