Suppose we have the following scheme:
$x_{in} \rightarrow f(x_{in}) = y \rightarrow g(y) = x_{out}$ Now we are able to measure $x_{in}, y$ and $x_{out}$ (all set/measured $N$ times) besides that the function $f(x)$ is known (but not easy computable) and unfortunately $g(y)$ is unknown.

The problem is here is about minimizing $||\vec{x}_{in} - \vec{x}_{out}||$, with the constraint that $abs(x_{in}-x_{out}) < 0.02x_{in}$. We are able to modify the input of $g(y)$ and set a new $y$ as $y_{new} = a\times y + b$

Problem:
$\min_{a,b}||\vec{x}_{in} - \vec{x}_{out} || $ s.t. $abs(x_{in,n} - g(y_{new,n})) < 0.02x_{in,n}$ $\forall n \in \{1,...,N\}$
Where $x_{out,n} = g(y_{new,n}) = g(a\times y_{n} + b) \geq 0, x_{in,n} \geq 0$

Now I've read some things about

• Simulated Annealing
• Black box optimization
• Surrogate modelling

But I can't get quite my finger on it whether I'm searching in the right direction and if there even are solutions/algorithms to solve this sort of problem.

Is there someone who can give me some good references on this type of problems or maybe possiblities to transform it such that the optimal values of $a$ and $b$ are found. There is no real need to find the function $g(y)$.