Suppose we have the following nonlinear programming problem (NLP):
\begin{equation} \begin{split} \min_{x\in\mathcal{X}}\quad &f(x)\\\\ \mathrm{s.t.}\quad &g(x)\leq0\\\\ &0\leq x\leq1 \end{split} \end{equation}
Where the values of $x$ are between 0 and 1, and the objective and constraint functions are already scaled to be close to 1.
Despite the scalings, I have the problem that the absolute value of the derivative of the objective function $f(x)$ wrt to some of variables is $10^8$ larger than that of other variables. In mathematical terms:
$$\exists i,j, i\neq j; \left\vert \dfrac{df(x)}{dx_i} \right\vert \approx 10^8 \left\vert\dfrac{df(x)}{dx_j}\right\vert $$
The question: Is there a way to scale the derivatives, without losing the other "nice" scalings.
I've been trying to apply some sort of parameter scaling to "stretch" the design space $\mathcal{X}$ by doing e.g. $y=Dx$, where $D$ is a diagonal positive definite matrix and solve for $y$. But it seems that you cannot mess around with one thing without messing another. E.g., in this case the bounds of $y$ would not be between 0 and 1. This may not be bad, but the optimization algorithm I am using works best when everything is scaled "nicely". I have tried non-linear scalings, but I couldn't get it to work (e.g. using a $log$ for a reparametrized variable).