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 $i\in [n]$ 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 $i\in [n]$

$\frac{\frac{{a}^{2}}{2n}+max\{0,1-{x}_{i}\}}{\frac{{a}^{2}}{2}+\sum _{j\in [n]}max\{0,1-{x}_{j}\}}$

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... ?

Please advise and thanks in advance.

P.s. A more advanced question would be, what can be said on the fraction of two convex function in general?

I need your expertise in understanding the following:

Let $n\in \mathbb{N}$, ${x}_{i}\in \mathbb{R}$ for every $i\in [n]$ 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 $i\in [n]$

$\frac{\frac{{a}^{2}}{2n}+max\{0,1-{x}_{i}\}}{\frac{{a}^{2}}{2}+\sum _{j\in [n]}max\{0,1-{x}_{j}\}}$

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... ?

Please advise and thanks in advance.

P.s. A more advanced question would be, what can be said on the fraction of two convex function in general?

Gianna Walsh

Beginner2022-09-08Added 7 answers

For simplicity, consider the case where $f$ and $g$ are convex, twice differentiable functions on an interval and $g>0$. We have

${\left(\frac{f}{g}\right)}^{\u2033}=\frac{{f}^{\u2033}{g}^{2}-2{f}^{\prime}g{g}^{\prime}-fg{g}^{\u2033}+2f({g}^{\prime}{)}^{2}}{{g}^{3}}$

and the condition for $f/g$ to be convex is that the numerator is always nonnegative. Unfortunately, not a very nice condition!

${\left(\frac{f}{g}\right)}^{\u2033}=\frac{{f}^{\u2033}{g}^{2}-2{f}^{\prime}g{g}^{\prime}-fg{g}^{\u2033}+2f({g}^{\prime}{)}^{2}}{{g}^{3}}$

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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