Find a set of $K_1, K_2,....K_n$ such that $K_1x_1 +K_2x_2+....+K_n x_n \geq f(x_1,x_2,...,x_n)$ I would like to ask how can I find a set of $K_i$, i=1,....,n such that
$K_1x_1 +K_2x_2+....+K_n x_n \geq f(x_1,x_2,...,x_n)$
and the volume of the regions lying between  $K_1x_1 +K_2x_2+....+K_n x_n$ and $f(x)$ is minimum,
where $f(x_1, x_2, ..., x_n)$ is a known nonlinear Lipschitz continuous function such that $f(0)=0$ and $x \in \mathcal{X}$, where $\mathcal{X}$ is a compact set.
For example, given $x_1 \in [-5,5], x_2 \in [-3,3]$ how can I find $K_1, K_2$ such that
$K_1x_1 +K_2x_2 \geq f(x_1,x_2)$.
Thank you for your suggestions.
I am not an expert on Maths, so I would appreciate if you can be patient and explain a little more details to me.

Under some conditions in the set $\mathcal{X}$ and $f(x)$. You can write your problem as $$\left\{\begin{array}{rl}
 \min_{k\in\mathbb{R}^n}& \int\cdots\int_{\mathcal{X}}(k\cdot x-f(x))dx_1\cdots dx_n \\\\\text{subject to} & k\cdot x-f(x)\geq 0, \text{ to } x\in \mathcal{X} 
\end{array}\right.$$ in which $k\cdot x=k_1x_1 +k_2x_2+....+k_n x_n$.
You can, for a moment, apply the constraints just to a finite subset of $\mathcal{X}$, let's say $k\cdot y_i-f(y_i)\geq 0$, to $n$ vectors $y_i\in \mathcal{X}$. Then you has a linear problem in $k$ that you can solve numerically, if the solution exists.
Please see also Lagrange Multipliers and Constrained Optimization.
Perhaps it helps you.
You can find more seaaching for "\(\min_x (Ax-b)\) " on SearchOnMath, for instance.
