Is these two optimization problems share the same solution?

Question

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 $X^o$?

Answer 1

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 $t=2$, $X=(1/2)11^T$.