Numerically find the minimum distance between given point and the curve of given function.

Question

Let $f\colon\Bbb{R}^n\to\Bbb{R}$ and $\mathbf{x}_0\in\Bbb{R}^n$. How could I (numerically) find the minimum Euclidean distance between the curve $f(\mathbf{x})=0$ and $\mathbf{x}_0$, granted that $f$ is given analytically. I am looking for an numerical solution in $C$ programming language.

Apologies if this question does not fit the site's requirements, but I think I would have more luck in MathSE, rather than in StackOverflow.

Thanks in advance!

Answer 1

This is an optimization problem with on equality constraint. By the method of Lagrange multipliers, you have to consider the function $$ L(x,\lambda)=\frac12\|x-x_0\|^2 + λ·f(x) $$ and find critical points for it. These satisfy $$ 0=f(x)\\ 0=\frac{\partial L}{\partial x}=x-x_0-λ·\nabla f(x) $$ which you can try to solve via Newtons method.