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Thread: Convexity of function

  1. #1
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    Convexity of function

    Hi all,

    Can anybody show that the function $\displaystyle f: R^N \rightarrow R $ is convex, where:

    $\displaystyle
    f(x) = \frac{K + \sum_{i=1}^N x_i^2 }{ \sum_{i=1}^N x_i }
    $

    And $\displaystyle x_i \geq 0, \forall i$ and $\displaystyle K > 0$

    Any help is welcome (this is not homework!).


    Thanks!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by leoemil View Post
    Hi all,

    Can anybody show that the function $\displaystyle f: R^N \rightarrow R $ is convex, where:

    $\displaystyle
    f(x) = \frac{K + \sum_{i=1}^N x_i^2 }{ \sum_{i=1}^N x_i }
    $

    And $\displaystyle x_i \geq 0, \forall i$ and $\displaystyle K > 0$

    Any help is welcome (this is not homework!).


    Thanks!
    The not thinking way is to show that the Hessian $\displaystyle D_iD_jf(x)$ is positive semi-definite.
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