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Math Help - Conditions for concavity/convexity - multivariable case

  1. #1
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    Mar 2011
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    Conditions for concavity/convexity - multivariable case

    Hi,

    Here's the problem:
    Give an informal argument to show that a function F which satisfi es F(0) = F(1) = 0
    and F''(t) >=0 must satisfy F(t) <= 0 for 0 <= t <= 1.

    The function F is de fined for fixed x1 and x2 by
    F(t) = f(tx1 + (1 - t)x2) - tf (x1) - (1 + t)f(x2);
    where f is a smooth function which satisfi es f '' (x) >= 0. Show that F satisfi es the conditions of the fi rst paragraph and deduce that f is convex.

    How can this result be extended to the case where f is a function of many variables (i.e. where x1 is replaced by the vector x1, etc)?

    For the first part you can give a geometric argument or alternatively use Rolle's Theorem to formalise it. The second part I simply have to use the chain rule and I get the definition of convexity for f(.)

    I have a bit of difficulty with the last part - generalizing the result.

    I am not sure how to get the second derivative of F''(t). If i am correct i get

    F''(t) = Σ_i [ (X1_i-X2_i) Σ_j (X1_j-X2_j) (d^2f/dX_i dX_j) ]

    I need conditions for F''(t0)<= 0 to be equivalent to "the hessian matrix is positive definite"


    i am not sure that the above expression is right. Even if it is, there are some simplifications which can be made i think. Anyways i am not sure how to get "the hessian is positive definite" as a condition for convexity, from this expression.


    Thanks a lot!
    Last edited by Val92; December 26th 2011 at 12:58 AM.
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  2. #2
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    Joined
    Mar 2011
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    Re: Conditions for concavity/convexity - multivariable case

    NEVERMIND:

    I figured it out.

    F''(t) can be written as X'HX where X=(X1_i -X_2_i) For i = 1,2.....n

    and H is the hessian

    Therefore F''(t) >= 0 is equivalent to H being positive definite

    which in turn implies that F(t)<= 0 for 0<=t<=1... This gives us convexity of f(.)
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