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Math Help - Concave function

  1. #1
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    Concave function

    Let f(x) be a concave function of N variables defined over a convex set,S \in R^NLet g be an increasing and concave function of one var.

    prove that g(f(x)) is concave over S
    Last edited by altave86; January 16th 2010 at 01:04 PM.
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  2. #2
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    Probably obvious, but,...

    It seems the second derivative should get us somehwere...

    g''(f(x)) \cdot f'(x) + g'(f(x)) \cdot f''(x)

    The information in the problem statement, relating to g'' and f'' is too compelling to pass up. Also, with f' positive, it seems we've only g' to argue. Will the convexity of S do that?

    For what it's worth...
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  3. #3
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    Quote Originally Posted by TKHunny View Post
    Probably obvious, but,...

    It seems the second derivative should get us somehwere...

    g''(f(x)) \cdot f'(x) + g'(f(x)) \cdot f''(x)

    The information in the problem statement, relating to g'' and f'' is too compelling to pass up. Also, with f' positive, it seems we've only g' to argue. Will the convexity of S do that?

    For what it's worth...
    Thanks! But the fact that f(x) is a function of many variables would't force us to evaluate the second derivative with the gradient of f? In other words remember that x is a vector
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  4. #4
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    Quote Originally Posted by altave86 View Post
    Let f(x) be a concave function of N variables defined over a convex set,S \in R^NLet g be an increasing and concave function of one var.

    prove that g(f(x)) is concave over S
    The proof by TKHunny is actually not true since we don't know that the functions are differentiable.
    However, as he mentioned the proof is obvious.
    Let me give it below.
    Proof. Since f:\mathbb{R}^{n}\to\mathbb{R} is concave, we have
    f\big(\lambda x+(1-\lambda)y\big)\geq\lambda f(x)+(1-\lambda)f(y) for all x,y\in\mathbb{R}^{n} and all \lambda\in[0,1]. (*)
    As g:\mathbb{R}\to\mathbb{R} is increasing, we use (*) to get
    g\Big(f\big(\lambda x+(1-\lambda)y\big)\Big)\geq g\big(\lambda f(x)+(1-\lambda)f(y)\big).........( g is increasing)
    ............................... \geq \lambda g\big(f(x)\big)+(1-\lambda)g\big(f(y)\big).....( g is concave)
    for all x,y\in\mathbb{R}^{n} and all \lambda\in[0,1].
    This completes the proof that g\circ f is concave on \mathbb{R}^{n}..... \rule{0.3cm}{0.3cm}
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  5. #5
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    I dare you to show me where I offered a proof. I just said it was compelling, not useful.

    I often can be found not knowing that I have no idea what I'm talking about.
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