We consider the linear program
$$\min_{x \in R^n} \{c^Tx \mid a_1^Tx \le b_1, a_2^Tx \le b_2\}$$
where $c, a_1, a_2 \in \mathbb R^n$ and $b_1, b_2 \in \mathbb R$ are given. Now we need to reformulate this LP as an SDP.
Can someone help with this task? Thank you!
Use the following linear matrix inequality (LMI) instead of the two linear inequalities
$$\begin{bmatrix} b_1 - \mathrm a_1^{\top} \mathrm x & 0\\ 0 & b_2 - \mathrm a_2^{\top} \mathrm x\end{bmatrix} \succeq \mathrm O_2$$
Take a look at Sylvester's criterion for positive semidefiniteness.
You have to put it in the form $$ \min_{x\in \mathbb{R}^n} c^T x $$ $$ s.b. F_0 + \sum_i x_i F_i \succeq 0 $$ where $F_i,\, i=0...$ are symmetric matrices of order $2$.
Take $$ F_0= \begin{pmatrix} b_1 &0 \\ 0 & b_2 \end{pmatrix} $$ and $$ F_i= \begin{pmatrix} -a_{1i} &0 \\ 0 & -a_{2i} \end{pmatrix} $$ where $a_j=(a_{j1},\ldots a_{jn})^T, \; j=1,2$
