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From: Dan Cass on 10 Jul 2010 07:40
> My question is:
>
> the function f(x) = log(x) is concave
>
> the function h(z) = log(sum(exp(g_i)) is convex, as
> long as the functions g_i are convex (the log to a
> sum of exponentials to convex functions)
>
> How is the last true? The outer function log() is
> concave and nondecreasing, while the inner function
> is convex. There is no composition rule for this...
>
> Thansk :)
>
> Benjamin

The general case resembles the two-function case:
Write w for sum(exp(g_i))
Then the second derivative of log(w) has denominator w^2,
and the numerator, after some algebra, is
\sum_k (g_k)'' * exp(u_k) * w
+ \sum_{i<j} [g_i' - g_j']^2 * exp(g_i + g_j).
Hence if each g_k is convex, so that the (g_j)'' are
positive, then the second derivative of log(w) is
positive, making log(w) convex as stated.
From: Dan Cass on 10 Jul 2010 07:42
> > My question is:
> >
> > the function f(x) = log(x) is concave
> >
> > the function h(z) = log(sum(exp(g_i)) is convex,
> as
> > long as the functions g_i are convex (the log to a
> > sum of exponentials to convex functions)
> >
> > How is the last true? The outer function log() is
> > concave and nondecreasing, while the inner
> function
> > is convex. There is no composition rule for
> this...
> >
> > Thansk :)
> >
> > Benjamin
>
The following is a re-post with a typo fixed...
> The general case resembles the two-function case:
> Write w for sum(exp(g_i))
> Then the second derivative of log(w) has denominator
> w^2,
> and the numerator, after some algebra, is
> \sum_k (g_k)'' * exp(u_k) * w
> + \sum_{i<j} [g_i' - g_j']^2 * exp(g_i + g_j).
> Hence if each g_k is convex, so that the (g_k)'' are
> positive, then the second derivative of log(w) is
> positive, making log(w) convex as stated.
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