Parametric nonlinear programming

I believe, I have a parametric nonlinear optimization problem.
The non-convex constraints depend on some parameters, and I seek a solution that satisfies these constraints for all parameters in a specified continuous range.

Let $ {bf A}(vec{x},vec{theta}) := {bf A} inmathbb{R}^{ntimes n}$ be a matrix-valued function of $vec{x}in mathbb{R}^n$ and $vec{theta}in mathbb{R}^m$ (here $mll n$). Define
$ {bf L}(vec{x}):= {bf L} inmathbb{R}^{ntimes n}$ to be a matrix which is a function of $vec{x}$, and $ vec{F}(vec{x}):= vec{F} $ is a vector-valued function of $vec{x}$.

Main problem: The goal is to find any $vec{x}$ such that
$$
{bf A}(vec{x},vec{theta})vec{y} = vec{F}(vec{x}),

{bf L}(vec{x})vec{y}ge vec{0},
$$
holds for all $vec{theta}$ in some parameter space (ideally, $0<vec{theta}<infty$ or $0<ale vec{theta}le b$ for scalars $a,b$).

For a fixed $vec{theta}$ I am able to solve the problem using nonlinear programming. I have tried discretizing the parameter space and then solving the resulting coupled nonlinear programs, but this only leads to a solution that is valid for the discrete parameter values.

Is there a better way to think about this problem? Or any standard ways to solve it? There is no objective function in the formulation I posed above, and I don’t know if that is of any significance.