I'm trying to understand how to express a simple LP in the standard semidefinite programming (SDP) form.
In particular, consider the following linear program: $$ \begin{aligned} \min_{x_1, x_2} \;& -2x_1 - x_2 \\ \text{s.t. } & \begin{cases} -x_1 + x_2 \le 1,\\ x_1 - 2x_2 \le 2,\\ x_1 \ge 0,\\ x_2 \ge 0. \end{cases} \end{aligned} $$ I know that a general SDP can be written as:
As $F_i$ symmetric matrix and ⪯0 (non positive definitive)
How can this LP be represented in that matrix form?
What I know, that I first should do: $$ \begin{aligned} \min_{x_1, x_2} \;& -2x_1 - x_2 \\ \text{s.t. } & \begin{cases} -x_1 + x_2 - 1 \le 0,\\ x_1 - 2x_2 - 2 \le 0,\\ -x_1 \le 0,\\ -x_2 \le 0. \end{cases} \end{aligned} $$ So, from the form: $$ \min_x \; c^T x \quad \text{s.t. } \quad F(x) = F_0 + \sum_i x_i F_i \preceq 0. $$
Sadly my professor didn't explain this very well.. I tried to learn using google, but I don't think I understand how to do it... I created 4 matrices of the diagonal, but I need to have it summed or something like that.
