Division of convex functionsI 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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Division of convex functionsI 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&#039;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 · Accepted Answer

For simplicity, consider the case where   f and   g are convex, twice differentiable functions on an interval and   g  &amp;gt;  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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