Division of convex functions I need your expertise in understanding the following: Let in NN, x_i in RR for every i in [n] and let a in RR_+. What can be said about the following in term of convexity (let j be any arbitrary integer such that i in [n]: ((a^2)/(2n)+max{0, 1 - x_i})/((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?

Lisantiom 2022-09-07 Answered
Division of convex functions
I need your expertise in understanding the following:
Let n N , x i R for every i [ n ] and let a R +
What can be said about the following in term of convexity (let j be any arbitrary integer such that i [ n ]
a 2 2 n + max { 0 , 1 x i } a 2 2 + j [ 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?
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Answers (1)

Gianna Walsh
Answered 2022-09-08 Author has 7 answers
For simplicity, consider the case where f and g are convex, twice differentiable functions on an interval and g > 0. We have
( f g ) = f g 2 2 f g g f g g + 2 f ( g ) 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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