I have a linear programming problem

min $c^T x$

$Ax\leq b$

However, in my problem, $A$ contains also some variables $y$, e.g.

$$A = \begin{pmatrix} y_1 & 4 \\ 3 & y_2 \end{pmatrix}$$

I want to find a value of $y$ such that the solution $x$ of the LP, for that fixed choice of $y$, is positive.

This question is different than normal formulation of parametric LP which typically only involves parameters in the cost vector or right-hand side of the constraints, and I do not want to simply use nonlinear programing. Any good solution?