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? .

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.

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.