Solving equations with composed constrained functions

I was lately curious about an iterative approach that would solve maths equations containing composed functions with contraints.

For example, if I have the following equation:

$$ f(g(h(w))) = 0 text{, with } w = begin{pmatrix}
a_{11} & 0 & ldots & a_{1n}\
0 & a_{22} & ldots & a_{2n}\
vdots & vdots & ddots & vdots\
0 & 0 &ldots & a_{nn}
end{pmatrix}
$$

Along additional constraints on the $3$ functions like
$ f < g $;
$ h > 2 * g $; and
$ f, g,h $ not constant

The goal is to find the $3$ functions expressions given a specific matrix $w$ and the constraints.

What approach would be the most convenient to find a solution or solutions to this problem ? I was personally thinking about using reinforcement learning (machine learning) where each time a solutions is chosen, a positive or negative reward will be attributed to the solution generator.

Thank you

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