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Convex Function analyze

  1. Apr 16, 2008 #1
    Could anyone comment on the convexity of

    f(x,y) = (x^2) * exp(y) ...... i.e. x square into e to the power y.

    I did try to find Hessian of the same and the value I get is :

    Hessian(x,y) = -2 * x^2 * exp(2y).... which looks <= 0 for all x and y.

    I assume this should imply f(x,y) is concave. However when I plot this function using a 3D graph plotter it seems convex .


    for plotting function.

    I am sure I am making a simple mistake or something.

    Any help would be useful.
  2. jcsd
  3. Apr 16, 2008 #2


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    Science Advisor

    I don't know why you would say that, even a rough graph looks concave to me. Using the website you give (you are missing a few letters) it looks concave. Are you clear on the difference between "convex" and "concave"? If, for any two points on the graph, the line segment between this is above the graph, it is "concave". It the line segment between two points on the graph is below the graph, then it is "convex".

    (Some texts say "concave upward" and "concave downward" rather than "concave" and "convex".)
  4. Apr 16, 2008 #3

    Isn't f(x) convex if :

    f( a*x1 + (1-a) * x2 ) <= a * f(x1) + (1-a) * f(x2)

    this would imply graph must be below line connecting two points for function to be convex.
  5. Apr 17, 2008 #4
    Be careful about Hessian, that is a multivariable function ! and if you work it out that is exactly what it is called a saddle point.
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