# Transform semi-definite programming problem with linear matrix inequality(LMI) to SDP of standard form

For a SDP problem with LMI, we can write it as: $$minimize \quad c^Tx$$ $$s.t. \quad x_1F_1+x_2F_2+...+x_nF_n+G \preceq 0$$ $$Ax=b$$ where $G,F_1,F_2,...,F_n \in S^k,A \in R^{p \times n}$.

Now I want to transform above form to SDP of standard form: $$minimize \quad tr(CX)$$ $$s.t. \quad tr(A_iX)=b_i,i=1,2,...,p$$ $$X \succeq 0$$ Where $C,A_1,...,A_p \in S^n$.

Suppose $A=\begin{bmatrix} a^T_1 \\ a^T_2 \\ \vdots \\a^T_p\end{bmatrix}$, I tried to diagonalize $c,x,a_i$ such that: $$C=diag(c),X=diag(x),A_i=diag(a_i)$$

But I cannot figure out how satisfying constraint $X \succeq 0$. So what is the trick?

• Why $\preceq$?? Sep 11 '17 at 13:42
• Did you refer to "$\preceq$" in $x_1F_1+x_2F_2+...+x_nF_n+G \preceq 0$? Sep 11 '17 at 13:49
• Yes. Usually, $\succeq$ is used instead. Sep 11 '17 at 13:54
• This might help, but it's not a true duplicate: Convert Semidefinite program forms Sep 11 '17 at 14:12
• Is this an academic exercise or do you think you need to do this to actually solve it? Because if it's the latter, you should stop right now, and use a proper modeling framework like YALMIP or CVX (disclaimer: mine) so you don't have to. Sep 11 '17 at 14:13