I am looking for a method to solve (if it is possible) the following problem
Find the best increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$ fitting the conditions
$$f(x_i)^2 - f(x_j)^2 = c_{ij}, \tag{1}$$ in terms of $\mathcal{l}^2$ norm. Assume that $x_i, x_j,c_{ij}$, with $i,j \in \Omega \subset \mathbb{N},$ are known.
A candidate tool to tackle this is the isotonic regression, but we should know the values of $f(x_i), ~\forall i \in \Omega$, which is not the case. Moreover, the method should be adapted to take the constraints into account.
I have also tried to rewrite the problem as a functional optimization problem by assuming a differentiate function $f$ such that $f'(x) \geq 0$ and, thus, use some tool such as the Euler-Lagrange equation to deal with, but I stuck in the formulation of the problem.
Insted of $(1)$, the problem can be generalized for any conditions $$g_n\left(f(x_i),f(x_j), f(x_k),\ldots\right) = 0,~~n=1,\ldots N.$$ and I am curious to know what kind of mathematical tool can be employed in these problems.
I appreciate any help. Thanks in advance!
We might as well assume the $x_i$ are increasing, otherwise we can relabel them to be so. Then as you only care about the value of $f$ at these points we might as well define $f$ to be linear between the $x_i$. Now if we start counting $i$ from $1$ we have defined $f$ by $f(x_1)$ and the slopes $m_i=\frac {f(x_{i+1})-f(x_i)}{x_{i+1}-x_i}$. You have a constrained multidimensional optimization problem with error function $\sum (c_{ij}-(f(x_i)^2-f(x_j)^2))^2$ where the constraints are $m_i \ge 0$. You can feed it to your favorite minimizer. It seems the only thing special is choosing where $f$ crosses $0$ which might induce some strange behavior of your error function. If that seems to be a problem you could try adding in terms $Am_i^2$ when $m_i \lt 0$ and removing the constraints. If you make $A$ large it will have a similar effect to your constraints but everything will be differentiable. You will then find where $f(x)$ should cross zero and can then impose that and refit.