Lisantiom

2022-09-07

Division of convex functions
I need your expertise in understanding the following:
Let $n\in \mathbb{N}$, ${x}_{i}\in \mathbb{R}$ for every $i\in \left[n\right]$ and let $a\in {\mathbb{R}}_{+}$
What can be said about the following in term of convexity (let $j$ be any arbitrary integer such that $i\in \left[n\right]$
$\frac{\frac{{a}^{2}}{2n}+max\left\{0,1-{x}_{i}\right\}}{\frac{{a}^{2}}{2}+\sum _{j\in \left[n\right]}max\left\{0,1-{x}_{j}\right\}}$
I am asking this since, it's easy to see that both the denominator and nominator are convex (it resembles the objective function of SVM), however is this fraction convex, or quasi-convex, concave, etc... ?
P.s. A more advanced question would be, what can be said on the fraction of two convex function in general?

Gianna Walsh

For simplicity, consider the case where $f$ and $g$ are convex, twice differentiable functions on an interval and $g>0$. We have
${\left(\frac{f}{g}\right)}^{″}=\frac{{f}^{″}{g}^{2}-2{f}^{\prime }g{g}^{\prime }-fg{g}^{″}+2f\left({g}^{\prime }{\right)}^{2}}{{g}^{3}}$
and the condition for $f/g$ to be convex is that the numerator is always nonnegative. Unfortunately, not a very nice condition!

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