Optimization for non linear function

Question

I have an injective function $f:N\mapsto \mathbb{R}^k$ with $N\subset \mathbb{R}^{k-d}$ open. Let $v_n\in \mathbb{R}^k$ is a sequence of random variable. I'd like to find $\hat x\in N$ such that \begin{equation*} \|f(\hat x)-v_n\| \end{equation*} as small as possible when $\|.\|$ denotes the Euclidean norm.

I know that if $f$ is linear then , i.e., $f(x)=Ax$ then the least square $\hat x=(A^TA)^{-1}A^Tv_n$ is the solution of my problem. Is it correct?

My question is: if $f$ is not necessary linear, how will I solve my problem? Any suggestion? Thank you in advance

Does $F$ have a specified form? For example, known to be quadratic, or is it simply general? — Stella Biderman, Mar 10 '14 at 09:40
No @StellaBiderman, $F$ has no special form. It can be general function — Jlamprong, Mar 10 '14 at 09:44
Thanks @ChristianBlatter for your correction. I've edited it — Jlamprong, Mar 16 '14 at 18:00

score 0 · Answer 1 · answered Mar 10 '14 at 08:12

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The point $\hat x$ where $$x\longmapsto \|f(x)-v\|$$ reaches the minimum (if $\exists$) is the same that the point where $$x\longmapsto \|f(x)-v\|^2=\langle f(x)-v,f(x)-v\rangle$$ reaches the minimum. The difference is that the second function is differentiable and the minimun will be a critical point.

answered Mar 10 '14 at 08:12

Martín-Blas Pérez Pinilla

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Thank @Martin-Blas. But, I forgot to write $v$ is random. Sorry for my forget. – Jlamprong Mar 10 '14 at 09:36

Optimization for non linear function

1 Answers1