I am trying to get started with semi-definite programming following Boyd's notes. I was trying to solve a problem as below:
I have an SDP of the following primal form:
$\operatorname{min}\langle C, X\rangle$
subject to $\left\langle A_{i}, X\right\rangle= b_{i}, \quad i=1,2$
$X$ is positive-semidefinite with $$ C=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right], A_{1}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right], A_{2}=\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right], b_{1}=1, b_{2}=0 $$
- How can I check if this satisfies slater's condition?
- How to find the optimal value and optimal X corresponding to this problem?
My attempt: Assume $$ X=\left[\begin{array}{ll} a & b \\ b & c \end{array}\right]$$
$\left\langle A_{1}, X\right\rangle= 1$ implies $a=1$ $\left\langle A_{2}, X\right\rangle= 0$ implies $c=0$
Now the determinant is $-b^{2}$, hence positive-semidefiniteness implies $b=0$
But for strict feasibility, we need X to be positive-definite, which is impossible in this case. Hence slater's condition is not satisfied.