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Thread: Unique minimum of a simple function

  1. #1
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    Unique minimum of a simple function

    Hi,

    I want to prove that the following function has a unique minimum … and I'm surprised at how hard this is!

    $\displaystyle f(x,y) = \frac{\frac{a}{x} + \frac{b}{y} + c}{d - e(x+y)} + \frac{u}{x} + \frac{v}{y}$

    The scalars $\displaystyle a,b,c,d,e,u,v$ are all positive. The function is defined on the open triangle $\displaystyle x > 0$, $\displaystyle y > 0$, $\displaystyle x+y < \frac{d}{e}$.

    Can you think of some simple proof?

    I've tried a few things:
    – take the logarithm
    – try to show quasi-convexity (which is a stronger result) by several means
    – try to rule out saddle points (see my post http://www.mathhelpforum.com/math-he...ts-194122.html and answer by xxp9)
    – try to identify a composition of elementary functions
    …to no avail

    Below is a contour plot of the inverse $\displaystyle 1/f$ (because it's nicer to plot than $\displaystyle f$). The function tends to zero on the border of the triangle. You clearly see there's a unique maximum of $\displaystyle 1/f$ (minimum of $\displaystyle f$) somewhere inside the triangle.

    Unique minimum of a simple function-contour_plot.jpg
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  2. #2
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    Re: Unique minimum of a simple function

    Hi again,

    I figured out that my function $\displaystyle f$ is actually convex. One can show in a few steps that $\displaystyle f$ is a sum of functions that are all convex on the mentioned triangle.

    For some reason (which I won't explain in detail) I had convinced myself that $\displaystyle f$ couldn't be convex. I failed to see the wood for the trees. You can consider this issue as closed (sorry for posting!).
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