I know that if f is convex and strictly non-negative this is true, additionally if f is affine, then we should also have convexity, however I was hoping for something a bit better. I currently have a function that is strictly convex but not strictly non-negative, does that mean the L_2 norm of this function is neither concave nor convex? Or does that simply mean that we can not say anything about the functions convexity?
Additionally, even if the problem is not convex, if the initial guess for f(x) results in values that are strictly over-estimates, i.e.
$$\underset{x}{\text{min}} \big(\sum(f(x_i) - y_i)^2\big)^{1/2}$$
is ensured to have the property that
$$ f(x_i) - y_i > 0 $$
for all i. Then will the optimization procedure have a better chance of finding a global minimum?
Finally, if I add a constraint to the optimization problem requiring that $ f(x_i) > y_i $, such that I am now solving the problem,
$$\underset{x}{\text{min}} \big(\sum(f(x_i) - y_i)^2\big)^{1/2}$$ subject to: $ f(x) > y $
Will convexity of f(x) be enough to ensure the primal problem is convex? If so are there ways to solve this for a general anonymous function f?