Hello all,


I am dealing with some SDP optimization, and I come across the following problem.


The optimization problem is given by


\begin{aligned}
&\operatorname\*{min}\_{t\_1,\ldots,t\_m,X}\ \sum t\_i \\\
&\;\text{s.t.}\ \ 
\begin{bmatrix}
 A(X)+B & e\_i \\\
 e\_i^T & t\_i 
\end{bmatrix} \succeq 0,\ \ \ i=1,\ldots,m\ \ \ \ \ \\\
&\ \ \ \ \ \ \ \;X\succeq 0,\ \ \ \ \ \ \ \text{trace}(X) = C
\end{aligned}


where $X\in\mathbb{R}^n$, $(t\_i)\_{i=1}^m$ are postive scalar, and $e\_i$ is the unit vector in $\mathbb{R}^m$. It is known that the both matrices $A$ (for every $X$) and $B$ are positive definite. Also, it's known that $A(X)$ is a linear function in $X$, and hence this optimization problem is convex. Note that $B$ is independent on $X$ and on $(t\_i)\_i$.


My question: is it true that the above minimization has the same optimal solution, $X^o$, which the following minimization problem has ("just" ignoring $B$)


\begin{aligned}
&\operatorname\*{min}\_{t\_1,\ldots,t\_m,X}\ \sum t\_i \\\
&\;\text{s.t.}\ \ 
\begin{bmatrix}
 A(X) & e\_i \\\
 e\_i^T & t\_i 
\end{bmatrix} \succeq 0,\ \ \ i=1,\ldots,m\ \ \ \ \ \\\
&\ \ \ \ \ \ \ \;X\succeq 0,\ \ \ \ \ \ \ \text{trace}(X) = C.
\end{aligned}


If I am not wrong, a feasible solution of the second problem is also feasible for the first one. But, are these two problems has the same solution $X^o$?



Is these two optimization problems share the same solution?
I don't see any reason why the two problems should have the same optimal solution.


Consider both SDPs with $A(X)=X$, $B=[25\ \ {-10};\ {-10}\ \ 20]$, and $C=1$, and $m=1$. Let the matrices be $2\times 2$.


Then, the first SDP has the optimal solution
\begin{equation\*}
t = .1502,\qquad X = \begin{bmatrix} 0.4257 & 0.4945\\\\
 0.4945 & 0.5743
\end{bmatrix}
\end{equation\*} 


While the second SDP has the optimal solution $t=2$, $X=(1/2)11^T$.


