Consider a smooth function $f:\mathbb{R}^n\to\mathbb{R}^m$ with $n > m$ and the level set $$ L = \{x\in\mathbb{R}^n\,:\, f(x) = 0\}. $$ Consider $\epsilon > 0$ fixed. Given a point $x\in\mathbb{R}^n$, I would like to find an expression for a point $y(x)$ such that:
- $y(x)$ is "along" the Jacobian: $$ y(x) = x + J_f(x)^\top a \qquad a\in\mathbb{R}^m $$
- $y(x)$ satisfies $\|f(y(x))\| \leq \epsilon$
- $y(x)$ maximises the distance $\|y(x) - x\|$.
Attempted Solution
I basically need to solve the following problem $$ \max_{a\in\mathbb{R}^m} \|y(x) - x\|^2 \qquad \text{such that} \qquad \|f(y(x))\|\leq \epsilon $$ I tried constructing the Lagrangian $$ \|y(x) - x\|^2 + \lambda(\epsilon^2 - \|f(y(x))\|^2) $$