Problem with nonlinear lmi in cvx

I have a problem with this situation:

is a nonlinear lmi (P is symmetric an unknown, L is unknown variable) to convert it to linear LMI,
I choose L^T PL=rho1 and PL=rho2 but cvx running doesn’t work suitable, because I must multiply of two convex variables in constraint, if I ignore it, my problem does not work what do I do? .

sorry new image is correct

You can make this appear linear in rho1, rho2, and P, but the constraints on rho1 and rho2 would be nonlinear and non-convex…

This nonlinear matrix inequality appears non-convex to me unless you have some clever reformulation or relaxation into an LMI, which your introduction of rho1 and rho2 does not accomplish.

As with your previous question, please read Why isn't CVX accepting my model? READ THIS FIRST!

By the way, it is a nonlinear matrix inequality, not a nonlinear LMI. The “L” in LMI stands for linear, which this is not. CVX can handle LMI’s, but not nonlinear matrix inequalities.