In the wikipedia page of LMI, the standard form is given by
$$A_0+y_1A_1+y_2A_2+\cdots+y_mA_m \succeq 0,$$
where $A_i$ are $m\times m$ symmetric matrices and $y_i$ are real vectors, $i=1,2,\ldots m.$
I have few problems in this LMI standard form that I would like to solve. I know $A_i's$ and want to find if there exist $y_i's$ that satisfy the above inequality. So it is a feasibility problem. Sometimes I require some $y_i$ to take only positive values.
However, when searching for LMI Matlab or CVX solvers, I always find some other forms of LMI, like $A^T P + P A \prec 0$ ex1, which I cannot connect with standard form from wikipedia.
Is it compulsory to rewrite standard form of LMI to be able to solve it? I wonder if there is ready Matlab or cvx programs that can solve standard form.
Example: Given $A\in\mathbb{C}^{n\times n}$, check if there exists $k\in\mathbb{R}$ and $\alpha\in[0,\pi]$, such that
$$(A^*A-I)+\frac{k}{2}(A+A^*)+k\cos\alpha I \succ 0,$$
Let $A_0=A^*A-I$, $A_1=A+A^*$, $A_2=I$, $y_1=\frac{k}{2}$ and $y_2=k\cos\alpha$, then we obtain standard LMI problem
$$A_0+y_1A_1+y_2A_2 \succ 0$$
