# Hessian of Euclidean norm of vector function

I'm not familiar with Matrix algebra. I'm trying to minimize an $\ell_2$ (Euclidean) norm of vector function. I found out the gradient of the function, using the previous questions and answers in this site. Could anyone help me to find the Hessian?

My function is:

$$f(x) = |\phi(x)|$$ |.| denotes a vector norm and $x$ is vector.

Another doubts:
My objective is to minimize $\phi(x)$s to get $x$ values. If I use $|\phi(x)|^2$ (square of $\ell_2$ norm instead of $|\phi(x)|$, would the objective function still be convex?
Which function is better choice?
Which algorithm (SQP or Interior Point) is the best suit for this type of problems? (I'm trying to use Ipopt)

• Do you mean to minimize $f$ or $\phi$? – Elsa Jul 31 '17 at 7:41
• @Elsa, I want to minimize all $\phi$s so I choose $f$, the norm of $\phi$s – user8153630 Jul 31 '17 at 8:11